Basic Stats

Warner, R. M. (2021). Applied sta s cs I: Basic bivariate techniques (3rd ed.). Thousand Oaks, CA: Sage Publica ons. ISBN: 978-1-5063-5280-0.

CHAPTER 3 FREQUENCY DISTRIBUTION TABLES 3.1 INTRODUCTION Most published research examines rela onships between variables. However, before you can examine how scores on two or more variables are related, you need to examine scores for each variable separately. In this chapter, you’ll learn how frequency distribu on tables (or, simply, frequency tables) are used to understand the behavior of scores for just one variable using frequency distribu on tables set up by hand or using SPSS. Examples in this book use IBM SPSS® Version 25 unless otherwise noted. SPSS data sets used in this book can be downloaded from the textbook website at edge.sagepub.com/warner3e. Instruc ons for most analyses described in this book using the R open-access program are provided in Rasco (2020). Appendix 3A at the end of this chapter provides a brief introduc on to SPSS and basic file management. The following examples use the small set of hypothe cal data in the SPSS data file named temphr10.sav that appears in Figure 3.1. File structure is similar for many other sta s cal programs. The data set in Figure 3.1 has 10 cases (rows 1 through 10) and five variables (columns 1 through 5). In most of my examples, each case corresponds to one person. Cases can be other kinds of en es (such as trees, asteroids, or na ons). Each column contains the 10 scores for the variable named at the top (for example, the variable sex). Each row lists the scores for one case on all five variables.

Figure 3.1 SPSS Data File (temphr10.sav) This hypothe cal data set includes five variables as examples of different types of variables. The categorical variable has codes of 1 = male and 2 = female (addi onal codes to represent categories such as nonbinary could have been included.) Heart rate (hr) is a quan ta ve variable (number of beats per minute); integer values are reported. Temp_Fahrenheit and temp_Celsius are also quan ta ve variables; these represent body temperature reported to one decimal place (e.g., 97.1°F). Likert_ra ng is a score on a five-point scale indica ng degree of agreement for this ques on: “I believe the president is doing a good job,” with the following response op ons: 1 = strongly disagree (SD), 2 = disagree (D), 3 = don’t know or neutral (N), 4 = agree (A), and 5 = strongly agree (SA). No ce the top-level menu bar in Figure 3.1; the four top-level SPSS menus discussed in examples are circled. Before se ng up tables or graphs or doing analyses, always pause and ask, What ques ons can this answer? For categorical variables such as sex we can ask: Were any group membership codes “impossible” values? Did some individuals report scores that were not included in the possible response op ons? Do some responses iden fy types of people the researcher does not intend to include in the study? For example, if a study requires par cipants to have normal vision, and some poten al par cipants report limited vision, data for those persons may be excluded from later analyses. How many categories or groups does the categorical variable have? What are the rela ve sizes of these groups? Which score is most common or frequent? Which group has the most cases? The mode corresponds to the group with the largest number of cases. (There can be more than one mode.) Do all groups have large enough numbers of cases (for example, more than 10 cases) to be used in later analyses that compare groups? Were there missing values? Did some individuals not provide responses? Frequency tables provide answers to all these ques ons. In Figure 3.1, the categorical variable sex has two categories or groups (addi onal groups such as nonbinary sex could have been included). You can see that there are seven male respondents (people with scores of 1 on sex) and three female respondents (people with scores of 2 on sex). This tells you (obviously) that there are two groups in the sample and that there are more male than female respondents. For quan ta ve variables such as heart rate, we can ask: What are the lowest and highest scores?

What is the range of scores? Range is the difference between highest and lowest score. This provides a preliminary idea of variability. Are all the values plausible (for example, in a frequency table for heart rate, are all scores plausible values for heart rate)? Were there any missing values? What is an average or a typical score? Loca ng a score in the “middle” of a frequency table provides a preliminary idea about something more formally called central tendency. For selected score values, what are loca ons of these scores compared with the distribu on in the table? For example, consider your own heart rate. What percentage of scores in the sample were lower than your heart rate? Is your heart rate unusually low or unusually high, or near the middle? Frequency tables for quan ta ve variables provide preliminary informa on about variability and central tendency. In Figure 3.1, you can see that the lowest score for the quan ta ve variable heart rate (hr) is 62; the highest score is 82. The difference between highest and lowest scores, 20, is the range. This tells us something about variability in hr. Further informa on about central tendency (also called average) and about variability for quan ta ve variables can be obtained from descrip ve sta s cs such as standard devia on and variance and from graphs such as histograms. These are discussed in the next two chapters. A frequency table of scores for quan ta ve variables provides the context we need to evaluate an individual score (e.g., to decide whether that score is high rela ve to the sample). If your score on hr is 95, you can see that it is unusually high compared with this sample. In addi on, thinking about frequency distribu ons is an important skill needed to understand later topics throughout the book. 3.2 USE OF FREQUENCY TABLES FOR DATA SCREENING We look at frequency tables to get to know the data and to iden fy poten al errors and problems with data before we do other analyses. This process is called preliminary data screening. Introductory sta s cs textbooks o en present students with sample data sets that are assumed not to have errors or missing informa on. In real-world applica ons, data o en have problems, and it is important to look for them. These problems include: Informa on is some mes missing for some members of a sample. Some scores can be unusually large or small; unusual or extreme scores can be problema c in some analyses. Some groups contain too few cases for meaningful analyses. Real data sets o en contain mistakes (incorrect, or even impossible, score values). Implausible or incorrect score values can arise in many ways. If a person is asked to report hair color and reports “plaid,” that is an unlikely response. If a heart rate is recorded as 275 beats per minute, the heart rate monitor is probably malfunc oning. However, a score value can appear plausible and s ll be incorrect; if a heart rate monitor is not properly calibrated, a person whose heart rate is given as 110 beats per minute might really have a heart rate of 95 beats per minute.

In an ideal world, researchers would proofread every single number in the data file against original data sources, if these exist. However, original data sources are some mes not available, and complete proofreading of data may be extremely me-consuming and costly. At a minimum, spot checks (checking some score values in SPSS against original sources of data) provide an opportunity to detect problems that might be more widespread throughout the data set and would require much closer checking. If you find scores that are clearly impossible or at least highly unlikely, the best op on is to obtain valid scores from other sources if that is possible. If a student reports a grade point average (GPA) of 6 when college GPAs are on a 0-to-4 scale, and you have access to university records and can find that student’s GPA, you could use the university record to replace the incorrect self-reported value. If a respondent reports large numbers of silly or impossible values, you might decide to drop that person’s data en rely. There is increasing concern about completeness and transparency in data repor ng (Simmons, Nelson, & Simonsohn, 2011). Research reports should include informa on about problems detected during preliminary screening. This informa on is o en obtained from frequency tables and graphs (such as histograms). The numbers or percentages of incorrect scores, extreme scores, and missing values should be reported. Authors also need to specify what, if anything, was done to remedy these problems. You might say something like “Data for five students were dropped because they reported unlikely or inconsistent informa on” or “Data from three sessions had to be dropped because of equipment malfunc on.” Whatever problems with data you find, and whatever ac ons you take, you need to keep a detailed record and include this informa on in published research reports. Problems (such as missing values) are discussed in greater detail elsewhere (e.g., Volume II [Warner, 2020]). 3.3 FREQUENCY TABLES FOR CATEGORICAL VARIABLES A frequency distribu on table for a categorical variable provides informa on about the number of groups and the number of cases in each group. There is one line in the frequency table for each possible score value; that line includes informa on about the number of cases and other informa on based on those numbers (such as percentages). To set up a frequency table by hand, the first step is to list all possible score values; you need one line for each score value. For the categorical variable sex, the ini al table setup is of the form shown in Table 3.1. Each score value iden fies a group. The table is completed by coun ng the number of persons who have scores of 1 versus scores of 2 on the variable sex. For small data sets this can easily be done by hand. For larger data sets it is more convenient to use a program such as SPSS. In Table 3.1, on the basis of the data in Figure 3.1, you would enter 7 as the number of male par cipants and 3 as the number of female par cipants. The total number of cases in the sample, denoted N, is 10. 3.4 ELEMENTS OF FREQUENCY TABLES The names of the elements of frequency tables are usually given as follows; there is some varia on among computer programs and textbooks.

3.4.1 Frequency Counts (n or f) The frequency is the number of scores in each group, for a categorical variable, or the number of persons who have a specific score value on a quan ta ve variable, such as hr = 73. Here are three common nota ons for frequency; these can be used interchangeably. Lowercase n is the most common nota on for number of scores per group in research reports. For the data in Figure 3.1, there were n = 7 male and n = 3 female par cipants. Numerical or word subscripts can be used to indicate the name or code number for each group, for example, n1 or nmale = 7, and n2 or nfemale = 3.

Lowercase f, which stands for frequency, is a common nota on in textbooks; f is the same as n. In Figure 3.1, fmale = 7 and ffemale = 3. You will rarely see f in later discussions of sta s cs; group size is usually given as n. Frequency (the en re word) is the nota on SPSS uses to refer to n or f. In Figure 3.1, the frequency of male par cipants is 7; for female par cipants, the frequency is 3. 3.4.2 Total Number of Scores in a Sample (N) Uppercase N is used in research reports to represent the total number of cases in a sample. Total (the word) is used in SPSS frequency tables to represent N. N is the sum of ns (or fs) across all groups. For the temphr10.sav file in Figure 3.1, N = 10 cases: N = nmale + nfemale = 7 + 3 = 10. 3.4.3 Missing Values (if Any)

If scores are not available for some cases, the cells for those scores are usually le blank. (It is possible to iden fy some numerical value, such as 999, as an indicator of missing data.) SPSS counts frequencies for missing values and includes this count in the frequency table. 3.4.4 Propor ons Values of n (or f) are usually converted into propor ons (P) rela ve to N. Propor on is some mes called rf (rela ve frequency; i.e., the frequency in one group rela ve to or as a part of total N). SPSS omits propor on and instead reports percentages. A propor on is obtained for each group by dividing the number of persons in that group (ni or fi or frequency) by the total number of people in the en re data set (N). Propor ons are a useful way to summarize group size informa on for categorical variables. To compute propor ons for groups: Find ni or fi for each group by coun ng cases in each group. This count can be called ni or fi or frequency. To find the propor on P (or rf) for each group: Pi = ni/N or fi/N. The propor on P for each group is the number of people in that group divided by the total number of people in the data set or sample. For the data in Figure 3.1, the propor ons of people in the male and female groups are:

3.4.6 Cumula ve Frequencies or Cumula ve Percentages These are useful only for quan ta ve variables (not for categorical variables), and they are discussed later.

3.5 USING SPSS TO OBTAIN A FREQUENCY TABLE To use SPSS to obtain a frequency table for the data in Figure 3.1, open the temphr10.sav file. (See Appendix 3A if you need to know more about ge ng started in SPSS.) At the top-level menu bar, make the following menu selec ons, as shown in Figure 3.2: <Analyze> → <Descrip ve Sta s cs> → <Frequencies>.

Figure 3.2 SPSS Menu Selec ons to Use the Frequencies Procedure These menu selec ons open the Frequencies dialog box that appears in Figure 3.3. The list of all variables in the data set appears on the le side. Move the variable sex to the pane on the right under the heading Variable(s). To do this, highlight the variable name sex using the cursor. An arrow appears that indicates movement from le to right; click the arrow to move the variable sex into the Variable(s) pane. The list of one or more variables in the Variable(s) pane heading tells SPSS which variable or variables you want to examine. You can obtain frequencies for more than one variable at a me; this example uses just one variable. To run the analysis, click OK in the main Frequencies dialog box. Part of the SPSS frequency output appears in Figure 3.4. In this example, focus on the columns enclosed in the box; ignore the columns with headings “Valid Percent” and “Cumula ve Percent.” Cumula ve percentage is meaningful for quan ta ve variables, but it is not meaningful for categorical variables. The “Valid Percent” column provides informa on that differs from the “Percent” column only when there are missing values for the variable, as discussed in the next example and in Appendix 3B. The SPSS results in Figure 3.4 confirm results obtained from by-hand coun ng of cases. There are two groups; the group sizes are 7 male (70%) and 3 female (30%) par cipants.

To summarize informa on about a categorical variable, a table could include the informa on in Table 3.2; however, this is more informa on than required for a research report. Given N, readers can deduce any one column from other columns. When propor on or percentage is reported, total N must also be reported. Unless we know N, we can’t reproduce the numbers (ns) in groups. In journal ar cles, it is common to report percentages to characterize a sample (in this example, we would just say that the sample is 70% male). In addi on, readers need to know when percentages are based on very small samples. Suppose someone claims that “80% of den sts prefer Gooey toothpaste.” That makes it sound as if at least 100 den sts were asked about preference, and 80 of them preferred Gooey. However, if 4 out of 5 den sts prefer Gooey, that is also 80%.

A research report might include informa on about the sex composi on of the sample in its “Par cipants” sec on. This can be in sentence form: “The sample of N = 10 persons was 70% male and 30% female; there were no missing values for sex.” If only two or three categorical variables are used to describe the kinds of people in the study, a few sentences suffice. If a study includes a larger number of categorical variables, percentages for each categorical variable may be summarized in a table.

3.6 MODE, IMPOSSIBLE SCORE VALUES, AND MISSING VALUES Consider the data for a different hypothe cal categorical variable, hair color, in Figure 3.5. Only the first 20 out of 50 lines in the file hair color.sav are displayed. This example illustrates addi onal things to look for in frequency tables. This file illustrates the use of case numbers to iden fy individual par cipants. A case number is a unique iden fying number for each person in the study; these numbers can be arbitrary. It is good prac ce to include case numbers for several reasons. Case numbers are needed if: You plan to check scores in the SPSS file against original data sources. You want to follow up, or contact, individual cases. You want to match scores from two or more data sources for each person (such as data from surveys done at different points in me). You want to iden fy persons with extreme or implausible scores. You want to iden fy persons who have missing values. In Figure 3.5, note that the case number 206 appears 3 mes. This indicates a problem in record keeping. For online surveys, duplicate case numbers can indicate that the same person logged in and completed the survey mul ple mes. No ce that the case in line 14 has an empty cell for hair color. This indicates a missing value; the person did not provide an answer to the survey ques on. Variables with large numbers of missing values are a problem. Pink and blue are not natural hair colors, but a few people dye their hair these colors. Next, examine the frequency table obtained for the hair color data set (using the same SPSS procedures as in the earlier example) in Figure 3.6.

I crossed out the “Cumula ve Percent” column in Figure 3.6 as a reminder that this is not applicable for categorical variables. In the le -hand column, each hair color numerical score (1, 2, 3, etc.) is replaced by a verbal label. Appendix 3A explains how verbal labels can be assigned to numerical score values. Excluding the hair color response “12,” seven different hair colors were reported. The modal hair color was brown. For a categorical variable, the mode is the score that has the highest frequency. It is possible to report secondary modes; the next most frequent hair type, black, could be characterized as the second highest mode. If you decide that you need a minimum of 10 cases in a group to use that group in later analyses, the only groups that qualify are brown and black hair. Each of the other hair colors (blond, red, gray, pink, and blue) was reported by fewer than 10 persons. Two lines near the bo om of the table indicate problems. A score of “12” for hair color did not have a verbal label. Assume that a response of “12” was not an cipated by the researcher and that “12” was not presented to par cipants as a possible response op on. This kind of error can be avoided during data collec on by requiring fixed-choice responses instead of allowing people to write in open-ended responses. This response is nonsense or out of range.

Figure 3.5 Hypothe cal Data for Self-Reported Hair Color The line labeled “Missing System,” with a frequency of 4, tells us that four persons did not answer the hair color ques on. When more than a few (let’s say more than 5%) of persons don’t provide data, this signals a problem. For analyses covered in this volume, we will usually allow SPSS to exclude missing values from analyses automa cally (using listwise dele on).

(However, this is not best prac ce; see Volume II [Warner, 2020] for discussion.) The percentage of missing values should be included in research reports for all variables. To understand why percentages in the “Valid Percent” column differ from those in the “Percent” column, see Appendix 3B.

Figure 3.6 Frequency Table for Hypothe cal Hair Color Data 3.7 REPORTING DATA SCREENING FOR CATEGORICAL VARIABLES On the basis of Figure 3.6, you could report the following. The total N for the study was 50 (you would also include other informa on, such as how the sample was selected, age and other demographic or background variables, and so forth). The sample included these hair color groups: 32% brown, 24% black, 16% blond, 6% red, 6% gray, 4% pink, 2% blue, 2% out-of-range scores, and 8% missing values. 3.8 FREQUENCY TABLES FOR QUANTITATIVE VARIABLES 3.8.1 Ungrouped Frequency Distribu on For quan ta ve variables, SPSS sets up a frequency distribu on table with one row for each possible score value. This is called an ungrouped frequency distribu on. Each line provides informa on about the number of persons who had the same score, for instance, the number of persons whose heart rate score was 72. Ungrouped frequency distribu on tables can have many lines and some mes don’t provide the clearest sense of pa ern in your data. However, they can

answer the same ques ons as frequency tables for categorical variables, that is, are there out- of-range or impossible or missing values? To illustrate frequency tables for quan ta ve variables, it is useful to consider a larger data set. The data for the next example is in the SPSS data file temphr130.sav. To obtain the output shown in Figure 3.7, follow the instruc ons to run the frequencies procedure in Figures 3.2 and 3.3; choose the variable hr instead of the variable sex. Examina on of this frequency table tells us the following things:

 There were 130 persons in the sample (total N = 130).  The smallest (minimum) value for heart rate was 57 beats per minute.  The highest (maximum) value for heart rate was 89 beats per minute.

Figure 3.7 Ungrouped Frequency Distribu on Table for Values of hr in Data File temphr130.sav

The range of heart rate values (89 – 57) was 32. All scores are plausible values for heart rate. If there were hr values below 40 or above 150, we would suspect that the heart rate monitor malfunc oned or that there were some extremely unusual people in the sample. There were no missing values in this data set (there is not a line for missing values). 3.8.2 Evalua on of Score Loca on Using Cumula ve Percentage The “Cumula ve Percent” column is not useful for categorical variables; however, it is useful for quan ta ve variables. Cumula ve percentage is the percentage of cases who have scores equal to or below a specific score value. For example, the cumula ve percentage for a heart rate score of 61 is the sum of the percentages for all lower hr scores and cases with a score of 61 (i.e., we sum the percentages of cases for hr scores of 57, 58, 59, 60, and 61). The cumula ve percentage for 61 = 1.5% + .8% + .8% + 1.5% = 4.6%. This tells us that 4.6% of the people in this sample had hr scores of 61 or lower. For a specific value of heart rate such as 77 beats per minute, we can ask whether that score is low or high, compared with other scores in the sample. There are two different ways to word the ques on, and they have slightly different answers. We can ask: What percentage of people have hr scores of 77 or lower? (The answer is 65.4%.) What percentage of people have hr scores below 77 (i.e., 76 or below)? (The answer is 60%.) No ce that these two ques ons differ. To answer the first ques on, we include people with scores of 77. To answer the second ques on, we exclude people with scores of 77. We answer the first ques on (what percentage of people have hr scores of 77 or lower?) by repor ng the cumula ve percentage for 77 from Figure 3.7; 65.4% of cases had hr scores of 77 or lower. The cumula ve percentage of scores at or below a specific score value (such as 77) can also be called the percen le rank of that score. We answer the second ques on (what percentage of people have hr scores below 77?) by looking at the cumula ve percentage for the first score that is smaller than 77, in this case, the cumula ve percentage for a score of 76. From Figure 3.7, 60% of cases had hr scores lower than 77. For variables that apply to humans, you can personalize your experience with sta s cs by thinking about your own scores. If you know that your heart rate is 92, it is higher than all hr scores in the hypothe cal data in Figure 3.7. This tells you (in the parlance of some high school students) whether your heart rate is “weird” or unusual rela ve to this set of scores. If your heart rate is 75, your score is near the middle of these scores. Later you will learn more formal methods to evaluate score loca ons. O en researchers want to know a “typical” or average value. For now, we’ll define this as a score with a cumula ve percentage at or near 50% (this is close to the median, discussed in the next chapter). A score of 73 had a cumula ve percentage of 49.2, so we can say that a score of 73 was about average or typical for this sample.

3.8.3 Grouped or Binned Frequency Distribu ons When the number of different score values is large, it may be more convenient to set up a grouped frequency distribu on, that is, to report the number of cases for groups defined by ranges of score values; this process is known as binning. A grouped frequency distribu on could show numbers of cases for ranges of heart rate scores, such as: 61 to 65 66 to 70 71 to 75 SPSS provides ungrouped (rather than grouped) frequency tables, by default. Default procedures are the “decisions” SPSS makes unless you specify something different. Ungrouped frequency distribu on tables provide much of the informa on you need to evaluate poten al problems with scores on quan ta ve variables. When we begin to look at graphs in the next few chapters, it is o en helpful to have grouped scores. SPSS automa cally groups data when it sets up histograms. Grouped frequency tables are rarely necessary. However, methods to create these by hand are provided in Appendix 3C if you want to do this. 3.9 FREQUENCY TABLES FOR CATEGORICAL VERSUS QUANTITATIVE VARIABLES Frequency distribu on tables are very useful for categorical variables; they provide all the informa on you need to describe the pa ern of scores and iden fy poten al problems with data. However, for quan ta ve variables, informa on beyond frequency tables is needed. Addi onal summary informa on for quan ta ve variables (descrip ve sta s cs such as the mean and graphs such as histograms) is described in the next two chapters. 3.10 REPORTING DATA SCREENING FOR QUANTITATIVE VARIABLES Frequency tables don’t tell us everything we need to know when screening quan ta ve variables. Informa on from graphs such as histograms and boxplots will also be needed. Therefore, discussion of repor ng data screening for quan ta ve variables is presented a er these topics. 3.11 WHAT WE HOPE TO SEE IN FREQUENCY TABLES FOR CATEGORICAL VARIABLES Here are the things we hope to see in frequency tables: Groups that correspond to all the groups we wanted to include in the study. A reasonable minimum number of cases in all groups that will be compared in later analyses. Standards for minimum group size vary depending on the cost and difficulty of obtaining cases. For a variety of reasons, I suggest that 30 per group is a reasonable minimum in many situa ons, but that is not an ironclad rule. There should not be groups that correspond to kinds of cases we planned to exclude from the study. There should be few missing values (less than 5% missing is a reasonable standard). There should be no “impossible” responses to group membership ques ons.

Categorical variables can correspond to naturally occurring groups or to groups formed by a researcher (o en in the context of an experiment). 3.11.1 Categorical Variables That Represent Naturally Occurring Groups O en, scores for naturally occurring groups are used to characterize the sample (i.e., describe the kinds of cases included in the sample). If researchers want to generalize results to some hypothe cal popula on, the composi on of the sample (e.g., propor ons male and female) should be similar in the sample and popula on; the sample should be representa ve of the popula on of interest. Some mes researchers want to compare naturally occurring groups exposed to different risk or protec ve factors by self-selec on into the situa on (for instance, smokers vs. nonsmokers; meditators vs. nonmeditators). If this is the goal, these naturally occurring groups should be as similar as possible on other variables (e.g., similar propor ons of male and female, similar average age). Groups should have large enough ns to make comparisons reasonable. Suppose a researcher wants to compare level of educa on across religious groups. The categorical variable religion could have many categories. Depending on where the sample is obtained, some religious groups may contain very small numbers. It may not be possible to include these in later analyses. 3.11.2 Categorical Variables That Represent Treatment Groups For an experiment, all groups should have sufficient cases for analyses to be believable. A minimum of about 10 cases per group is desirable. In experiments, approximately equal group sizes are usually preferred, although unequal group sizes can be handled by most sta s cal procedures. 3.12 WHAT WE HOPE TO SEE IN FREQUENCY TABLES FOR QUANTITATIVE VARIABLES We hope to see few or no missing or implausible values. The range of score values in the sample should correspond at least approximately to the hypothe cal popula on of interest. A very small range some mes makes it difficult to find any associa on between variables in later analysis. For example, consider amount of television watching me as a quan ta ve variable. Suppose a researcher obtains a convenience sample of college students and that within the sample, the minimum amount of me is 0 hours and the maximum amount is 4 hours per week. However, the researcher would like to know something about the effects of TV exposure on mood in a broader adult popula on, some of whom watch 3 or 4 hours of TV each day (and thus 21 to 28 hours a week). If the sample does not include any people who watch TV this much, results won’t be generalizable to popula ons with much higher viewing me. On the other hand, if a researcher wants to hold a variable (fairly) constant within a study, a narrow range of scores may be desirable. If the popula on of interest is college students between the ages of 18 and 22, then a range of 18 to 22 in the sample would be desirable. Much higher ages (37, 55, and so forth) would be problema c. 3.13 SUMMARY

Researchers should make decisions about cases and values they want to include or exclude from their data before data collec on. For example, persons with some kinds of vision limita ons may not provide useful informa on about responses to op cal illusions. On the other hand, some studies might be set up to examine persons with specific types of vision limita ons. Inclusion and exclusion criteria can be set up in terms of categories (e.g., exclude smokers) or in terms of quan ta ve scores (e.g., include only ages between 18 and 65). Excluding cases a er data analyses have been performed is a ques onable research prac ce that can lead to incorrect conclusions (John, Loewenstein, & Prelec, 2012). Research reports should include informa on about problems with data (e.g., the numbers of percentages of impossible or missing or extreme values). Anything done to remedy problems (such as dele ng scores from analysis) should be stated clearly, and the ra onale for the decisions should be provided. (Don’t make up a different story for each case you take out of a data set; have consistent rules.) Readers need this informa on to evaluate generalizability of results, poten al limita ons of the study, and poten al problems in data analysis. You need to “clean up” your data before you do any addi onal analyses. There’s an old maxim in computer programming: Garbage in, garbage out. If your scores have errors or come from types of cases that were not supposed to be included in your study, results of analyses will be incorrect. The me to look for errors is at the beginning, before you have invested a lot of me in running analyses. APPENDIX 3A: GETTING STARTED IN IBM SPSS® VERSION 25 First, make sure your computer is set up to run SPSS. You can purchase or rent your own copy of SPSS (be sure to look for academic, educator, or student discounts, if you qualify). If you use SPSS through an organiza on site license in a business or university, ask your instructor or informa on technology department for site-specific instruc ons. When you have access to SPSS on your laptop, you should see the SPSS program icon on your desktop or in the start menu. If you can’t find it, ask someone to help you locate the icon in program file folders and create a shortcut on the desktop. Next, obtain the SPSS data files for this book. Data files for examples in this book can be downloaded from the SAGE textbook website at edge.sagepub.com/warner3e. I suggest that you create a folder (you might call it MySPSS) for your SPSS files. Ini ally you need to know about two types of files. File names that end in .sav are SPSS data files; the icon for an SPSS data file looks like this: File names that end in .spv are output files; the output file icon is . If you aren’t familiar with the management of computer folders and files, YouTube and Google can help. Search for “find files in Windows 10” (or whatever opera ng system you use). They can o en help answer ques ons about procedures both inside and outside SPSS. 3.A.1 The Bare Minimum: Using an Exis ng SPSS Data File to Obtain, Print, and Save Results For a minimal session in SPSS, you need to:

Open an SPSS data file. Run a sta s cs or graphics procedure. Print or save results. Exit from SPSS. To open an SPSS data file (and the SPSS program), click the icon for the exis ng SPSS data file. The icon for an SPSS input data file, in this case named heart rate study.sav, looks like this: . The first thing you see is the Data View, as shown in Figure 3.8. This data file includes two variables, named sex and hr (each variable corresponds to one column). It has 10 cases (each case corresponds to one row). In this example the categorical variable sex was coded 1 for male and 2 for female. For a minimal session you need only two of the top-level menu selec ons marked by rectangles in Figure 3.8. The pull-down menu for <File> is used to save and print files and exit from SPSS. The pull-down menu for <Analyze> provides a list of sta s cal procedures. To run the SPSS frequencies procedure, make the following menu selec ons: <Analyze> → <Descrip ve Sta s cs> → <Frequencies>, as in Figure 3.9. The Frequencies dialog box appears in Figure 3.10. SPSS sta s cal procedure dialog boxes have the following features. On the le is a list of all variables in the data set. On the right, under a pane headed Variable(s), is a list of the variables for which sta s cs will be obtained; ini ally this list is empty. To choose a variable, highlight a variable name in the le -hand list of all variables, then click the right-poin ng arrow to move it into the list under Variable(s). For many procedures you can select more than one variable. The frequencies procedure sets up a table of values for the selected variables. You can run the procedure by clicking the OK bu on near the bo om le . Bu ons along the right-hand side of the window can be used to select addi onal op onal informa on. You’ll learn more about other bu ons as we go along. A er you have selected one or more variables and clicked the OK bu on, a new window appears: an output window with sta s cs and/or graphical results, like Figure 3.11 (see page 54). To save the results (the frequency table) in Figure 3.11, make the following menu selec on: <File> → <Save As>, as shown in Figure 3.12 (see page 54).

Figure 3.8 Data View for the SPSS File heart rate study.sav The Save Output As dialog box (Figure 3.13, page 55) allows you to select the loca on for the saved output file and a name for the file. I suggest that the name should tell you what data you are working with (heart rate study) and the date when you created the file (June 6). (Note that the <Save As> command is applied to the currently open window. When you are looking at the data file, the <Save As> command will save that data file.) You can print the output file by using the menu selec on <File> → <Print> (or just use Ctrl + P). When you are finished, you can make the menu selec on <File> → <Exit> to leave SPSS.

Figure 3.9 Menu Selec ons to Open the SPSS Frequencies Procedure Dialog Box

Figure 3.12 Use of Pull-Down Menu to Save, Print, and Exit You need to know how to navigate to different windows. Figure 3.14 shows the three kinds of windows. In the Data window we can look at either Data View (the scores) or Variable View (the proper es of variables). When you open a new or an exis ng SPSS data file, you are in Data View (bo om le in Figure 3.14). If you are crea ng or edi ng a file, you will need to switch back and forth between Data View (where you can type in and modify score values) and Variable View (where you specify proper es of variables, such as type of measurement and

number of decimal places). To switch between Data View and Variable View, use the tabs at the bo om of the Data window, shown in Figure 3.14.

Figure 3.13 Save Output As Dialog Box

Figure 3.14 Three of the Windows in SPSS A er you run an analysis, SPSS creates a new window for output (the Output window displays the results of an analysis, such as tables or descrip ve sta s cs). The Output window opens automa cally when an analysis is run. Start at the top-level menu bar: Select <Window> (shown in Figure 3.15). From the pull-down menu, you can toggle between <Output1 [Document1]>, to view the results of analyses you have run, and <temphr10.sav [DataSet1]>, to view the original data in either Data View or Variable View. There are addi onal SPSS windows and file types; you do not need to know these to get started. 3.A.3 Crea ng a File and Entering Data In this sec on you’ll see how to create an SPSS data file like the one in Figure 3.16 and enter data.

Figure 3.15 SPSS Window Pull-Down Menu to Change View Between Data and Output Files

Figure 3.16 SPSS Data File temphr10.sav Whether you plan to type scores into SPSS or import data from a different program, you need to open a blank or new SPSS file first. Open SPSS by clicking the program icon . You will see the

Welcome to IBM SPSS Sta s cs window in Figure 3.17. Click <New Dataset> to open a blank data file. In Figure 3.18, you can see that this file does not yet have a name in the top le -hand corner, and it has no variable names across the top. You can import data from a different program, such as Microso Excel, into this blank file, or type scores in by hand. (If you type scores in by hand, I suggest that you create variable names first, as described in the next sec on.) If you have data in a different file format (such as Excel), you can import your data into SPSS by making this menu selec on: <File> → <Import Data> → <Excel>, as shown in Figure 3.19 (see page 58).

Figure 3.17 SPSS Welcome Window

Figure 3.18 A Blank (New) SPSS Data Set and the Data View and Variable View Selec on Tabs If you need to enter your own data instead of impor ng from another file, you can type in score values for each case, for each variable, in Data View. If you are crea ng a new data file, the cells in Data View are ini ally empty. Enter one score per cell, just as in spreadsheet so ware such as Excel. Leave cells blank for any missing scores. Remember to save the data file every me you add new informa on. It is a good idea to use a new or different file name for each save so that if you make a mistake, you can recover earlier versions of your files. 3.A.4 Defining Variable Names and Proper es of Variables A er you import a file or type in scores, you may need to create or change variable names and proper es. Ini ally you will probably be in Data View. To work with variable names and proper es, click the Variable View tab (see Figure 3.14). This opens Variable View, as shown in Figure 3.20. The Variable View window is the place where you create (or modify) variable names and specify (or change) informa on about variables such as measurement type. Within Variable View, type a name for each variable in the “Name” column (that is, type in the names sex, hr, …,

LikertRa ng). A er you type in these names, your new SPSS Data View will appear, as shown in Figure 3.21. You can select op ons from the menu bar in the Variable View window to add (or modify) the following informa on about variable proper es.

 How many decimal places should be displayed? (This can make output easier to read, but it is not essen al.)

Figure 3.19 SPSS Menu Selec ons to Import Files For categorical variables, what labels are associated with each score value (e.g., 1 = male, 2 = female)? I strongly recommend the use of value labels. If you return to a data set later, you may forget the meanings of group codes. Do any numerical values need to be iden fied as missing? (Blank cells are always treated as missing, but some mes specific score values are also treated as missing.) This is some mes

necessary when you work with archival data in which missing values were represented several ways. Blank cells in SPSS data are treated as missing (not as zeros). What is the type of measurement for each variable? Addi onal proper es are usually not needed at the beginning. SPSS documenta on (accessed in the <Help> menu on the top-level SPSS menu bar) will tell you about other proper es that you can specify. Variable Property 1: Decimal Places The “Decimals” column in the Variable View worksheet is used to set the number of decimal places to display for each variable. By default, two decimal places are used for all variables; for example, a heart rate score of 82 appears as 82.00 in Data View. A variable such as sex is coded using integer values (with no decimal places) and should be displayed with zero decimal places. To change the number of decimal places for sex to zero, look at the headings in the Variable View window (in Figure 3.22, page 60) and locate the column heading “Decimals.” Find the row that corresponds to the variable of interest (sex is in row 1) and the column for the property of interest (decimals is in column 4); click that cell. This cell is highlighted in Figure 3.22. When you click the right-hand edge of this highlighted cell, up and down arrows appear; these can be used to increase or decrease the number of decimal places. I set number of decimal places to zero for sex; in Figure 3.22, that change has been made. (Decimal places for the other variables have not yet been changed.)

Variable Property 2: Value Labels

It is useful to a ach value labels to scores for categorical variables. If you return to a data set that includes a categorical variable, such as sex, weeks or months a er you created it, you may forget which number code you used for each sex. The inclusion of value labels also makes output easier to read. To create value labels for sex, highlight and click on the cell in the row for sex and the column “Values” (as shown in Figure 3.23). Ini ally this cell says “None” because the values do not yet have labels. This opens the Value Labels dialog box in Figure 3.24. In the Value Labels dialog box in Figure 3.24, type each score value in the Value box and its corresponding label in the Label box; then click the Add bu on. For sex, I typed in “1” as the score value, then “male” as the label for this score value, then clicked the Add bu on to enter 1 = “male” into the list of labels. I then typed in “2” and “female.” This is the point at which I took the screen shot in Figure 3.24. The next step is to click Add (to move 2 = “female” into the list of value labels). When the list of labels is complete, click OK to exit from the dialog box and return to the main Variable View window. Many useful videos that show SPSS “in ac on” for mul ple- step procedures like this are available on YouTube.

Variable Property 3: Missing Values In real data, some score values can be missing values. In most SPSS files, missingness is indicated by a blank cell in Data View. In other words, when you enter data, you can indicate

missing scores by leaving the cells in the data worksheet blank. If you do this, the scores are not treated as zeros when sta s cs are calculated; blanks are omi ed when you request sta s cs and graphs. If you work with downloadable data files produced by survey organiza ons or other archival data sources, you may find that specific number codes are some mes used in addi on to or instead of blanks to represent scores that should be treated as missing. In a survey ques on about number of children, the response to the ques on regarding number of children might be coded 77 if the interviewer accidentally skipped the ques on, 88 if the respondent refused to answer, or 99 if the family situa on was too complicated to describe. If specific numerical values are used to represent missing values in your data set (in addi on to or instead of blanks), iden fy the score values that should be treated as missing by entering them in the “Missing” column of the Data View worksheet. For example, for the variable hr, you could enter a value of 999 into the “Missing” column, as in Figure 3.25 (see page 62). SPSS would then treat cases with blanks, and also cases with hr scores of 999, as missing values. Variable Property 4: Type of Measurement This book dis nguishes between categorical and quan ta ve variables (see Chapter 2). SPSS makes a similar dis nc on. SPSS has three types of measurement: Scale corresponds to quan ta ve variables, ordinal corresponds to ranks, and nominal is another name for categorical. The SPSS icons for these three levels of measurement appear in Figure 3.26 (see page 62). To assign a level of measurement, click the cell in the “Measure” column for the variable you want to specify. A pull-down menu with three measure types will appear. For each variable, select the appropriate measure type using this pull-down menu, as shown in Figure 3.26.

Figure 3.24 Value Labels Dialog Box

A er variable names and all four of these proper es have been set, the final Variable View window appears as in Figure 3.27. It is important to save your work frequently. A new data file is given a name when it is saved. To save a file, click these menu op ons: <File> → <Save>. In the Save Data As dialog box (in Figure 3.28), use the “Look in” box (near the top) to indicate the loca on where your file will be saved. Use the “File name” box to provide a name for your file. Then click Save. Your SPSS file should now have a name in the top le -hand corner.

Figure 3.25 Iden fica on of 999 as Indica ng a Missing Value

Figure 3.26 Selec on of Measure Type for Each Variable

Figure 3.27 Final Variable View for Sample Data Set

Figure 3.28 Save Data As Dialog Box APPENDIX 3B: MISSING VALUES IN FREQUENCY TABLES Real data have problems that aren’t usually seen in textbook examples. For example, missing responses are common in real data; these appear as blank cells in SPSS data files. Evalua on of the amount of missing data is an important issue when working with real data. The frequency table columns headed “Percent” and “Valid Percent” use different values of total N when compu ng percentages. In the “Percent” column, the total N for the file is used. In the “Valid Percent” column, the divisor used for percentages is N – number of missing values. Suppose you have a data file with total N = 60 cases and 2 missing scores on a ques on about pet preference. In the “Percent” column, missing values are treated as a group; percentages for both missing and nonmissing values are computed rela ve to the total N of 60 in the data set. In the “Valid Percent” column, the missing cases are essen ally removed or ignored; the

percentage of people in each group is obtained by dividing each group n by the number of people with valid (nonmissing) responses, 58 (see the example in Figure 3.29). The pet preference ques on has three response op ons: 1 = cat, 2 = neither, and 3 = dog. Note that there are blank cells in lines 6 and 11 for two imaginary subjects who did not answer this ques on. SPSS automa cally recognizes blank cells as missing. Output from the SPSS frequencies procedure (in Figure 3.30) presents group membership informa on two ways: The “Percent” column includes informa on about all cases (N = 60 in this example). The “Valid Percent” column reports only informa on for cases with valid (nonmissing) responses to the pet preference ques on (valid N = 58). The row labeled “Missing System” indicates cases with system missing scores (blank cells). In this example, 2 cases had no response to the pet preference ques on. The “Percent” column in Figure 3.30 includes all 60 cases in the data file and essen ally includes the “missing system” people as a fourth group that has a frequency of 2. The percentages in the “Percent” column show how the total of 60 cases were divided up among these four groups (the three pet preference choices and missing). The divisor for these percentages was 60 (the total number of cases in the file).

Figure 3.29 Frequency Table for Hypothe cal Scores: What Type of Pet Do You Want?

Figure 3.30 Frequency Table for Pet Preference (Two Scores Missing) The numbers in the column headed “Valid Percent” use only the “valid” cases (i.e., those that did not have system missing scores). The total number of valid cases is the total number of cases in the file (60) minus the number of system missing cases (2), which yields 58 valid cases. The percentages in the “Valid Percent” column show how these 58 people were divided up among the three pet preference groups; 58 was the divisor for each percentage. Beginning data analysts should, at least, no ce how many missing values there are and report that informa on (i.e., state that two people did not answer the pet preference ques on). SPSS automa cally omits cases with missing scores from analyses. However, advanced data analysts are expected to pay more a en on to the amount and pa ern of missing data, and may replace missing scores with es mated values (see Volume II [Warner, 2020]). APPENDIX 3C: DIVIDING SCORES INTO GROUPS OR BINS Rules for se ng up bins are as follows. Bins should be equal in width, if possible. (Occasionally data reports such as U.S. Census Bureau income data use unequal width bins. This can be confusing to readers who may not no ce that, other factors being equal, a wider bin is likely to contain more scores than a narrower bin.) The bins must include all scores. No score should be le out; the lowest bin must include the minimum score; the highest bin must contain the maximum score. In addi on, scores must not “fall through the cracks.” If one bin extends from 60 to 65, and the next bin extends from 66 to

70, a score of 65.5 would not be included in either bin. If scores are given to one decimal place, then the end points of bins must be given to one addi onal decimal place (two decimal places). To make sure that a score of 65.5 falls in one and only one bin, assuming that scores are given to only one decimal point, you could use bins that extend from 60 to 65.50, and 65.51 to 70. The data in this example are given in whole numbers, so the use of bins from 51 to 55, 56 to 60, and so forth, includes all possible values. The number of bins should not be too small (in many situa ons, between 5 and 10 bins are reasonable). The number of bins should not be too large (if you have one bin for each set of two adjacent score values, there is li le to no advantage over the ungrouped frequency distribu on). Exercises in introductory textbooks some mes require students to create groups or bins by hand and use these to set up grouped frequency distribu ons. Mechanics of se ng this up by hand can be tedious. By default, SPSS sets up ungrouped frequency distribu ons (discussed in this chapter) and grouped or binned histograms (discussed in the next chapter). I think these defaults are reasonable and useful in most situa ons. For the following example I use the heart rate data from Figure 3.7. The lowest hr score was 57 and the highest was 89. It is o en convenient to have bins for which either the lower or upper bound is a score that ends in 5 or 10. I decided to use seven bins; each bin has an upper bound that corresponds to a score that ends in 5 or 0. The lowest bin must include 57. The lowest bin corresponds to hr scores of 56 to 60 (this includes five score values). The next bin is for scores between 61 and 65; the next, scores between 66 and 70; and so forth, up to the last (seventh) bin that corresponds to hr scores between 86 and 90. Here is the procedure to set up your own preferred bins, or to modify the groups or bins SPSS sets up, for quan ta ve variable scores. The bin number can be treated as a categorical variable. Use the <Transform> → <Recode into Different Variables> menu selec ons (Figure 3.31, le ) to open the Recode into Different Variables dialog box (Figure 3.31, right), as in the example below. (It is never a good idea to select <Recode into Same Variables>, because if you make a mistake, you can’t reverse the opera on to get back to the original scores.) The Recode into Different Variables procedure requires three steps. You need to: Iden fy the “old” or exis ng variable. Provide a name for the new (different) variable. Systema cally list the score(s) on the old variable that correspond to scores on the new variable. The Recode into Different Variables dialog box in Figure 3.32 is used to iden fy the names of the old and new variables. All variables in the data set ini ally appear in the pane on the le -hand side in Figure 3.32. In the first step, when you highlight the name of an exis ng variable, a right

arrow appears; clicking on this arrow moves the name of the exis ng variable into the middle pane, under the heading “Numeric Variable -> Output Variable.” For the second step, you type in a new (different variable) name into the box on the right under the heading “Output Variable.” For the third step, you click the Change bu on. A er you do this, the middle window will contain this informa on: hr --> hr_binned. To open the next SPSS dialog box, click the Old and New Values bu on in Figure 3.32. The next dialog box, Recode into Different Variables: Old and New Values, appears in Figure.3.33. There are many ways to recode scores within the Recode into Different Variables: Old and New Values dialog box in Figure 3.34. For this example, I recoded small ranges of scores on hr (e.g., 56 through 60) into single scores for hr_binned (such as Bin 1). The following steps were needed to create each bin, as shown in Figure 3.33. On the le -hand side of the dialog box, I selected the radio bu on labeled “Range.” I typed the values that corresponded to the lowest value and the highest value for each bin; the values 71 and 75 appear in Figure 3.33. In the top right-hand corner of the Recode into Different Variables: Old and New Values dialog box, I typed in the number 4 to indicate that the score on the variable hr_binned will be 4 for persons who had hr scores from 71 to 75. Then click the Add bu on to move the command for this recode into the list of commands in the bo om right-hand window under the heading “Old --> New.” Con nue entering old and new values un l you have created all the bins that you will want. The final list of all old and new values appears in Figure 3.34.

Figure 3.31 Menu Selec ons to Recode Scores for Quan ta ve Variables Into Grouped Bins or Categories

Figure 3.32 SPSS Dialog Box to Recode Scores Into New (Output) Variable

Figure 3.33 SPSS Recode into Different Variables: Old and New Values Dialog Box

Figure 3.34 The Final List of All Old and New Values To obtain the grouped frequency distribu on, run the frequencies procedure for the new variable hr_binned. Results appear in Figure 3.35. Value labels could be assigned to each score for hr_binned to specify the range of hr in each bin; for example, the label for Bin 1 could be “56 through 60.” For further discussion of issues in binning, such as choice of number of bins and score limits for bins, see Doane (2017), Glen (2013), Grande (2015), or Kumar (2015). The Recode into Different Variables command has other uses. For example, this command could be used to change the groups in the hair color data in Figure 3.6. We could decide to combine people who reported uncommon hair colors (red, gray, pink, and blue) into an “other” category. We can also change that incorrect score of 12 on hair color into a missing value. A possible set of recodes appears in Figure 3.36. These recodes are summarized in Table 3.3. A er doing this recode, I added value labels for scores on the variable newgrouphaircolor. The frequency table for the new scores appears in Figure 3.37. Now each group has at least eight members. Of course, combining groups may not always make sense; we don’t want to combine apples and oranges.

Figure 3.35 Binned (or Grouped) Frequency Table for Heart Rates

Figure 3.36 Old and New Values for Hair Color Table 3.3 Summary of Recodes

Figure 3.37 Frequency Distribu on for Hair Color Scores A er Recode Used to Combine Groups COMPREHENSION QUESTIONS

1. Frequency (f) is the same as number of cases or persons in a group (n). True or false? 2. A percentage is obtained by mul plying a propor on by the total N in the data set. True

or false? 3. A percentage is obtained by mul plying a propor on by 100. True or false? 4. Total percentage should sum to 1.0 when you add percentages across all groups in a

table. True or false? 5. If the sum of percentages across groups in a table does not equal 100% (within rounding

error), there must be a mistake, more than just rounding error, in the calcula on of at least one within-group percentage. True or false?

6. Suppose you are told that in a sample of N = 50, 40% of the people are male. What is the n of people who are male?

7. Here is an example of a survey ques on: Which religion do you belong to? 1 = Buddhist, 2 = Catholic, 3 = Hindu, 4 = Jewish, 5 = Islam, 6 = Protestant, 7 = other.

Suppose that 500 people answer this ques on by selec ng one of the responses. Does it make sense (yes or no) to do each of the following things with scores for this ques on? Why or why not?

 Compute a mean religion score by summing all scores and dividing by 500  Find the median religion.  Find the modal religion.  Count the number of persons in each group and summarize this informa on in a

frequency table.  Report percentages of people in each group.

8. Table 3.4 is a frequency table based on real data from the sinking of the Titanic.  Would you call survival status (0 = died, 1 = survived) a quan ta ve or a categorical

variable?

 Fill in the two blank cells in Table 3.4.  Explain the results presented in this table in one sentence. If you choose to report

percentage rather than n or f, make sure that your sentence includes the total N. What was the modal outcome? In a sentence a nonsta s cian could understand, what happened?

Table 3.4 Outcomes for the N = 324 First-Class Passengers on the Titanic

9. Now consider the survival outcomes for the third-class (steerage) passengers (see Table 3.5).

 Fill in the two blank cells in Table 3.5.  What was the modal (or most common) outcome for third-class passengers on the

Titanic: death or survival? Describe this mode by repor ng the percentage. Table 3.5 Third-Class (Steerage) Passengers on the Titanic

10. Compare the data in Tables 3.4 and 3.5. Did first-class passengers have worse, the same, or be er outcomes than third-class passengers? (If you have seen any of the Titanic films, you already know the answer and the reasons why this happened.) For complete data about outcomes for all passengers, see Sta s cal Consultants Ltd. (2012). DIGITAL RESOURCES Find free study tools to support your learning, including eFlashcards, data sets, and web resources, on the accompanying website at edge.sagepub.com/warner3e. Descrip ons of Images and Figures

The image is a SPSS frequencies dialog box that has two columns. A list of all variables in the data set is on the le side namely; hr, temp underscore Fahrenheit, temp underscore Celsius, Likert underscore ra ng. The right side shows the selected variable sex. An arrow poin ng to the right between the two columns shows how variables move from the le to the right. On the extreme right are the bu ons for Sta s cs, Charts, Format and Style. Below the columns, there is a check box to Display frequency tables. At the bo om of the dialog box are op ons bu ons for the following; OK, Paste, Reset, Cancel and Help. Back to Figure The image is a frequency table with five columns; valid count, frequency, percent, valid percent, and cumula ve percent. There are just three rows; for males, females and the total. Details are below: Valid Male; 7; 70; 70; 70 Female; 3; 30; 30; 100 Total; 10; 100; 100 The valid count, frequency and percent have been marked in a rectangle. Back to Figure The image is a frequency table for hypothe cal hair color data with valid count, frequency, percent, valid percent, and cumula ve percent for eight pieces of data. Valid Brown; 16; 32; 34.8; 34.8 Black; 12; 24; 26.1; 60.9 Blond; 8; 16; 17.4; 78.3 Red; 3; 6; 6.5; 84.8 Grey; 3; 6; 6.5; 91.3 Pink; 2; 4; 4.3; 95.7 Blue; 1; 2; 2.2; 97.8 12; 1; 2; 2.2; 100 Total; 46; 92; 100 Missing System; 4; 8 Total; 50; 100 The last column of cumula ve percent has been crossed out. Back to Figure

The image is a SPSS frequencies dialog box that has two columns. A list of variables in the data set is on the le side which in the screenshot is the variable hr. The right side shows the selected variable sex. An arrow poin ng to the right between the two columns shows how variables move from the le to the right. On the extreme right are tabs for Sta s cs, Charts, Format and Style. Below the columns, there is a check box to Display frequency tables. This has been selected. At the bo om of the dialog box are tabs for the following: OK, Paste, Reset, Cancel and Help. The OK tab is selected. Back to Figure The image is a screenshot of the frequency table output window. At the top are the following statements: Frequencies variables equals sex. Divided by order equals analysis. Frequencies Sta s cs Sex N: Valid – 10 N Missing – 0 Below this is a table tled Sex which has the following columns; valid count, frequency, percent, valid percent, cumula ve percent valid: male, 7, 70, 70, 70 valid: female, 3, 30, 30, 100 total, 10, 100, 100 Back to Figure The image shows the usage of a pull down menu to save, print and exit SPSS. At the top of the spreadsheet are the following menu bu ons; file, edit, view, data, transform, analyze, graphs, u li es, extensions, window and help. The File bu on has an arrow next to it.

The file drop-down menu shows the following op ons; new, open, import data, close, save, save as, export as a web report, export, display data file informa on, switch server, repository, collect variable informa on, page a ributes, page setup, print preview, print, recently used data, recently used files, and exit. There are arrows next to save as, print and exit. Back to Figure The image is a picture of the Save output as dialog box. At the top of the user interface is a list of drives and directories, under the head Look in. Alongside this are op ons to open other folders and directories. Below is a list of file and folder names within the drive or directory specified in the Look in op on. Under the Look in op on, the Desktop folder has been selected. The file names displayed here are the following;

 Personal files  Finished books  Older sage files  2nd edi on all files  Ac vi es  Allnovel may 2019  Benny  Haldt  Handymen  Health and medical  Learning R  New Folder  Rarely used shortcuts  Sage June 5  Sta s cs references  UNH and department  Heart rate study June 6.spv

A File name head allows for naming of the file. This name has been shown as Heart rate study June 6.spv in the image. Below this is the Save as type op on along with available choice of file types. Here, the viewer files open parenthesis into spv close parenthesis has been selected. There are check boxes below this to Lock file to prevent edi ng in smartreader and to encrypt file with password. Both are unchecked.

At the right are bu ons to Save, Paste, Cancel and Help. At the bo om of the dialog box is a Store file to repository op on. The image is a combina on one that shows three windows of SPSS. At the top is the drop-down menu op on for the Window tab on the screen. This has the following op ons; Split, minimize all windows, go to designated viewer window, go to designated syntax window, reset dialog sizes and posi ons, output1 document 1 dash IBM SPSS sta s cs viewer, and temphr10.sav dataset1 dash IBM SPSS sta s cs data editor. The last two op ons have check boxes of which temphr10.sav dataset1 dash IBM SPSS sta s cs data editor has been cked. Two arrows lead to the next images, named Data input and edit window and Output window – results of analysis. The first shows a spreadsheet with empty rows and columns, and two op ons at the bo om; data view and variable view. The data view bu on has been depressed. An arrow leads from this to the next image, which shows what the data view window does. These states; type in or edit score values. An arrow also leads from the data input and edit window to another image that shows what the variable view window does. This states;

Modify proper es of variables:  Number of decimal points  Type of variable, for example, nominal versus scale  Value labels  Iden fica on of missing vales other than blanks.

At the top is the drop-down menu op on for the Window tab on the screen. This has been circled and has the following op ons: Split, minimize all windows, go to designated viewer window, go to designated syntax window, reset dialog sizes and posi ons, output1 document 1 dash IBM SPSS sta s cs viewer, and temphr10.sav dataset1 dash IBM SPSS sta s cs data editor. The last two op ons have check boxes of which temphr10.sav dataset1 dash IBM SPSS sta s cs data editor has been cked. Back to Figure The image is an SPSS welcome window that shows op ons to open old as well as new files. The first choice is New Files that has two op ons: New Dataset and New Database query. New Dataset op on has been depressed.

The next sec on is named Recent Files and shows two files: Ch underscore 03 underscore data backslash temphr10.sav and the second states Open another file. Back to Figure The image shows the drop-down op ons for the File menu bu on. On being depressed, the file drop-down menu shows the following op ons; new, open, import data, close, save, save as, save all data, export, mark file read only, revert to saved file, rename dataset, display data file informa on, cache data, collect variable informa on, stop processor, switch server, repository, print preview, print, welcome dialog, recently used data, recently used files, and exit. The Import data bu on has been clicked, leading to another pull-down menu with the following op ons; Database, Excel, CSV data, Text data, SAS, Stata, dBase, Lotus, SYLK, Cognos TM1, and Cognos Business Intelligence. There are arrows next to the File menu bu on, Import data and Excel tabs. Back to Figure The image is of a SPSS variable view window that shows how to set the number of decimal places to display. At the top are the menu bu ons such as; file, edit, view, data, transform, analyze, and graphs. Below these bu ons are icon bu ons to open a file, save, print, go back and forward, and other table edi ng op ons. The table columns are named the following; name, type, width, decimals. The sheet has the following data in the cells:

 name, type, width, decimals  sex, numeric, 8, 0  hr, numeric, 8, 2  temp underscore Fahrenheit, numeric, 8, 2  temp underscore Celsius, numeric, 8, 2  LikertRa ng, numeric, 8, 2

The cell with 0 in the first row has been highlighted. The image is a screenshot of the data editor page demonstra ng how to open the value labels dialog box for categorical variables. The tles of the columns as well as some of the column data have been shown in the image. They are men oned below:

name, type, width, decimals, label, values sex, numeric, 8, 0, blank, none hr, numeric, 8, 2, blank, none The values cell for the sex row has been highlighted. Back to Figure The image is the value labels dialog box that demonstrates how to add values to labels. Under the value labels head are two open cells: the first is for value and the second for label. Here 2 has been filled for the value and the label has been filled as female. Below this is an empty box in which 1 equals open quotes male close quotes has been typed in. On the le , Add, Change and Remove tabs are available, and the Add tab has been highlighted. A spelling tab is on the top right. The bo om has three tabs: OK, Cancel and Help. Back to Figure The image shows how to indicate missing values. Under the missing values head, there are three radio bu ons: No missing values Discrete missing values Here 999 has been entered in the empty cell Range plus one op onal discrete missing value Empty cells exist for the low value, high value and discrete value. The second op on of discrete missing values is selected. At the bo om are tabs for OK, Cancel and Help. Back to Figure The image is a screenshot of the save as dialog box. At the top of the user interface is a list of drives and directories, under the head Look in. Alongside this are op ons to open other folders and directories. Below is a list of file and folder names within the drive or directory specified in the Look in op on. Under the Look in op on, which has been highlighted, the Ch underscore 03 underscore data folder has been selected. The file names displayed here are the following;

 Old data folder  Birth pr by day.sav  either categorical or quant example ch 3.sav  female height.sav  fv data for textbook example.sav  hr devia on worksheet.sav  male height.sav  PET.sav  skewkurt.sav  temphr10.sav  temphr130.sav

Below this there is a statement: Keeping 5 of 5 variables. The file name head allows for naming of the file. This name has been shown as temphr10. The Save as type pull-down menu is next, and the selec on is SPSS sta s cs open parenthesis star dot sav close parenthesis. Below, there is a check box that states: Encrypt file with password. This has not been checked. At the right are the following bu ons; variables, save, paste, cancel and help. At the bo om is a bu on that states: Store file to repository. Back to Figure The image shows an example of a frequency table where two scores are missing. There are five columns with the following names; valid name, frequency, percent, valid percent and cumula ve percent. Details are below:

 valid name, frequency, percent, valid percent, cumula ve percent  cat, 24, 40, 41.4, 41,4  neither, 14, 23.3, 24.1, 65.5  dog, 20, 33.3, 34.5, 100  total, 58, 96.7, 100  missing system, 2, 3.3  total, 60, 100

The columns showing percent and valid percent are highlighted. Back to Figure The image is a screenshot of how to use the menu op ons to recode data.

At the top of the spreadsheet, tled temphr130.sav, are the following menu tabs; file, edit, view, data, transform, analyze, graphs, u li es, extensions, window and help. Below these tabs are icon tabs to open a file, save, print, go back and forward, and other table edi ng op ons. The transform menu tab has been highlighted, and a drop-down menu with the following op ons is visible: compute variable, programmability transforma on, count values within cases, shi values, recode into same variables, recode into different variables, automa c recode, create dummy variables, visual binning, rank cases, data and me wizard, create me series, replace missing values, random number generators and run pending transforms. The Recode into different variables choice has been highlighted and its details are provided below. On the le is the list of variables which are; sex, hr, temp underscore Fahrenheit and temp underscore Celsius. An arrow shows how once they are selected, they move to the next bin, termed Input variable right arrow output variable. An op on on the right allows for name and label to be a ached to the output variable. A tab below this bin states Old and new values. A IF tab is present below this, for op onal cases selec on condi on. At the bo om of the dialog box are op ons tabs for the following; OK, Paste, Reset, Cancel and Help. Back to Figure The image is a screenshot of how to use the menu op ons to recode data into new output variable. On the le is the list of variables which are; sex, temp underscore Fahrenheit and temp underscore Celsius. An arrow shows how once they are selected, they move to the next bin, termed Numeric variable right arrow output variable. The variable hr has been shi ed here. An op on on the right allows for name and label to be a ached to the output variable. The name is hr underscore binned and the label is blank. A change tab below this has been highlighted.

A tab below this bin states Old and new values. A IF tab is present below this, for op onal cases selec on condi on. At the bo om of the dialog box are op ons tabs for the following; OK, Paste, Reset, Cancel and Help. Back to Figure The image shows the Old and new values dialog box under the Recode into different variables op on. On the top le the old value of the variable can be input. There are seven check boxes here: value; system-missing; system or user missing; range; through; range, lowest through value; range, value through highest and all other values. The fourth choice Range has been checked. The value input there is 71 and the value input in the next sec on Through is 75. On the right the new value sec on has the following check-box op ons: Value, systems-missing, and copy old values. The first op on value has been input as 4. Below this is a box to recode old to new values. The figures shown here are: 56 thru 60 right arrow 1 61 thru 65 right arrow 2 66 thru 70 right arrow 3 Below this box are two check op ons: output variable are strings and convert numeric strings to numbers. The bo om has op ons tabs for the following; Con nue, cancel and help. Back to Figure The image shows the final list of al old and new values. On the top le the old value of the variable can be input. There are seven check boxes here: value; system-missing; system or user missing; range; through; range, lowest through value; range, value through highest and all other values. The fourth choice Range has been checked. On the right the new value sec on has the following check-box op ons: Value, systems-missing, and copy old values. The first op on value has been checked. Below this is a box to recode old to new values. The figures shown here are:

56 thru 60 right arrow 1 61 thru 65 right arrow 2 66 thru 70 right arrow 3 71 thru 75 right arrow 4 76 thru 80 right arrow 5 81 thru 85 right arrow 6 86 thru 90 right arrow 7 Below this box are two check op ons: output variable are strings and convert numeric strings to numbers. The bo om has op ons tabs for the following; Con nue, cancel and help. Back to Figure The image consists of a binned frequency table for heart rates. There are five columns; valid count, frequency, percent, valid percent and cumula ve percent. Details are below:

 valid count, frequency, percent, valid percent, cumula ve percent  1, 4, 3.1, 3.1, 3.1  2, 14, 10.8, 10.8, 13.8  3, 25, 19.2, 19.2, 33.1  4, 32, 24.6, 24.6, 57.7  5, 31, 23.8, 23.8, 81.5  6, 19, 14.6, 14.6, 96.2  7, 5, 3.8, 3.8, 100  Total, 130, 100, 100

Back to Figure The image is a dialog box that displays the old and new values for hair color. On the top le the old value of the variable can be input. There are seven check boxes here: value; system-missing; system or user missing; range; through; range, lowest through value; range, value through highest and all other values. The first choice Value has been checked. On the right the new value sec on has the following check-box op ons: Value, systems-missing, and copy old values. The second op on System-missing has been checked.

Below this is a box to recode old to new values. The figures shown here are: 1 right arrow 1 2 right arrow 2 3 right arrow 3 12 right arrow SYMIS 4 thru 7 right arrow 4 Below this box are two check op ons: output variable are strings and convert numeric strings to numbers. The bo om has op ons bu ons for the following; Con nue, cancel and help. Back to Figure The image shows the frequency distribu on a er the recode has been used to combine groups. There are five columns; valid count, frequency, percent, valid percent and cumula ve percent. Details are below:

 valid count, frequency, percent, valid percent, cumula ve percent  Brown, 16, 32, 35.6, 35.6  Black, 12, 24, 26.7, 62.2  Blond, 8, 16, 17.8, 80  Other open bracket red, grey, pink, blue close bracket, 9, 18, 20, 100  Total, 45, 90, 100  Missing system, 5, 10  Total, 50, 100

CHAPTER 4 DESCRIPTIVE STATISTICS 4.1 INTRODUCTION Informa on about quan ta ve variables can be summarized using simple descrip ve sta s cs, including the mode, median, and mean. These describe central tendency or “typical” responses. We obtain informa on about varia on of scores by examining minimum and maximum values, range, variance, and standard devia on. These descrip ve sta s cs cannot be applied to categorical variables, because scores on quan ta ve variables are only labels for group membership; it would not make sense to rank or add scores for a categorical variable such as hair color.

Descrip ve sta s cs such as means tell us what kinds of cases are included in a sample. For example, what was a typical age for a person in the study, and how much did people vary in age? Readers need to know the characteris cs of the sample to assess poten al generalizability of results. For example, if the average age of persons in a sample is 18, results may not be generalizable to persons between 40 and 80 years of age. 4.2 QUESTIONS ABOUT QUANTITATIVE VARIABLES You have already seen that a frequency table for a quan ta ve variable provides informa on about the number of missing values and the presence of implausible scores. The minimum, maximum, and range can also be obtained (preliminary informa on about variability), and a “middle” can be iden fied approximately by examining the score that has a cumula ve frequency close to 50% (to describe a typical or average score). One or more modes can be iden fied, although modes are not always useful when obtained from ungrouped frequency tables. Addi onal informa on about central tendency and variability can be obtained by compu ng descrip ve sta s cs on the basis of the values of all scores. These descrip ve sta s cs answer the following ques ons: What is a typical or common response (central tendency)? The sample mean and median describe central tendency for quan ta ve variables. How much do responses vary or differ? Addi onal informa on about variability is obtained from variance and standard devia on, introduced in the next sec ons. 4.3 NOTATION Here is all the nota on you need for now: N represents the total number of cases in a data set; n refers to the number of cases within a group. X is used to represent scores for a variable. Subscripts can be used to make it clear when a sta s c belongs to an individual person. The scores for persons 1, 2, and 3 in a sample can be denoted X1, X2, and X3. When a list of scores in a data set or group are summed, as in X1 + X2 + X3 + … + XN, the shorthand nota on ∑ is used. (This is capital le er S set in Greek symbol font, and it is read as “summa on.”) “∑X” means “add all scores for variable X.” 4.4 SAMPLE MEDIAN The mode, discussed earlier for categorical variables, can also be used with quan ta ve scores; the mode is the score with the highest frequency. The median, discussed in this sec on, defines the average as the score value that has 50% of people’s scores above it and 50% of people’s scores below it. The mean, discussed in the next sec on, defines the average as the sum of all the scores divided by the number of scores. For some samples, the values of the mode, median, and mean are equal. When the values of the mean, median, and mode differ, the ques on arises: Which one or two of these sta s cs do the best job of communica ng informa on about typical or “average” values?

Readers of research reports and mass media ar cles need to ask, Did the author define average using the mode, median, or mean? Readers need to understand the circumstances under which these numbers provide different percep ons of what is “average.” This example uses data that appeared in Figure 3.1 in the preceding chapter. To obtain the sample median (denoted “Mdn”), we first need to rank scores from highest to lowest, as shown in Figure 4.1. We then count the numbers of cases up from the bo om of the list and down from the top of the list. It is easiest to explain the median by looking at the list of individual scores, as in the following example. (There are also procedures to locate the median in grouped frequency tables, not included here.) To obtain the sample median for a set of N scores in a small data set:

1. Arrange the X scores in rank order from highest to lowest (as in Figure 4.1). If you have data in an SPSS or Excel file, there are commands to sort or rank scores in a column.

2. Determine the number of scores, c, that corresponds to half of N, the total number of scores. The value c tells you how many scores you need to count (up from the bo om and down from the top) to locate the median.

If N is an odd number, c = (N – 1)/2. If N is an even number, c = (N – 2)/2.

3. Count the number of scores (down from the top and up from the bo om of the list). If N is an odd number, this coun ng procedure iden fies one score in the middle of the ranked list of scores, and that score is the median. If N is an even number, this will iden fy two scores. When there are two scores in the middle of the ranked list, add them and divide the sum by 2 to obtain the median.

Figure 4.1 Finding the Median for an Even Number of Scores Consider the list of N = 10 heart rate scores in Figure 4.1. For N = 10, c = (10 – 2)/2 = 4. Count down four scores from the top of the list of scores; count up four scores from the

bo om, as shown by the arrows in Figure 4.1. Because N is an even number, two scores remain in the middle (scores of 73 and 74). Add these and divide the sum by 2; in this example, the median = (73 + 74)/2 = 73.5. Figure 4.1 shows a list of the 10 scores; each line is the score for one case. One of the score values, the score of 75, appears twice in this list because two persons had heart rates of 75. This figure helps make the interpreta on of a median clear: A sample median is the score value that separates cases into two equally sized groups: Half of the people in the sample have scores above the median, and half the cases have scores below the median. We can also say the median defines “average” as a value for which 50% of people have scores below and 50% of the persons have scores above that value. The mechanics of finding a sample median can be tedious if the number of scores is large, if scores must be rank-ordered by hand, or if the median must be obtained from a grouped frequency distribu on. I prefer that students focus on the way the median defines average instead of details of by-hand methods to obtain the median. 4.5 SAMPLE MEAN (M) The sample mean (denoted M) is obtained by first summing all the X score values in a sample of N scores and then dividing that sum by N, the number of scores: Other (4.1)

Here and in later chapters, we’ll think about each equa on in terms of the informa on included, how that informa on is combined, and the condi ons under which the value of the sta s c tends to be larger or smaller. Adding the X scores summarizes score informa on across all par cipants. The size of ∑ X depends on two things. Other factors being equal, the larger the X scores are, the larger ∑ X will be. Other things being equal, the larger N is, the larger ∑ X will be. To obtain a sample mean that represents the size of a typical score and that is independent of N, we correct for sample size by dividing ∑ X by N. The “bag of tricks” (frequently used arithme c opera ons) in basic sta s cs is small. The formula for the mean illustrates two of these tricks. First, to summarize informa on for a set of scores, we sum them. Second, to correct for the effect of sample size, we divide the sum by the number of pieces of informa on included in the sum. As you encounter more advanced sta s cs, you will see these tricks used again. Learn to look at ∑X and think, this term summarizes informa on about values of all the scores across cases. When you see “divide by N,” remember that dividing by N corrects for the effect of sample size on the sum. Equa ons can be converted into sentences in everyday

language that tell you what informa on is included and what is being done to that informa on. Equa on 4.1 is more than just instruc ons for computa on. It is also a statement or “sentence” that tells us the following: 1. What informa on is the sample sta s c M based on? It includes all the individual X

scores and the N of cases in the sample. 2. Under what circumstances will the sta s c (M) turn out to have a large or small value?

M is large when individual X scores are large and posi ve. Because we divide by N when compu ng M to correct for sample size, the magnitude of M is independent of N.

4.6 AN IMPORTANT CHARACTERISTIC OF M: THE SUM OF DEVIATIONS FROM M = 0 For each score in a sample, we can compute a devia on (or distance from the sample mean). If you know or can guess your heart rate, you can calculate your own devia on from the mean of the sample data in temphr10.sav. For example, if your hr score X = 75, and the mean M for the sample is 73.1, then the devia on of your score from the mean is (X – M) = (75 – 73.1) = +1.9 beats per minute. This devia on tells you that your heart rate is 1.9 beats per minute higher than the mean of this sample. We can find a devia on from the mean for every score in a sample. We find the devia on of an individual X score from M by subtrac ng M from X: Other (4.2)

For each case, this devia on tells us two things. On the basis of the sign of the devia on, we know whether a person’s X score was below or above the mean. From the absolute magnitude of the devia on, we can see whether the score was close to M (small devia on) or far from M (large devia on). Figure 4.2 shows the heart rate scores and the devia on from the mean for each score (the variable named devmean is X – M for each case). For the person in line 1 in Figure 4.2, hr = 70, and devmean, devia on of heart rate score from the mean of heart rate, is (70 – 73.1) = –3.1. For the person in line 1, heart rate is about 3 beats per minute lower than the sample mean. What happens if we sum the 10 devmean scores? Their sum (–3.1 – 4.1 – 2.1 – 11.1 + 1.9 + .9 + 6.9 – .1 +1.9 + 8.9) is 0. In fact, the sum of devia ons of scores from the mean in a sample is always 0. This is not a formal proof, only a demonstra on. (Proofs are presented in mathema cal sta s cs books.) The mean is the value rela ve to which devia ons of scores in a sample sum to 0. We can state this property of the sum of the (X – M) terms in equa on form:

Figure 4.2 Devia ons of Individual Heart Rate Scores From the Mean Other (4.3)

The mean and the median are both in the “middle” of the distribu on, but in different ways. The median is the score value that divides the top 50% of cases from the bo om 50% of cases. The mean is the score value for which the sum of the posi ve devia ons equals the sum of the nega ve devia ons. The mean some mes equals the median, and both of these can equal the mode; however, these values are not always equal. Why is it useful to think about “average” on the basis of the sum of devia ons from the mean? Devia ons from means are basic building blocks for almost every analysis you will learn. In a later sec on of this chapter, you will see that sample variance and standard devia on are also calculated using devia ons from the mean. You will see in later chapters that most of the analyses you will learn to evaluate rela onships between pairs of values are calculated using devia ons from means (and squared devia ons from means). These analyses may appear to be quite different from one another, but you will be able to see later that they are a “family” of techniques based on the same informa on about data (they are all constructed using devia ons from the mean). The mean is part of this family of analyses. This is a major reason why researchers o en report sample means (rather than medians or modes). The median and mode, while o en useful on their own, are not members of this family of analyses.

The mean has two advantages. First, it is related to many other popular types of sta s cal analyses. Second, it is the measure of central tendency or average most o en included in research reports; it is what readers of research reports usually expect to see. However, means have poten al disadvantages, as discussed in the next sec ons. First, means can be influenced greatly by extreme scores; and second, for some kinds of distribu on shapes, the mean is not a good indica on of typical or common responses. 4.7 DISADVANTAGE OF M: IT IS NOT ROBUST AGAINST INFLUENCE OF EXTREME SCORES The sample mean is not robust against the influence of extreme scores or outliers. Informally, we can say that a sta s c is robust if it yields reasonable results even when its assump ons are violated. Sample means and some other sta s cs do not “behave well” when outliers are present. Ideally, we do not want the value of a sta s c to change substan ally if we add or drop a few extreme scores. Unfortunately, adding a few extreme scores to a sample can greatly change the value of the mean. It is not desirable for any sta s c to depend on the values of one or a few scores. To demonstrate the impact of one extreme score on the mean, we’ll look at the heart rate data with and without an extremely high score at the upper end. The first column in Figure 4.3 shows the original hr scores from temphr10.sav (which you saw in Figure 3.1). In the column to the right, the new variable called hrOutlier has the same scores for hr for the first nine cases, but the last score is changed from 82 to a more extreme value of 160. Here are the values of mean, median, and mode for the two sets of scores in Figure 4.3:

Figure 4.3 Demonstra on of Effect of High-End Outlier on Sample Mean

We can evaluate the robustness of mean, median, and mode by asking whether each central tendency sta s c changes when one score value is changed. The median and the mode did not change. The value of the mean changed substan ally. With the new score of 160 added to the data, the sample mean is ∑ X/N = (62 + 69 + 70 + … + 160)/10 = 809/10 = 80.9. This is much larger than the mean obtained without the score of 160 (which was 73.1). In the original data (hr scores in the first column), the value of M was in the “middle”; there were five scores above and five scores below M. M was a “central” value that was reasonably close to most people’s individual scores and close to the median. On the other hand, for the hrOutlier scores, there are nine heart rate scores below and one heart rate score above the mean of 80.9. This example illustrates two things:

 When one very high score is added to this sample, the value of M increases (while the value of the median and mode do not change). This demonstrates that the mean is less robust against the impact of extreme scores than the median and mode.

 With one or more extremely high scores added, the value of the sample mean M is higher than the median; and in this example, M is actually higher than the majority of the individual scores in the sample. Under these circumstances the sample mean M is not a very good way to describe “average” or typical responses. Note that adding an extremely low score will make the mean smaller than the median.

4.8 BEHAVIOR OF MEAN, MEDIAN, AND MODE IN COMMON REAL-WORLD SITUATIONS This sec on previews the use of graphs to represent score frequencies for quan ta ve variables (graphs are discussed more extensively in Chapter 5). Figure 4.4 shows a frequency table for a set of hypothe cal scores. A corresponding histogram presents the same informa on graphically; the height of each bar in the histogram corresponds to the frequency of that score (i.e., the number of people who had that score value). 4.8.1 Example 1: Bell-Shaped Distribu on First let’s consider a hypothe cal batch of scores for which the mean, median, and mode have similar values. Suppose you have a survey ques on that asks people to rate their degree of agreement with this statement: “I think that the U.S. economy is doing well.” Response op ons are scores of 1 = strongly disagree (SD), 2 = disagree (D), 3 = neutral (N), 4 = agree (A), and 5 = strongly agree (SA). We might obtain a frequency distribu on like the one in Figure 4.4. Note that the answer given by the largest number of people corresponds to 3 (neutral), the next highest frequency responses were 2 (disagree) and 4 (agree), and the most extreme responses, 1 (strongly disagree) and 5 (strongly agree), were uncommon. For now, we will call this pa ern a “bell-shaped” distribu on. (Later, we’ll talk more formally about normal distribu ons.) Bell-shaped distribu ons tend to have values of the mean, median, and mode that are close to one another.

In the graph in the lower part of Figure 4.4, the number above the bar for each score value (such as 0) corresponds to the frequency of that score in the table (in the upper part of Figure 4.4). For example, in this hypothe cal data set, a score of 1 had a frequency of 6. A score of 3 had a frequency of 33 (i.e., 33 people chose the answer 3). The histogram or graph at the bo om of Figure 4.4 represents the same informa on about frequencies using bars with heights that correspond to frequency. This distribu on can be informally defined as bell shaped; there is a peak in the middle, and the pa ern is symmetrical; that is, the le -hand side of the distribu on is approximately a mirror image of the right-hand side.

Figure 4.4 Hypothe cal Likert Scale Ra ngs With Bell-Shaped Frequency Distribu on: (a) Frequency Table and (b) Corresponding Histogram When a distribu on is approximately bell shaped, the values of the mean, median, and mode are close together. For the data in Figure 4.4, the mean, median, and mode all have values close to 3 (and all are good descrip ons of “typical” score values). In Figure 4.4, most

scores are near 3; the nearby values of 2, 3, and 4 include about 80% of the scores. For a bell-shaped distribu on, any of these values (mean, median, or mode) provides a good sense of average or typical scores. Choice among these sta s cs is not a problem when we have an approximately bell-shaped distribu on. Data that have this bell-shaped distribu on typically work well in most bivariate analyses covered later in this book. 4.8.2 Example 2: Bimodal or Polarized Distribu on Next consider a set of hypothe cal Likert scale ra ngs for this statement: “I am a poli cal liberal.” A Likert ques onnaire item has two components: a statement of opinion or belief and a set of response op ons that represent different degrees of agreement with that statement. A 1-to-5 scale is o en used. The frequency distribu on of ra ngs for a hypothe cal Likert format ques on in Figure 4.5 is an example of bimodal or polarized ra ngs (i.e., scores tend to be at one extreme or the other). Most people gave a ra ng of 1 (strongly disagree) or 5 (strongly agree). The highest mode is for a ra ng of 5. The frequency for a ra ng of 1 was almost as high. Very few people gave ra ngs between these extremes. Figure 4.5 shows the frequency table and histogram for this hypothe cal outcome.

Figure 4.5 Hypothe cal Likert Scale Ra ngs for Polarized or Bimodal Responses: (a) Frequency Table and (b) Histogram

In this example, because the distribu on in Figure 4.5 is bimodal, with one mode at the highest possible score and a second mode at the lowest possible score, neither the mean (M = 3.11) nor the median (Mdn = 3) describes typical or average response very well. In fact, very few people gave ra ngs close to 3. We get a be er sense of “typical” responses if we report the two modes. People either love liberal policies or hate them. The point of this example is that in some frequency distribu ons, the mean and median may not be good ways to describe typical or average response. 4.8.3 Example 3: Skewed Distribu on Some variables represent counts of events or behaviors, for example, How many children do you have? How many speeding ckets have you received? Distribu ons for variables like these o en have many responses of 0, 1, or 2 (with a smallest possible value of 0). However, the highest responses can be 8, 10, or more. For these types of variables, the shape of a distribu on is o en asymmetrical or skewed. A frequency table of hypothe cal answers to the ques on “How many children do you want to have in the future?” appears in Figure 4.6.

Figure 4.6 Frequency Distribu on for Hypothe cal Scores on Number of Children Wanted: (a) Frequency Table and (b) Histogram The distribu on in Figure 4.6 is described as “posi vely skewed” because there is a longer (and thinner) tail at the posi ve end of the distribu on. In this posi vely skewed

distribu on, there are a few extreme scores at the high end (e.g., the persons who said they wanted 11 and 16 children). In this example, the mean of 2 is not a good indica on of typical responses (more than half of the people in this sample reported wan ng fewer than 2 children, i.e., either 1 or 0 children). As noted earlier, the mean is not robust against the effect of outliers; the extreme high scores of 11 and 16 made the mean (M = 2) for this set of scores higher than the median (Mdn = 1). We could call the scores of 0 and 1 both modes. These two modes are a be er way to describe “typical” scores. We could say that there is a mode at 0 and a smaller mode at 1; large percentages of people reported wan ng 0 children (31.8%) or 1 child (25.8%). In this situa on, the most informa ve way to report data would be to report one or two modes, or perhaps the en re frequency table. In this situa on, the median would also be a reasonable way to describe “typical” responses. The mean is somewhat high, although not so high that it would be completely unreasonable to report it. 4.8.4 Example 4: No Clear Mode It is possible for the numbers of cases to be about the same for all score values. Suppose the list of scores is as follows: [2, 4, 5, 7, 8, 8, 9]. The value 8 could be called the mode because it has a frequency of 2 and all other scores have a frequency of 1. Calling 8 a mode would not make sense, and using it to describe a typical score value would not make sense in this situa on. When in doubt, report more informa on. The en re frequency distribu on table, or the values of mode, mean, and median along with a graph, provide the most complete informa on to readers. The goal of repor ng should be to impart the clearest possible understanding of pa ern in the data. Despite poten al problems with sample means, it is more common to see reports of means (than of medians or modes) in science research reports. This happens because sample means are the basis for computa on of the most widely used bivariate sta s cs (such as t tests and analysis of variance). If you see a report of an “average” in mass media or in a research report, you need to know whether the descrip on of average is based on a mean, median, or mode. For some kinds of frequency distribu ons, such as the bimodal distribu on in Figure 4.5, a mean can be misleading informa on. 4.8.4 Example 4: No Clear Mode It is possible for the numbers of cases to be about the same for all score values. Suppose the list of scores is as follows: [2, 4, 5, 7, 8, 8, 9]. The value 8 could be called the mode because it has a frequency of 2 and all other scores have a frequency of 1. Calling 8 a mode would not make sense, and using it to describe a typical score value would not make sense in this situa on.

When in doubt, report more informa on. The en re frequency distribu on table, or the values of mode, mean, and median along with a graph, provide the most complete informa on to readers. The goal of repor ng should be to impart the clearest possible understanding of pa ern in the data. Despite poten al problems with sample means, it is more common to see reports of means (than of medians or modes) in science research reports. This happens because sample means are the basis for computa on of the most widely used bivariate sta s cs (such as t tests and analysis of variance). If you see a report of an “average” in mass media or in a research report, you need to know whether the descrip on of average is based on a mean, median, or mode. For some kinds of frequency distribu ons, such as the bimodal distribu on in Figure 4.5, a mean can be misleading informa on. 4.9 CHOOSING AMONG MEAN, MEDIAN, AND MODE Students some mes ask ques ons such as “Which is be er: The mean, the median, or the mode?” This is usually not the best kind of ques on to ask when choosing among sta s cs. It is more useful to ask, Under what circumstances does the mean provide the most useful informa on? Under what circumstances is the median (or mode) a be er choice? Here are some guidelines for choice among the mean, median, and mode:

1. If a frequency distribu on or histogram has one or more modes that are not near the center of the distribu on, the mean may not be a good way to describe typical response. It may be be er to report one or more modal scores. In Figure 4.5, we could say that there was a polariza on of opinion; people either strongly agreed with liberal policies (score of 5) or strongly disagreed (score of 1), with very few persons repor ng neutral feelings.

2. If a frequency distribu on is skewed (with a long thin tail on one side), the median, and one or more modes, may be a be er way to describe what is typical or average than the mean. Posi ve skewness (with extreme scores on the high end) is common in social science data. Nega ve skewness is possible (with a few extreme scores at the low end) but less common.

3. If a distribu on is bell shaped or approximately normal, the values of the mean, median, and mode will be close together. The mean is a good way to describe central tendency for bell-shaped distribu ons; the median and mode will have similar values.

4. When in doubt, or if the situa on is complicated, it may be be er to report the en re frequency distribu on (and/or histogram) along with values for the mean, median, and one or more modes.

Good prac ce:

 Do preliminary data screening by examining a frequency distribu on table and graph to evaluate whether the mean, median, and/or mode(s) are be er ways to describe central tendency.

 If implausible score values appear, go back and reexamine the data to correct errors.  Note the number of missing values.  State whether extreme scores or mul ple modes were detected (or whether the

distribu on is approximately normal).  State clearly what sta s c is used (mean, median, or mode) to describe average

responses. Bad prac ce:

 Obtain a mean, median, or mode without examining a frequency table or graph.  Select the index of central tendency value that “fits the narra ve.” For example, if

you want to report a high average, you can select whichever of these three sta s cs has the highest value, whether it makes sense or not. This is decep ve.

 Fail to make clear which index of central tendency is reported, and fail to note poten al problems with it.

Chapter 1 men oned “lying with sta s cs.” Reports of central tendency can be decep ve when they present only selected informa on that creates the impression the author wants to create. When an author wants readers to think, “Wow, that average is really high,” the author might choose to report the highest of the three values (mean, median, or mode). Conversely, if the author wants readers to think, “Wow, that average is really low,” the author might choose to report the lowest value among mean, median, and mode. An author who cherry-picks the -highest “average” is presen ng misleading (although perhaps not technically false) informa on. 4.10 USING SPSS TO OBTAIN DESCRIPTIVE STATISTICS FOR A QUANTITATIVE VARIABLE Previous sec ons discussed sta s cs for central tendency; the following sec ons discuss sta s cs to describe variability. In this sec on, SPSS is used to obtain all these descrip ve sta s cs (to describe both central tendency and variability) from data in the file named temphr10.sav using the SPSS frequencies procedure. To run Frequencies, make these menu selec ons (as in the example in Chapter 3): <Analyze> → <Descrip ve Sta s cs> → <Frequencies>. This opens the main dialog box for the frequencies procedure; in this window, move the variable hr into the Variables window. Click the Sta s cs bu on in the top right-hand corner of the main dialog box for the frequencies procedure to open the Frequencies: Sta s cs dialog box (shown on the right-hand side of Figure 4.7). There is a checkbox menu; click these checkboxes as shown to select central tendency sta s cs and sta s cs to describe variability (in the area headed “Dispersion”), as shown. The sta s cs that describe variability are explained in upcoming sec ons. Click Con nue to exit from the Frequencies: Sta s cs box and return to the main Frequencies dialog box; and click OK in the main dialog box to run the analysis. Output appears in Figure 4.8.

Figure 4.7 SPSS Frequencies: Sta s cs Dialog Box to Obtain Descrip ve Sta s cs for Quan ta ve Variables The values for mean, median, and mode in Figure 4.8 agree with the values obtained in earlier sec ons by hand, and they are close together. This example verifies the by-hand computa ons for mean and median done in previous sec ons for the same set of scores.

Figure 4.8 Output for Descrip ve Sta s cs for Hypothe cal Heart Rate Data in temphr10.sav The next sec on describes variability or varia on in quan ta ve scores. You will see how descrip ve sta s cs for varia on (including minimum, maximum, range, variance, and standard devia on) can be obtained by hand and how they are interpreted. 4.11 MINIMUM, MAXIMUM, AND RANGE: VARIATION AMONG SCORES The simplest way to describe varia on among scores begins by rank-ordering scores from lowest to highest. The lowest score value is the minimum (o en abbreviated as Min); the highest score value is the maximum (Max). As noted in Chapter 3, the range is maximum – minimum. For the heart rate data in Figure 4.1, Min = 62, Max = 82, and range = 20. Why does this informa on ma er? It helps us characterize the variety of people we have in the sample. When a variable has real-world uses, clinical or other interpreta on guidelines can help us understand what the minimum and maximum scores in a sample tell us. For example, guidelines published by the Mayo Clinic state that the normal adult res ng heart rate ranges from approximately 60 to 100 beats per minute. A well-condi oned athlete might have a heart rate of about 50 beats per minute. The people in this hypothe cal sample all have hr scores within the lower half of the normal range. This tells us that the sample consisted of people with heart rates in the low normal range, and this suggests a sample of persons with good cardiovascular fitness. If the sample had Min hr = 90 and Max hr = 120, this would indicate that many or most of the members of the sample have unusually high heart rates. When a frame of reference for the evalua on of scores is available, it should be used when

characterizing the sample. For example, if depression is assessed, one might ask, Are some scores high enough to warrant diagnoses of mild, moderate, or severe depression? In a study of a new an depressant drug, for example, readers would want to know whether most pa ents were mildly or severely depressed. 4.12 THE SAMPLE VARIANCE S2 We can obtain more useful informa on about variability by using informa on for all the individual scores. If all people had the same heart rate score, there would be no variance (e.g., a sample with hr scores of 72, 72, 72, …, 72 will have variance of 0). Variance in hr exists when people have different values of hr. Variability is evaluated by examining how far individual people’s scores are from the mean. 4.12.1 Step 1: Devia on of Each Score From the Mean Equa on 4.2 appeared earlier, and it is repeated here as Equa on 4.4. The first step in calcula on of variance is to compute the devia on of each person’s score from the sample mean M. (X – M) answers the ques on, How far is a person’s X score above or below the mean? Other (4.4)

For the data in temphr10, the devia on of the first X score from the mean is (70 – 73.1), that is, the score for the first case minus the mean of hr scores. Why do some people have higher, and some people lower, hr scores? Because people have different characteris cs, such as physical fitness, smoking, and anxiety, that make their heart rates higher or lower than other people’s. Sta s cal analyses you will learn later in the course provide ways to evaluate how much of the individual differences in hr might be related to each variable, such as anxiety. 4.12.2 Step 2: Sum of Squared Devia ons Next, we need to summarize informa on about distances from the mean across all the people in the sample. You might think that you could summarize informa on by summing the devia ons, the values of (X – M), across all people in the data set. However, recall from Sec on 4.6 that this sum of devia ons from the mean is always zero. It might occur to you that this problem could be avoided by summing the absolute values of these devia ons. However, there is another approach that yields more useful results. Here we introduce another tool in the sta s cian’s bag of tricks. When devia ons sum to 0, we get around that problem by squaring the devia ons before we sum them. Squaring devia ons makes all the terms in this sum posi ve.

To summarize informa on about individual score distances from the mean: First, we square each person’s devia on from the mean. (Squaring a nega ve value yields a posi ve value, so squaring devia ons gets rid of the problem that posi ve and nega ve devia ons would cancel each other out by summing to 0.) Then we sum those squared devia ons. The resul ng sum is called the sum of squares (or sum of squared devia ons), abbreviated SS. In upcoming steps, SS will be used to compute sample variance and standard devia on. We return to the ques on: How much do people’s scores in a sample vary or differ rela ve to the sample mean? In words, the answer to this ques on is: We find out how far each X score is from the mean by compu ng a devia on, we square each devia on, then we sum the squared devia ons to summarize informa on about distance from the mean. This gives the formula for SS, the sum of squared devia ons of scores from their mean: Other (4.5)

A different version of the formula for SS is o en given in introductory textbooks: Other (4.6)

Equa on 4.5 makes it easier to see what informa on about scores is included when you compute SS. Equa on 4.6 is easier for by-hand computa on of SS from scores. They yield the same results. No ce that we square each individual devia on first; then we add those squared devia ons. Appendix 4A describes rules about precedence in the order of arithme c opera ons. Opera ons that are enclosed in parentheses are done before opera ons outside the parentheses. For example, if you see the expression ∑(Y 2), you square the value of each Y, and then sum the squared values. If you see the expression (∑Y )2, you sum the values of Y and then square that sum. Some mes textbook examples use numbers that give a whole-number result for SS; however, in real data, SS is usually not a whole number. Appendix 4B reviews rounding. I suggest that you retain at least three decimal places during computa ons. Final results for most sta s cs are o en rounded to two decimal places. See Appendix 4B for a discussion of rounding. In Figure 4.9 (data from temphr10.sav) the squared devia on from the mean for each individual person appears in the last column (the variable named devia onsq). Adding the

scores for devia onsq gives the value of SS for this data set: SS = 288.90. For larger data sets, it is more convenient to have a computer program do this.

Figure 4.9 Devia ons and Squared Devia ons of Heart Rate Scores From Mean Note that SS cannot be a nega ve number (because we are summing squared devia ons, and squared numbers cannot be nega ve). Other factors being equal, SS tends to be larger when:

1. The individual (X – M) devia ons from the mean are larger in absolute value. 2. The number of squared devia ons included in the sum increases.

The minimum possible value of SS (which is 0) occurs when all the X scores are equal and, therefore, equal to M. For example, in the set of scores [73, 73, 73, 73, 73], the SS term would equal 0. There is no limit, in prac ce, for the maximum value of SS. To interpret SS as informa on about variability, we need to correct for the fact that SS tends to be larger when the number of squared devia ons included in the sum is large. Dividing by N, the number of scores in the sample, seems like the obvious solu on. However, this does not provide the best answer. 4.12.3 Step 3: Degrees of Freedom It might seem logical to divide SS by N to correct for the increase in size of SS as N increases. However, this yields values that are slightly too small; Gosset (discussed in Tankard, 1984) worked out the reason for the problem and discovered a simple solu on. When we look at

the pieces of informa on used to compute SS (i.e., the devia on of each score from the sample mean), it is possible to see that we do not have N independent devia ons (or pieces of informa on) available to compute the SS; in fact, we have only (N – 1) pieces of informa on. To explain why devia ons from the mean in a sample of N scores provide only (N – 1) independent pieces of informa on about distance from the mean, recall that the sum of all devia ons of scores from the mean must equal 0. Suppose we have N = 3 scores in a sample (call these scores X1, X2, and X3) and that their mean is M. First, we convert each X score into a devia on by subtrac ng the sample mean M. We know that the sum of these devia ons must equal zero. That yields this simple equa on: Other

When we compute (X1 – M) + (X2 – M) (on the le side of the equa on), this gives us the value that the remaining devia on, (X3 – M), must have. Only the first two devia ons are “free to vary,” that is, free to take on any possible value. Once we know the value of any two of the devia ons, the value of the last devia on is determined (it must be whatever number is needed to make the sum of all devia ons equal 0). This is only a demonstra on, not a formal proof. A degrees of freedom (df) term tells us how many independent pieces of informa on we have available when we compute SS or another sta s c, such as a variance (in other words, how many devia ons from the mean are free to vary). This modified divisor, N – 1, is called the degrees of freedom (df). The df term tells us how many of the devia ons are “free to vary.” The use of df instead of N as a divisor is another frequently used tool in the sta s cian’s bag of tricks. Later analyses also use df terms, although df o en has different values than (N – 1) in other situa ons. Degrees of freedom for the SS and sample variance are obtained using Equa on 4.7:

(Some textbooks use ŝ2 to denote a sample variance calculated as SS/N. In actual prac ce, this nota on is almost never used when sta s cs are applied to real-world data, and you will not see ŝ2 again in this book.) Return to the data in Figure 4.9. The first column shows heart rate scores for each person. The second column shows the devia on of each person’s score from the mean (the variable name is devia on). The third column shows each person’s squared devia on (the variable name is devia onsq). If we sum the squared devia ons, we obtain SS = 288.90. For this sample of N = 10 cases, df = (N – 1) = 9. For the hr data in Figure 4.9, s2 = 288.90/9 = 32.1. It is useful to think about situa ons that would make the sample variance s2 take on larger or smaller values. The smallest possible value of s2 occurs when all the scores in the sample have the same value; for example, the set of scores [73, 73, 73, 73, 73, 73] would have SS = 0 and s2 = 0. The value of s2 is larger for a sample in which individual devia ons from the sample mean are rela vely large. SS will be larger for a batch of data in which scores take on a wide range of different values, such as [44, 52, 66, 97, 101, 119, 120], than in a data set in which the scores differ by very small amounts, such as [60, 65, 66, 68, 71, 74, 74]. SS cannot be nega ve; however, there is no fixed upper limit for possible values of SS. In large data sets, and for variables that have values in the thousands or tens of thousands, values of SS can be extremely large. The informa on s2 provides about differences among hr score values is in terms of “squared hr in beats per minute”; however, the original hr scores were in beats per minute. It would be useful to convert variance back into the units of the original data. Now we use another tool in the sta s cian’s bag of tricks: To convert something given in X2 units back into original X units, we take the square root. 4.13 SAMPLE STANDARD DEVIATION (S OR SD) The sample standard devia on is usually denoted s in textbooks. In research reports it is o en denoted SD. Taking the square root of s2 gives the value of s. Another way to find s is to compute it from the values of SS and df: Other

(4.9)

The shorter ver cal arrow next to the frequency table in Figure 4.10 extends from M – (1 × SD) to M + (1 × SD). This corresponds to the frequencies enclosed in the smaller ellipse. The longer ver cal arrow ranges from M – (2 × SD) to M + (2 × SD), score values from 59.5 to 69.5. This corresponds to scores in the larger ellipse. Most women in the sample had heights that were included in the range M – (2 × SD) to M + (2 × SD); only three women (2.5%) had scores below 59.5, and only two women (1.7%) had scores above 69.5. In words: When we combine informa on about distance from the mean (SD) with the loca on of the mean (M), we obtain informa on about the range of values within which most of the X scores lie; this is called the range rule. The range rule works only for bell- shaped distribu ons, as in the present example.

Figure 4.10 Hypothe cal Data for Female Height in Inches for N = 120 Women With M = 64.5 and SD = 2.5 Here are some approximate (not exact) rela onships of SD with data values that can help you understand what SD = 2.5 tells us. In the preceding example, the range for height scores (70 – 58) was 12. The range rule suggests that, for a bell-shaped distribu on, the range is o en a li le less than 4 × SD. For these data, 4 × SD = 4 × 2.5 = 10. Turning this statement around, the range rule suggests that SD is o en a li le less than one quarter of the range. Knowing that SD is related to range may help you understand SD. Remember that the range rule works only for bell-shaped distribu ons.

 The value of SD tells us about typical distances of scores from the sample mean.  Few scores are lower than 2 × SD units below M or higher than 2 × SD units above M.

In other words, 2 × SD is a large distance from the mean; only a small percentage of scores are that far away from M.

 If a research report tells you that the distribu on of scores is close to normal with known values for M and SD, this is sufficient informa on for you to guess the range.

 Using SD = 2.5, individual devia ons of height from a mean height of 2.5 inches or less (either posi ve or nega ve devia ons) were very common.

 Almost all people had devia ons from the mean that were less than 2 × SD in absolute value; 2 × SD = 2 × 2.5 = 5 inches. To say this another way, most women had heights between 62 and 67 inches.

Good prac ce: To choose the most appropriate sta s cs to describe central tendency and variability, the data analyst should examine a frequency distribu on table or graph. If the distribu on is approximately normal, M and SD are good ways to describe these. If the distribu on is clearly non-normal, Mdn and interquar le range may be preferred. For distribu ons that are not bell shaped, see the next chapter for be er ways to describe varia on among scores. 4.15 WHY IS THERE VARIANCE? Why do scores differ across people? This is the most fundamental ques on in applied sta s cs. For data about humans, the ques on becomes: What makes people different? Why do some people have higher, and some people lower, heart rates? Why are some people taller and others shorter? Some characteris cs do not differ across people (they are constant). Most people have five fingers on each hand. The rare excep ons are people who have genes for a different number of fingers, or people who have lost fingers because of injury. However, characteris cs such as heart rate do differ across persons and situa ons. Suppose you measure hr for all members of a group. Some persons will have low hr; their hr may be lower than average because they are physically fit and do not smoke. Others have high hr; these elevated hr scores might be due to anxiety or caffeine consump on.

A first goal of sta s cal analysis is to quan fy or describe how much people differ. Range, variance, and standard devia on provide this informa on. We will consider a more interes ng ques on in upcoming chapters: Can we explain or predict these differences in heart rate? Can we understand why people differ? You probably already have some intui ons about factors that are related to hr, for example, smoking and physical fitness. When we go on to bivariate analyses, we will ask how hr scores are sta s cally related to other variables, such as amount of anxiety or stress. Results of these analyses can lead to inferences that stress predicts, or perhaps influences, heart rate. In later chapters you’ll see that the overall variance for a variable such as hr can be divided (or par oned) into propor ons of variance that can be predicted from or are related to other variables (such as physical fitness, smoking, anxiety, and caffeine use). Some variables may predict large propor ons of variance in heart rate (possibly these are the variables that have the strongest influence on hr). For those of us who are excited about sta s cs, this is where the fun begins; this is where we can make discoveries or test past research claims about discoveries. Other variables may predict li le or none of the variance in hr. 4.16 REPORTS OF DESCRIPTIVE STATISTICS IN JOURNAL ARTICLES Most journal ar cles report descrip ve sta s cs for numerous variables. Informa on about categorical variables (that describe groups in the study) can usually be provided in sentence form. Usually informa on for numerous quan ta ve variables is summarized in table form. The following data are from Warner, Frye, Morrell, and Carey (2017). The predictor variable of most interest was number of servings of fruit and vegetables consumed per day (NCIfv, servings of fruits and vegetables from a Na onal Cancer Ins tute food frequency ques onnaire). Past research suggested that people who eat more fruits and vegetables tend to have higher scores on measures of well-being such as life sa sfac on and posi ve mood. The outcome variable of most interest was life sa sfac on (LS). Before doing analyses to evaluate whether NCIfv predicts LS, we need to know about the behavior of scores for each of these variables. This survey was completed by 492 students from a university in New England, including 152 male and 340 female students. They were recruited from introductory courses, 79 from a nutri on course and 413 from psychology classes. All par cipants were between ages 18 and 24; the modal age was 18. Descrip ve sta s cs for quan ta ve variables appear in Table 4.1. Tables of descrip ve sta s cs o en use abbreviated names for variables that are used throughout the paper. Notes at the bo om of the table iden fy the variables and provide addi onal informa on about them. Direc on of scoring must be clear (for example, we need to know that a score of 5 indicates be er sleep, rather than more sleep problems). It is helpful to list variables in sets (in this example, a list of well-being outcome measures, a list of behavioral predictors, and a list of dietary predictors). An earlier “Methods” sec on in the research report would provide more informa on about how variables were measured. Informa on about distribu on shapes should be included; this is discussed in Chapter 5.

4.17 ADDITIONAL ISSUES IN REPORTING DESCRIPTIVE STATISTICS Many addi onal kinds of informa on can be included in summary tables. The minimum informa on usually provided for each quan ta ve variable is M and SD. Table 4.1 included the possible minimum and maximum scores for each variable, on the basis of the way scores were obtained for these variables. Readers who are not familiar with the variables will find this helpful to evaluate the obtained scores. Here are other things a summary table might include: the minimum and maximum scores obtained in the sample, numbers of missing values for each variable, and informa on about reliability for each variable. If you do research in a specific area, look at tables in published research reports to see if addi onal informa on is usually included in summary tables for descrip ve sta s cs.

When a study includes many groups and/or many variables, all groups and all variables should be iden fied and reported in descrip ve tables. This lets readers know if you have selec vely excluded some groups or variables from the analyses you report later. 4.18 SUMMARY Research reports o en describe scores on quan ta ve data using the sample mean M, the standard devia on SD (or s), and the variance s2. Readers tend to assume that scores for quan ta ve variables have an approximately bell-shaped distribu on (if they are not informed otherwise), and they interpret the descrip ve sta s cs accordingly.

The “bag of tricks” used to compute many sta s cs is actually quite small, and you have seen several of these tricks in this chapter:

 When a sum of devia ons would be zero, square terms before summing them.  When correc ng for the number of devia ons (or pieces of informa on) included in

a sum, divide by df instead of by N.  To put informa on back into the original terms of measurement, take the square

root. These “tricks” are used again in many future analyses. You have seen that the sample mean is not always the best descrip on of central tendency. In some frequency distribu ons, M is much larger (or smaller) than the median, and the magnitude of the mean is influenced strongly by a few extreme scores. When frequencies have more than one mode, or are skewed, M is some mes not the best descrip on of the “typical” response. When you report a mean, you need to tell readers something about the shape of the frequency distribu on to provide the background informa on needed to understand poten al problems with the mean. Sta s cs books provide so many examples of bell-shaped distribu ons that students may assume that all data have this distribu on shape. However, many common kinds of variables do not have bell-shaped distribu ons. Graphs, discussed in Chapter 5, can be used to evaluate whether scores have a bell-shaped distribu on or some other distribu on shape. We should not assume that all distribu on shapes are bell shaped. When repor ng informa on about variables, remember that readers may assume a bell-shaped distribu on if you do not explain clearly that the distribu on shape is different. If you read mass media reports about “averages,” you need to know whether average was es mated using the mode, median, or mean; under some circumstances, these three descrip ve sta s cs can yield very different values. The next chapter provides further informa on about obtaining and interpre ng graphs of frequency distribu ons and addi onal ques ons we can ask about distribu ons of scores on a quan ta ve variable. APPENDIX 4A: ORDER OF ARITHMETIC OPERATIONS Many equa ons combine two or more arithme c opera ons, for example, ∑ X2 includes both squaring and summing X scores. When opera ons are combined, the result o en differs depending upon the order in which opera ons are done. Consider this set of scores: X = [1, 3, 5, 2]. If you square each X value and then sum the squared values, you would obtain (1 + 9 + 25 + 4) = 39 . If you sum the X’s and then square that sum, you would obtain (1 + 3 + 5 + 2)2 = 112 = 121. It is important to know which arithme c opera on to do first. There are rules of precedence (order) for arithme c opera ons (see h p://mathworld.wolfram.com/Precedence.html). When I present equa ons I explain in words the order in which computa ons should be done, and o en, I use extra parentheses

to make this clear in the equa on. When an expression appears within parentheses, such as (X – 5), do that opera on first. If you see ∑(X 2), square each X value first, and then sum the squared X values: (1 + 9 + 25 + 4) = 39. If you see (∑ X )2, sum the X values first, and then square the sum: (1 + 3 + 5 + 2)2 = 112 = 121. Be aware that if you do arithme c opera ons in the wrong order, you can obtain answers that are incorrect by huge amounts. APPENDIX 4B: ROUNDING Computer programs o en provide numbers given to several decimal places. Each number that comes a er a decimal point represents one decimal place. For example, the number 4.171 has three decimal places. If you do by-hand computa ons, you should retain at least three decimal places during your computa ons to minimize rounding error. Final results are usually rounded to a small number of decimal places, o en two decimal places. The preferred number of decimal places to report differs across disciplines and may differ across variables. Use common sense. It would be silly to say that the average American gets 7.481 hours of sleep per night; it would make more sense to report this as 7.5 hours. If you are in doubt, report more decimal points than you think reviewers or editors or readers will want; these can always be rounded later. Use past research in your area of interest as a guide for the number of decimal places to report. Here are simple rules for rounding. If a final digit is greater than 5, the digit before it is increased by one unit when you round (this is “rounding up”). For example, 3.86 would be rounded to 3.9. If the final digit is less than 5, the digit before it is le the same when rounding (this is called “rounding down”); for example, 3.83 would be rounded to 3.8. If the final digit is exactly 5, you can toss a coin to decide whether to round up or down. In many journal ar cles, and for many sta s cs such as M, results are presented to one or two decimal places. One excep on is that p values, introduced in later chapters, are o en reported to three decimal places. COMPREHENSION QUESTIONS

DIGITAL RESOURCES Find free study tools to support your learning, including eFlashcards, data sets, and web resources, on the accompanying website at edge.sagepub.com/warner3e. Descrip ons of Images and Figures Back to Figure The figure is a table that shows the hr and count for an even number of scores. On the le is a table that shows the hr values as well as the coun ng cases values.

 62: 1  69: 2  70: 3  71: 4  73: nil  74: nil  75: 4  75: 3  80: 2  82: 1

On the right are two arrows; one begins from the top and points down to the median in the center and the second begins from the bo om and points up to the median. The median has been given as the value calculated below: Open parenthesis 73 plus 74 close parenthesis divided by 2 equals 73.5. Back to Figure

The image has a table with 2 columns; one shows hr and the other hrOutlier alongside a count.

1. 62; 62 2. 69; 69 3. 70; 70 4. 71; 71 5. 73; 73 6. 74; 74 7. 75; 75 8. 75; 75 9. 80; 80 10. 82; 160

The M for column 1 is 73.1 and M for column 2 is 80.9. For the first column, there are 5 scores below and 5 scores above M. For the second column, there are 9 scores below and 1 score above M. A note below the table states: Column 1 lists the original scores. In column 2, the value of 82 in the original data is replaced with the higher value of 160. Back to Figure The image is a combina on of a table that shows Likert Scale ra ngs and a bell-shaped frequency distribu on. The first part of the image is a five-columned table that displays valid count, frequency, percent, valid percent, and cumula ve percent for five pieces of data. The details provided below are in the same order as men oned in the table:

1. 6; 9.1; 9.2; 9.2 2. 11; 16.7; 16.9; 26.2 3. 33; 50; 50.8; 76.9 4. 10; 15.2; 15.4; 92.3 5. 5; 7.6; 7.7; 100

 Total: 65; 98.5; 100  Missing System: 1; 1.5  Total: 66; 100

The numbers 16.7, 50 and 15.2 are circled from the percent column. The second part of the image is the same data as above represented in a bell curve.

The horizontal axis represents the degree of agreement and ranges from 1 to 5, in increments of 1. The ver cal axis represents the frequency and ranges from 0 to 40, in increments of 10. There are five bins in the graph and their heights are: 6, 11, 33, 10 and 5. A line passes through each of the bars to form the bell curve. A note below the graph states: Mean is equal to 2.95, median is equal to 3, and mode is equal to 3. Back to Figure The image is a combina on of a table that shows Likert Scale ra ngs and a histogram displaying bimodal or polarized ra ngs. The first part of the image is a five-columned table that displays valid count, frequency, percent, valid percent, and cumula ve percent for five pieces of data. The details provided below are in the same order as men oned in the table:

1. 24; 36.9; 36.9; 36.9 2. 5; 7.7; 7.7; 44.6 3. 4; 6.2; 6.2; 50.8 4. 4; 6.2; 6.2; 56.9 5. 28; 43.1; 43.1; 100

Total: 65; 100; 100 The numbers 36.9 and 43,1 are circled. The second part of the image is the same data as above represented in a histogram. The horizontal axis represents the degree of agreement and ranges from 1 to 5, in increments of 1. The ver cal axis represents the frequency and ranges from 0 to 30, in increments of 10. There are five bins on the graph and their heights are; 24, 5, 4, 4 and 28. A note below the graph states: Mean is equal to 3.11, median is equal to 3, and modes are 5 and 1. Back to Figure The image is a combina on of a table that shows a frequency distribu on and a histogram.

The first part of the image is a five-columned table that displays valid count, frequency, percent, valid percent, and cumula ve percent for ten pieces of data. The details provided below are in the same order as men oned in the table:

 0; 21; 31.8; 31.8; 31.8  1; 17; 25.8; 25.8; 57.6  2; 9; 13.6; 13.6; 71.2  3; 8; 12.1; 12.1; 83.3  4; 4; 6.1; 6.1; 89.4  5; 3; 4.5; 4.5; 93.9  7; 1; 1.5; 1.5; 95.5  8; 1; 1.5; 1.5; 97  11; 1; 1.5; 1.5; 98.5  16; 1; 1.5; 1.5; 100  Total; 66; 100; 100

The second part of the image is the same data as above represented in a histogram. The horizontal axis denotes the number of children and ranges from 0 to 15, in increments of 5. The ver cal axis denotes the frequency and ranges from 0 to 25, in increments of 5. There are 10 bins on the graph and their heights are: 21, 17, 9, 8, 4, 3, 1, 1, 1, and 1. A note below the graph reads: Mean is equal to 3.11, median is equal to 3, and modes are 5 and 1. Back to Figure The image is a combina on screenshot that shows how to select descrip ve sta s cs as well as a SPSS sta s cs dialog box. In the first part of the image, a closeup of the taskbar of a spreadsheet shows different naviga on bu ons including Analyze, graphs, u li es, extensions, window and help. On the clicking of the Analyze bu on, a drop down menu with the following op ons has opened; reports, descrip ve sta s cs, Bayesian sta s cs, tables, compare means, general linear model, generalized linear models, mixed models, correlate, regression, loglinear, classify, dimension reduc on, scale, non-parametric tests, forecas ng, survival, mul ple response, simula on, quality control, ROC curve, spa al and temporal modeling and IBM SPSS amos. The reports tab has been depressed leading to another menu with the following op ons; frequencies, descrip ves, explore, crosstabs, turf analysis, ra o, p-p plots, and q-q plots.

The second part of the image shows the frequencies: sta s cs dialog box. On the top le are the Percen le values with the following check boxes; Quar les, Cut points for equal groups that has an empty slot to fill in for the number of equal groups, and Percen le. The top right has a central tendency sec on with the following check op ons; mean, median, mode and sum. The first three have been checked. The bo om le has a dispersion segment with the following check op ons; std. devia on, minimum, variance, maximum, range, and S.E. mean. All except for S.E. mean have been checked. On the bo om right is a check box that states Values are group midpoints. Below this is a sec on tled characterize posterior dist with two check boxes; skewness and kurtosis. At the bo om are bu ons for con nue, cancel and help. Back to Figure The image is a table tled Sta s cs with the following informa on:

 Hr  N

o Valid – 10 o Missing – 0

 Mean – 73.1  Median – 73.5  Mode – 75  Std. Devia on – 5.666  Variance – 32.1  Range – 20  Minimum – 62  Maximum – 82

Back to Figure The image is a table that shows heart rate values, devia on and square of devia ons.

 hr; devia on; devia onsq  62; minus 11.1; 123.21  69; minus 4.1; 16.81  70; minus 3.1; 9.61  71; minus 2.1; 4.41  73; minus .1; .01  74; .9; .81

 75; 1.9; 3.61  75; 1.9; 3.61  80; 6.9; 47.61  82; 8.9; 79.21  Sum: 0; 288.9

Back to Figure The image is a combina on of a table and a graph that shows hypothe cal data for female height. The table has four columns; valid count, frequency, percent and cumula ve percent. Details are below;

 58; 1; .8; .8  59; 2; 1.7; 2.5  60; 3; 2.5; 5  61; 8; 6.7; 11.7  62; 14; 11.7; 23.3  63; 12; 10; 33.3  64; 20; 20; 16.7; 50  65; 16; 13.3; 63.3  66; 18; 15; 78.3  67; 15; 12.5; 90.8  68; 5; 4.2; 95  69; 4; 3.3; 98.3  70; 2; 1.7; 100  Total; 120; 100

There are 2 circles over the figures; one covers the percent values 11.7, 10, 16.7, 13.3, 15, and 12.5 and the second covers a larger set of percent values including 2.5, 6.7, 11.7, 10, 16.7, 13.3, 15, 12.5, 4.2 and 3.3. The graph in the second part of the image shows the X and Y axes as well as the 1 into SD and 2 into SD lines. The following figures are men oned alongside the graph:

 M minus 2 into SD equals 59.5  M minus 1 into SD equals 62  M equals 64.5  M plus 1 into SD equals 67  M plus 2 into SD equals 69.5

CHAPTER 7 SAMPLING ERROR AND CONFIDENCE INTERVALS 7.1 DESCRIPTIVE VERSUS INFERENTIAL USES OF STATISTICS

Up to this point, we have used sta s cs such as M and SD only to describe scores in small samples. In some real-life situa ons, such as evalua on of exam scores for a class of students, that is all the data analyst wants to do. For example, a teacher may report summary informa on such as the mean, median, minimum, maximum, and standard devia on of scores in his or her class. However, teachers typically do not use this informa on to make inferences about students outside the class. When the use of sta s cs is limited to descrip on of a sample, that is called a descrip ve use of sta s cs. When instructors engage in descrip ve use of sta s cs, they report their results something like this: “In the sample of students in my classroom, N = 36, M = 85, and SD = 10.” Then the instructor stops and makes no statements about larger -popula ons of students beyond the students included in the class. In scien fic studies, however, researchers almost always want to say something about a popula on of cases beyond the cases included in the study. Here is a simple hypothe cal example of an inferen al use of sta s cs. A researcher wants to es mate (make an inference about) popula on mean length of lizards for the en re popula on of lizards on an island. Suppose it is not possible to locate every lizard. A biologist captures a sample of N = 25 lizards and finds mean length M = 2 in. The researcher can say, “The mean length in my sample is 2 in.” However, the biologist probably wants to say something about the mean length for the popula on of all lizards on the island. Two problems must be considered when using informa on from a sample to make inferences about a popula on. One issue, discussed earlier, is representa veness of the sample. Is the sample similar to the popula on of interest? We should be careful not to generalize results from a study to a popula on that includes many kinds of people that were not included in the study. Now we consider a second problem that arises when using a sample to make inferences about a popula on: the problem of sampling error. Different samples, drawn from the same popula on, usually have different sample means. Varia on in values of M across different samples from the same popula on is called sampling error. How much can we believe the mean from any one sample? If M is close to the popula on mean, it is a good es mate of that popula on mean; if it is far, then it is not a good es mate. We need to have some idea how far any individual value of is M likely to be from the popula on mean. This may appear to be an unanswerable ques on. How can we say anything about the distance of a sample mean M from the popula on mean if we don’t know the - popula on mean? However, this ques on can be answered by crea ng -ar ficial (-imaginary) popula ons of scores for which we do know the popula on mean, -drawing many -different samples from those imaginary popula ons, and examining the -distribu ons of values of M across all of these samples (for example, by se ng up a histogram for values of M). 7.2 NOTATION FOR SAMPLES VERSUS POPULATIONS The sta s cs we compute within samples are used to es mate corresponding popula on parameters. (Parameters of a normal distribu on include the mean and variance.) Table 7.1 summarizes the nota on used for descrip ons of samples versus popula ons.

Prior to this chapter, we have focused on distribu ons of individual X scores for individual cases in samples. When we described varia on among individual X scores in a sample, we used M as the descrip on of average and SD (the sample standard devia on) to summarize informa on about distances of X scores from M. Now we shi to a new level of thinking. A distribu on of values of M across a large number of different samples from the same popula on is called the sampling distribu on of M. A sampling distribu on for many values of M has mean μ, and distances of values of M from μ are described by σM as shown in Figure 7.1. This chapter discusses distribu ons of values of M instead of distribu ons of individual X scores. The following sec ons consider these ques ons:

1. Why do we obtain different values of M when we look at different samples randomly selected from the same popula on? (Because of sampling error.)

2. How can we obtain informa on about the varia on of values of M across samples? (This can be done using classroom demonstra ons, Monte Carlo computer simula ons, and algebraic methods. We will focus on the more concrete approaches.)

3. How can varia on of values of M across samples be described? The central limit theorem (CLT) tells us that distribu ons of values of M across samples have a normal shape with mean μ. The CLT also tells us how to calculate the value of σM, which provides informa on about differences in values of M across many samples because of sampling error.

The most important ques on is:

4. Why are we doing all this? Table 7.1 Nota on for Sample Sta s cs and Corresponding Popula on Parameters

Figure 7.1 Comparison of Familiar Distribu on of Individual X Scores With New Kind of Distribu on: A Distribu on of Values of M Across Many Samples Precise informa on about the amount of varia on in values of M due to sampling error, denoted σM early in the chapter and SEM later in the chapter, is used to:

 Set up confidence intervals (later in this chapter). A confidence interval tells us about the amount of sampling error we need to take into account when we use one sample value of M to try to es mate μ for a popula on.

 Set up sta s cal significance tests (in the next chapter). Significance tests ask yes/no ques ons about possible values of μ and answer those ques ons using sample M and SEM.

Confidence intervals and sta s cal significance tests can be used with almost all sta s cal techniques you will learn. That is why these topics are essen al: You will con nue to use them in all future chapters! Let’s look at these ques ons one at a me. 7.3 SAMPLING ERROR AND THE SAMPLING DISTRIBUTION FOR VALUES OF M 7.3.1 What Is Sampling Error? The term error o en means something different in sta s cs than in everyday life. In everyday use, the term error o en means “mistake.” If you add up a list of numbers incorrectly, that is a mistake. (Sta s cians and students some mes make mistakes, for example, a sta s cs student could make an arithme c error when calcula ng M.)

In sta s cs, the term error has several addi onal technical meanings that differ from our everyday understanding of error. This chapter introduces sampling error. (Later chapters discuss other special types of errors in sta s cs.) These errors are not mistakes; they are problems that arise in the use of sta s cs even when a sta s cian is doing everything correctly. Let’s begin the discussion of sampling error by examining just two samples. Suppose that you and a classmate each want to es mate the mean emo onal intelligence (EI) for all the students in a large counseling program. The EI test costs $30 per administra on, so each of you can afford to have a sample of only N = 10. If you draw different random samples, your value of M is very likely to differ from the value obtained by your classmate. That outcome doesn’t mean that either of you made a mistake (although mistakes are possible). It probably happens because, through luck of the draw, one of you happened to get more students with high EI scores in your sample. We don’t have enough informa on in this situa on to know which sample mean is a be er es mate of μ (that is, which value of M is closer to μ). Examina on of two samples does not tell us much about sampling error (except that it happens). Looking at hundreds of samples will tell us much more; it will help us figure out how much sampling error arises. You will probably not be surprised to hear that the larger the number of cases in the samples, the smaller the amount of sampling error. 7.3.2 Sampling Error in a Classroom Demonstra on The first me I taught sta s cs, I was a teaching assistant at Harvard; we used a textbook called Statlab (Hodges, Krech, & Crutchfield, 1975). The book included a popula on of data scores printed at the back (1,296 cases from a study of infant birth weight). And the book came with a set of dice (one red, one green). To randomly select one infant case from this popula on, each student rolled the dice twice. The first roll of the dice determined which page, and the second roll of the dice determined which case on that page, to include in the sample. To draw a random sample of 10 cases, each student repeated this process 10 mes. The result was that each student obtained his or her own unique random sample from the popula on. Each student then calculated M, sample mean infant birth weight, for his or her unique sample. These values of M differed across students because of sampling error. The instructor collected these values of M from all class members and set up a histogram for the 300 values of M. We hope that sample means for birth weight will tell us something about popula on means for infant birth weight. Examina on of this histogram or sampling distribu on that summarized all the means reported by students in the class revealed three things. Two features of this distribu on were easy to no ce. These are included in the CLT:

1. The distribu on of values of M was approximately normal. 2. The distribu on of values of M was centered at (had a mean of) μ, the popula on mean

infant birth weight for all 1,296 cases. (The instructor knew the value of μ from the instructor’s manual; the students did not.)

From this we can begin to understand the nature of sampling error. The resul ng bell-shaped or normal distribu on with mean of μ tells us that most students obtained values of M fairly close to μ. However, a few students obtained values of M that were far from μ. This happened even if no one made a mistake calcula ng M. Some samples, just by chance, had a few unusually heavy or light infant birth weights. I remember my first reac on to this type of situa on: Why not just use the mean of all the sample means as the es mate of μ? That reac on misses the point. The exercise is meant to teach us how much values of M vary across samples (as described by the histogram of values of M, for example). Understanding how much values of M vary across samples provides informa on about typical distances of M from μ. In actual research, researchers usually have just one sample mean. In this situa on, it is helpful to know (on the basis of the CLT or classroom exercises) the typical distances of values of M from μ. This gives us the context to think about reasonable values of μ, given the informa on we see in the sample (N, M, and SD). We know that our one sample mean is probably not exactly equal to μ. We would like to know how far from μ our value of M is likely to be. The standard error of the mean (SEM) provides that informa on. 7.3.3 Sampling Error in Monte Carlo Simula ons The same informa on about values of M across hundreds of samples that we thought about as a classroom exercise can be obtained more easily by using a computer to draw numerous random samples from a specific popula on. This is called Monte Carlo simula on (because of the random selec on). To do a Monte Carlo simula on, a mathema cal sta s cian creates a popula on of quan ta ve X scores and defines that set of scores as the popula on. Usually these scores are created such that they are normally distributed. For the popula on in this ar ficial data set, the sta s cian knows the value of μ. He or she can draw hundreds or thousands of different random samples (of size N, such as N = 25) from that popula on and compute the sample mean M for each of those samples. Then the sta s cian can set up a histogram to evaluate the shape of the distribu on of values of M and to see how close values of M are to μ. The sta s cian can quan fy the distance of each M from μ by finding a devia on (M – μ). A Monte Carlo simula on is a quicker way to obtain informa on like the classroom exercise for much larger numbers of samples, the results are similar, and the computer simula on outcome will yield more precise results. In Monte Carlo simula ons to obtain sampling distribu ons for values of M, the distribu on of values of M has a normal shape and the mean of the distribu on becomes closer to μ as the number of samples increases. Values of M tend to be closer to μ when N is large than when N is small. 7.4 PREDICTION ERROR

When we use M from one sample to es mate μ, we almost always have predic on error. M is an inaccurate es mate of μ if M differs from μ. The magnitude of predic on error is given by the difference (M – μ). The distance of M from μ may be small or large. The distance or difference between M and μ is called predic on error, represented in Equa on 7.1. Other (7.1)

If the true mean length for the en re popula on of lizards men oned in an earlier example is μ = 3.5 in., and the researcher found M = 2.5 in., then the predic on error would be (M – μ) = (3.5 – 2.5) = 1; that is, the sample mean would overes mate the popula on mean by 1 in. The distances of values of M from μ in the sampling distribu on provide informa on about magnitudes of predic on errors across many different samples. 7.5 SAMPLE VERSUS POPULATION (REVISITED) A sample is a subset of some popula on of interest. However, the way subsets are obtained differs across research domains. This was discussed in greater detail in Chapter 2. 7.5.1 Representa ve Samples In applica ons such as industrial quality control and opinion polling, popula ons may be clearly defined, and sta s cians can o en obtain representa ve samples. An industrial quality control analyst could evaluate whether the mean diameter for all the widgets produced by a factory in a week (the en re popula on of widgets) equals a precise standard diameter, such as 5 mm. If it is too me-consuming or costly to measure every single widget, the analyst could select a random sample of widgets, measure their diameters, compute the sample mean, and use the mean diameter of the sample to es mate the popula on mean. The widgets included in the sample could be selected systema cally (e.g., every 100th widget) or randomly (by using random numbers to decide which of the widgets are included in the sample). In opinion polls, the popula on of interest might be “all registered voters in the United States.” Sampling methods used by polling organiza ons are complex; their goal is to obtain a sample that is representa ve of the popula on of interest. 7.5.2 Convenience Samples By contrast, laboratory researchers in fields such as psychology, medicine, and educa on o en begin with accidental or convenience samples. O en, they want to make inferences about a broader hypothe cal popula on. It is more difficult to evaluate representa veness of samples rela ve to popula ons of interest in situa ons involving convenience samples. Trochim (2006) suggested that researchers can rely on proximal similarity to decide what popula ons we can make inferences about, on the basis of the composi on of the sample. In other words, analysts should limit generaliza ons to popula ons that resemble the types of

cases included in the sample. If a new drug is tested only on a sample of young adult male pa ents, for example, it is not reasonable to generalize results to women, children, and elders. Limita ons are o en men oned in discussion sec ons of research reports; for example, if a drug study included only men, the authors should note that the conclusions of the study may not apply to women. These limita ons are usually described in formal research reports, but mass media reports about research o en fail to men on them. 7.6 THE CENTRAL LIMIT THEOREM: CHARACTERISTICS OF THE SAMPLING DISTRIBUTION OF M Ques ons about the sampling distribu on of M include the same ques ons we have asked about other distribu ons in the past. For values of individual X scores, obtained for the people in one sample, we asked, What is the distribu on shape, what is the mean, and how much do the scores vary? For a distribu on of values of M, obtained across hundreds of samples, we ask analogous ques ons:

 What is the shape of the distribu on of values of M?  What is the average value of M across many samples?  How much do values of M vary across samples?

Your intui on probably suggests that samples with large N’s usually provide “be er” es mates than smaller samples. That intui on is correct. Other factors being equal, larger values of N tend to yield values of M that are closer to μ. The central limit theorem (CLT) derived by mathema cal sta s cians tells us that ideal or theore cal sampling distribu ons of M have the following three characteris cs:

1. The distribu on shape for values of M, based on many different random samples of the same size from the same popula on, is normal.

2. The mean of the sampling distribu on for M equals μ, the popula on mean. 3. The dispersion or varia on of this sampling distribu on, called the popula on standard

error of M (denoted σM), depends on the popula on standard devia on σ and N (the number of cases in each sample). Equa on 7.2 shows how σM is calculated from σ and N:

Other (7.2)

We would like to have small values of σM because σM tells us about the distances of values of M from μ. Small distances indicate less predic on error; large distances indicate greater predic on error. We will con nue, for a while, to discuss the ar ficial situa on in which we know all about the popula on (specifically, we know the value of σ). In actual applica ons, we need analyses that we can use in situa ons where we don’t know σ; those analyses are presented in later sec ons. 7.8 EFFECT OF N ON VALUE OF THE POPULATION STANDARD ERROR An empirical example illustrates the way the dispersion or varia on of the sampling distribu on changes as N is increased (and with σ held constant). We will treat the 130 heart rate scores in temphr130.sav as the popula on; the histogram appears in Figure 7.2. For this popula on, μ and σ are known: μ = 73.76, σ = 7.062. This popula on is not normally distributed.

Figure 7.3 Results of Monte Carlo Simula on of Sampling Distribu on of M for (a) N = 9, (b) N = 25, and (c) N = 64 7.9 DESCRIBING THE LOCATION OF A SINGLE OUTCOME FOR M RELATIVE TO POPULATION SAMPLING DISTRIBUTION (SETTING UP A Z RATIO) We have not yet reached the procedures that you will use in actual data analysis, but we are ge ng close. A researcher who has one sample mean will want to know the loca on of that sample mean rela ve to the theore cal sampling distribu on. The CLT specifies the characteris cs of that distribu on. When we have complete informa on about the popula on, we know the values of μ and σ, and assuming the popula on is normally distributed, we can answer the ques on about distance of M from μ by using a z score. Recall that we could describe the loca on of an individual X score rela ve to a normally distributed sample by first compu ng a z score, then using a table of the standard normal distribu on to evaluate the percentage of scores that lie below or above z (Chapter 6). For an individual X score, z = (X – M)/SD. We know that a sampling distribu on is normal in shape (for sufficiently large numbers of samples) with mean μ and a standard error of σM. When N and μ and σ are known, we can compute σM and find a z score loca on for a specific value of M as follows: Other (7.3)

Then we set up a z ra o: (M – μ)/σM = (76.22 – 73.76)/2.354 = 1.045, which I will round to 1.0. You can evaluate a loca on of z = 1.00 using a diagram of areas for the standard normal distribu on or by looking up areas in the table of the standard normal distribu on in Appendix A at the end of this book. I have suggested that outcomes that lie in the middle 95% of a distribu on can be called “not very far from the mean.” In this example, z = 1.00 is not very far from the mean. This implies that the mean for this specific sample of N = 9 was not far from the popula on mean. If you (and/or your classmates) draw your own random samples of 9 cases from the hr130.sav file, your results will differ (because of sampling error). 7.10 WHAT WE DO WHEN Σ IS UNKNOWN The descrip on of the sampling distribu on of M used so far assumes that σ (the popula on standard devia on) is known. In actual research, data analysts typically do not know σ. Usually we only have this informa on from one sample: N, M, and SD. We can s ll make inferences about sampling error even when we don’t know σ, but we need to modify some procedures and nota on. We can use the sample SD to es mate σ. When we use SD to es mate σ, we change the name for the standard error. Recall that the equa on for popula on standard error (Equa on 7.2) was: Other

SEM depends on the values of N and SD in the same way that σM depended on N and σ. If SD is held constant, and N is increased, SEM becomes smaller. If N is held constant and SD is increased, SEM becomes larger. Smaller values of SEM are desirable (they indicate smaller amounts of sampling error). Usually the best way to reduce SEM is to increase N. It would be nice if changes in the name and in the equa on for standard error were the only changes needed. However, a new problem arises when we use SD to es mate σ. Our es mate of the sampling error sta s c is now affected by two sources of error: the errors that arise when

we use M to es mate μ and, in addi on, the errors that arise when we use SD to es mate σ. How do we take this addi onal source of sampling error into account? Gosset (discussed in Tankard, 1984) understood this problem and found a solu on. When σ is not known, and we use SD to es mate σ, values of M are more widely dispersed around μ than in a normal distribu on. We need a sort of “fudged” normal distribu on with fa er tails to represent this situa on. In fact, we need an en re family of distribu ons, because the amount of addi onal sampling error that comes from use of SD depends (as perhaps you have guessed) on the size of N. We need a different distribu on, called a t distribu on, for each value of N (for each value of df, which in this situa on is N – 1). To take sample size into account, we now call the ra o in Equa on 7.5 a t score instead of a z score. Use of the name t reminds us that we need to evaluate it rela ve to one of the t distribu ons instead of the standard normal distribu on. Here is the equa on for the t ra o to describe the standardized distance of one value of M from either an actual or a hypothesized value of μX: Other

7.11 THE FAMILY OF T DISTRIBUTIONS Recall that a normal distribu on has the following fixed rela onship between distance from the mean (in z-score units) and area under the curve (probability).

The bo om 2.5% of the area in the normal distribu on in Figure 7.5 corresponds to z values below –1.96. In a normal distribu on, 50% of the area lies between z > –1.96 and z < +1.96. As suggested in Chapter 5, outcomes or cases that fall in the middle 95% of the area can be called “not very far from the mean.” The top 2.5% of the area corresponds to z values above 1.96; these values can be called “far from the mean.”

The family of t distribu ons can be understood as modified normal distribu ons. To obtain a t distribu on, it is as if you pushed down on the peak of the normal distribu on to make it lower, and allowed the tails of the distribu on to become fa er, as in Figure 7.6. A different t distribu on is required for each possible value of df. Recall that df (degrees of freedom) for the sum of squares SS and the standard devia on SD is N – 1. The t distribu on was developed to correct for the problems that arise when we use SD to es mate σ. The sample es mate of SD is based on only N – 1 independent pieces of informa on in the sample (the first N – 1 devia ons of scores from the sample mean). We can use the diagram in Figure 7.6 to see how the distance from the mean that is required to capture the middle 95% of the area changes as N (or df ) decreases. The curve for df = ∞ (infinity; denoted by the line with the longest dashes) corresponds to the normal distribu on. As df increases, the shape of the t distribu on converges to normal. For all prac cal purposes, when df > 120, we can use a table of the normal distribu on to evaluate areas or probabili es. In effect, when df is larger than 120, the amount of sampling error contributed by using SD to es mate σ becomes negligible; we are close to the “σ is known” situa on when the sample size used for SD is large. We can compare this with a situa on in which N = 4 and df = 3; this t distribu on corresponds to the solid line in Figure 7.6. It is much fla er in the middle, and has thicker tails, compared with the normal distribu on. Let’s compare: What cutoff or cri cal values of t do we need to capture the middle 95% of each of the following distribu ons? Remember that t and z are both unit-free measures of distance.

7.12 TABLES FOR T DISTRIBUTIONS Recall that data analysts are o en interested in iden fying outcomes that are “close to the mean” versus “far from the mean” of distribu ons. The most widely used defini on for “close to the mean” is the middle 95% of a distribu on. (Other percentages can be used.) In a normal distribu on, the middle 95% of the area lies between z = –1.96 and z = +1.96. When we use t distribu ons and N is small, the values that bound the middle 95% of the distribu on correspond to values greater than 1.96 in absolute value; see Figure 7.7. Suppose that you want to find the values of t that bound the middle 95% of the area in the distribu on for N = 9. In this situa on, df for t = N – 1 = 8. A table of cri cal values for t appears in Appendix B at the end of this book. An excerpt from this table appears in Figure 7.8. To read this table, you need to do the following:

1. Decide what percentage of the area will you use to define “close to” the mean. In most situa ons, this area is 95%. Find the column for the value 95% (just under the heading “Confidence Intervals (%)”).

2. Find the df value for your sample. Locate the row that has your df value, such as df = 8. 3. Locate the cell in that row and column. In this example, in Figure 7.8, the value in the cell

for 95% and 8 df is 2.306. When you are learning, it is useful to draw a diagram that shows how values of t are related to areas. Figure 7.9 shows the middle 95% of area for a t distribu on with df of 8. Values of t are given for larger numbers of different df values in more detailed tables (for example, see h p://davidmlane.com/hyperstat/t-table.html). The term df = ∞ corresponds to situa ons in which df is very large (in prac ce, when df > 120, the t distribu on becomes very close to the standard normal distribu on). As df increases, the cri cal values of t decrease; by the me df > 120, cri cal values of t converge to the standard normal distribu on. For df > 120, 2.5% of the distribu on lies below –1.96, the middle 95% lies between –1.96 and +1.96, and 2.5% of the distribu on lies above +1.96. When N is large, the amount of addi onal sampling error created by using SD to es mate σ becomes negligible.

7.13 USING SAMPLING ERROR TO SET UP A CONFIDENCE INTERVAL The following pieces of informa on are needed to set up a confidence interval (CI): C, an arbitrarily selected level of confidence. The value of C is usually 95%; it corresponds to the percentage of area we use for the middle of the distribu on when we look up cutoff values for t. C can be other values, such as 90% or 99%. The values of M, SD, and N from the sample data. We need to do the following to find the lower and upper limits of a confidence interval on the basis of a sample mean M: Find df; df = N – 1. Look up the (absolute) cri cal value from a t distribu on that corresponds to the middle C% of the t distribu on with N – 1 df. The cri cal value of t can be obtained from the table in Appendix B at the end of this book and is some mes denoted tcri cal C%. Calculate SEM (using SD and N from the sample). Find the lower limit of the 95% CI: Other

 Choose a lower level of confidence (such as 90% or 68% instead of 95%). However,

researchers are reluctant to make the level of confidence too low; 95% confidence is the most widely used value.

 Increase the sample size N.  Decrease SD. (Chapter 12, on the independent-samples t test, describes things

researchers can do in some situa ons that may decrease SD. However, in many situa ons, researchers have li le control over SD.)

7.14 HOW TO INTERPRET A CONFIDENCE INTERVAL The language used to interpret CIs is tricky. It is incorrect to say that a 95% CI computed using data from a single sample has a 95% chance of including μ. (It either does or it doesn’t, and we have no way to be certain which situa on we have for an individual sample.) It is more accurate to think about a CI as a statement about expected outcomes in the long run, across hundreds or thousands of different samples from the same popula on. For a 95% CI, approximately 95% of the CIs that are set up using the procedures described in this chapter are

expected to include the true popula on mean μ between the lower and the upper limits. Approximately 5% of these CIs will not contain μ. Cumming and Finch (2005) suggested this as a way to think about CIs: “a range of plausible values for μ; values outside the CI are rela vely implausible … [the] data are compa ble with any value of μ within the CI but rela vely incompa ble with any value outside it.” A problem with CIs is that, like M and SD, they vary across samples. Here is a thought experiment that illustrates the problem. If you randomly select 18 samples (each of size 25) from the same popula on, the values of M and SD will vary across these samples. That implies that the upper and lower boundaries of the 95% CIs will also vary across samples, as in the hypothe cal example in Figure 7.10. Each ver cal line with whiskers represents the lower and upper bounds of the CI for 1 of the 18 samples. The circle in the middle of each CI represents the mean for that sample; the circle is filled if the CI for that sample includes the true value of μ and open if the CI for that sample does not include the true value of μ. The true value of μ for the popula on corresponds to the horizontal line. In this example, 16 of the 18 CIs included μ, while the other 2 CIs did not include μ. If we had CIs for thousands of samples, 95% of them would be expected to include μ; the other 5% would not include μ. The 95% confidence level is a predic on about how many CIs out of thousands would include μ.

Figure 7.10 Hypothe cal Outcomes for 18 Confidence Intervals 7.15 EMPIRICAL EXAMPLE: CONFIDENCE INTERVAL FOR BODY TEMPERATURE Most of us assume that normal or average healthy adult body temperature is 98.6°F. In 1868, Wunderlich (cited in Mackowiak, Wasserman, & Levine, 1992) summarized data from over 1 million temperature measurements for 25,000 pa ents; he concluded that the “normal healthy” body temperature was 98.6°F or 37°C. Un l fairly recently, that value has not been ques oned; few studies of normal body temperature have been done. Mackowiak et al. (1992) believed that it would be useful to examine new data because instrumenta on for taking body temperature has changed since the 19th century. Shoemaker (1996) created an ar ficial data set in which the score values led to conclusions like those of Mackowiak et al. Data adapted from Shoemaker are used for the analyses in this sec on. It might seem that finding average body temperature for human popula ons would be easy, but it’s a more complex ques on than it appears. For readable discussions, see Cook (2018) and Maril (2018). A more recent study of temperature data collected through smart phone crowdsourcing is reported by Hausman et al. (2018). Values of N, M, and SD for the Fahrenheit temperature scores in the file shoemaker.sav were obtained using the SPSS frequencies procedure (menu selec ons are not repeated from earlier chapters). Results appear in Figure 7.11. The first thing to no ce is that the sample mean in Figure 7.11, M = 98.25, is lower than the popula on mean that people generally believe (98.6). The difference is (98.25 – 98.6) = –.35. This sample mean is about a third of a degree lower than the generally accepted value. (Note that if you look up Shoemaker’s ar cle, numerical values differ from the ones I reported; I modified his data slightly.) Is this difference or inconsistency small enough that we can dismiss it, or large enough that we should pay a en on to it? Further informa on is needed. To evaluate whether the sample mean is consistent with an es mate of μ = 98.6, we will set up a 95% CI and ask whether 98.6 is included in that CI (or not). No ce that the standard error of the mean (SEM) reported by SPSS is .0667. You can confirm this by hand: SEM = SD/ = .0667. To obtain the limits of the 95% CI, a new SPSS procedure is introduced (the one-sample t test). This procedure will be used more extensively in Chapter 8. Make the following menu selec ons: <Analyze> → <Compare Means> → <One-Sample T Test>. When the dialog box for the one- sample t test appears, as in Figure 7.12, move the name of the variable of interest into the list of variables to be analyzed. Leave the box “Test Value” containing the default value of 0. Then click OK.

Figure 7.11 Descrip ve Sta s cs for Temperature in Fahrenheit in shoemaker.sav The output appears in Figure 7.13. The area enclosed in the ellipse in Figure 7.13 shows the lower and upper limits for the 95% CI for mean temperature in degrees Fahrenheit. Note that this CI does not include the value that most people think of as average body temperature (98.6). The Shoemaker temperature data suggest that the true popula on mean for body temperature may be lower than the conven onally assumed value of 98.6. We can create graphs to display confidence intervals. When CIs are graphed, they are called error bars. However, note that lines that are called “error bars” in published graphs do not always represent confidence intervals; some mes error bars correspond to SD or SEM. Titles for the graphs should make it clear what the error bars represent. From the main SPSS menu, select <Graphs> → <Legacy Dialogs> → <Error Bar>. The Error Bar dialog box appears in Figure 7.14. Choose “Simple” and “Summaries of separate variables,” then click Define to open the next dialog box, in Figure 7.15. Enter the name of the variable for which you want error bars. There is a pull-down menu, “Bars Represent,” ini ally set to “Confidence interval for mean,” that allows you to specify whether you want bars to represent the CI (or SD or SEM); leave this at the default selec on for CI. The output appears in Figure 7.16.

On the basis of this sample of N = 130 temperature measurements, with M = 98.254 and SD = .7603, the 95% CI for body temperature in degrees Fahrenheit was [98.12, 98.39]. The value that people usually assume for popula on mean body temperature, 98.6, does not fall within this range of plausible values for μ. The results of this (hypothe cal) study are inconsistent with the claim that μ = 98.6. However, they do not conclusively disprove that μ = 98.6 The low value of M in this sample might have occurred because of sampling error. Informa on from addi onal studies is needed to evaluate whether the true popula on mean for body temperature is lower than 98.6. We should always look for replica ons using large and representa ve samples before we draw conclusions; data from one small study do not prove that μ = 98.6 is incorrect. However, results reported by Mackowiak et al. (1992) were somewhat inconsistent with that belief. The assump on that mean body temperature in a normal healthy adult popula on is 98.6°F deserves a second look; that value was based on a kind of instrumenta on to measure body temperature that is no longer used.

Figure 7.16 Graph of 95% CI for Temperature Data (Degrees Fahrenheit) 7.16 OTHER APPLICATIONS FOR CONFIDENCE INTERVALS 7.16.1 CIs Can Be Obtained for Other Sample Sta s cs (Such as Propor ons) The sample mean, M, is not the only sta s c that has a sampling distribu on and a known standard error. The sampling distribu ons for many other sta s cs are known; thus, it is possible to iden fy an appropriate sampling distribu on and to es mate the standard error and set up CIs for many other sample sta s cs, such as Pearson’s r. Poli cal polls (and some mes opinion polls) o en report sta s cs such as percentages, and it is possible to set up CIs for percentage es mates. 7.16.2 Margin of Error in Poli cal Polls In many poli cal and opinion polls, respondents are asked to state which among two or more alterna ves they prefer (for example, pass or reject Proposi on 13, which calls for legaliza on of recrea onal use of marijuana; inten on to vote for Candidate A, B, or C). In these situa ons, the sample sta s c of interest is a percentage (e.g., the propor on of respondents who say that they intend to vote for A, B, or C or who say they don’t know). It is possible to set up a 95% CI

for a sample percentage taking N in the sample into account. However, a margin of error reported for polling results usually corresponds to a 68% CI. As N increases, margin of error decreases. The lower and upper limits of a 68% CI for a sample percentage are

It is possible for margin of error to be related to a different level of confidence. Unfortunately, the defini on of margin of error is o en not stated specifically in media reports. O en the margin of error reported in media corresponds to a 68% confidence interval. As an example, suppose that 54% of those polled say that they plan to vote for Candidate A, with a margin of error of ±2%. This implies that the 68% CI ranges from 52% to 56% in favor of Candidate A. Plausible es mates of the popula on propor on lie within this range. If more than 50% of the vote is required to win the elec on, a CI from 52% to 56% indicates that it is plausible (but not certain) that Candidate A will win. Consider a different scenario, in which the propor on of people who plan to vote for B is 33 ± 2%, and the propor on of those who plan to vote for C is 35 ± 3%. This translates into CIs of 31% to 35% (for Candidate B) and 32% to 38% (for Candidate C). Candidate C may be ahead of candidate B by a small amount, but that small difference could easily be due to sampling error. 7.17 ERROR BARS IN GRAPHS OF GROUP MEANS Error bars can be superimposed on bar graphs in which the heights of bars correspond to group means. Consider the hypothe cal example in Figure 7.17. Students are asked to rate their degree of agreement with the statement “I feel guilty when I eat foods I know are unhealthy” on a five-point scale (1 = strongly disagree to 5 = strongly agree). Mean scores are calculated for male and female students. For male students, N = 4, M = 3.25, SD = .957, and SEM = .479; for female students, N = 3, M = 4.33, SD = .577, and SEM = .333. (For each group, you should be able to calculate SEM from N and SD.) Look at the correspondence between the descrip ve sta s cs and the graph. The height of each bar corresponds to the mean ra ng for a group. The end points of the 95% CI error bars can be found by mul plying tcri cal 95% (using a t distribu on with df = 2 for the female and df = 3 for the male group) by the value of SEM and iden fying this distance below and above the group mean. This type of bar graph is very common in research reports in which group means are compared. Keep in mind that the error bars on this type of graph can represent a 95% CI but might represent SD or SEM; look for that informa on in the figure tle or note.

By now you should be able to understand the nature of the differences between female and male students either by comparing values of M or by examining the bar graph. Which group had higher mean guilt about unhealthy foods?

7.18 SUMMARY This chapter emphasizes the importance of repor ng and interpre ng CIs. Recommenda ons made by Wilkinson and Task Force on Sta s cal Inference, APA Board of Scien fic Affairs (1999), and in the Publica on Manual of the American Psychological Associa on (American Psychological Associa on, 2009) state that CI informa on should be provided for major outcomes whenever possible. SPSS provides CI informa on for many, but not all, outcome sta s cs of interest. For some sample sta s cs and effect sizes, researchers may need to calculate CIs by hand (Kline, 2013). The editors of some journals now require that CIs be reported. What you need to remember from this chapter:

 Sampling error always occurs, even when researchers do not make mistakes.  The central limit theorem explains how sampling error affects values of M across many

samples.  On the basis of the CLT, we can calculate SEM from SD and N. SEM provides informa on

about typical magnitude of predic on error when we try to predict μ from M.  Usually the only informa on a researcher has is N, M, and SD from one sample.

 You can calculate df from N and SEM from SD and N. (In this chapter, df = N – 1, and SEM = SD/

 N  ).  Given df and level of confidence C, which is o en 95%, you can locate values of t that

correspond to the middle C% of the t distribu on with (N – 1) df.  Given values of t, M, and SEM, you can find the lower and upper limits of the C%

confidence interval, using Equa ons 7.10 and 7.11.  Editors and reviewers increasingly expect CI informa on for all important results.  O en lower and upper CI limits can be obtained using SPSS. For some sta s cs, you

need to calculate the CI limits by hand (as explained in later chapters).  A CI helps you understand how far your value of M may be from μ.  In Chapter 8, the informa on used to compute a CI is used in a different way to answer

yes/no ques ons about possible values of μ, using sta s cal significance tests. COMPREHENSION QUESTIONS

1. What is a confidence interval (CI)? 2. What informa on from a sample is required to set up a CI? 3. What is a sampling distribu on? What do we know about the shape and characteris cs

of the theore cal sampling distribu on for M (when σ is known)? 4. What is σM? 5. How does SEM differ from σM? 6. What is SEM? What does the value of SEM tell you about the typical magnitude of

sampling error? o As SD increases, how does the size of SEM change (assuming N stays the same)? o As N increases, how does the size of SEM change (assuming SD stays the same)?

7. How is a t distribu on like a standard normal distribu on? How is it different? 8. Under what circumstances should a t distribu on be used rather than the standard

normal distribu on to look up areas or probabili es associated with distances from the mean?

9. Consider the following ques ons about CIs: A researcher tests emo onal intelligence (EI) for a random sample of children selected from a popula on of all students who are enrolled in a school for gi ed children. The researcher wants to es mate the mean EI for the en re school. Let’s suppose that a researcher wants to set up a 95% CI for IQ scores using the following informa on:

The sample mean M = 130. The sample standard devia on SD = 15. The sample size N = 120. df = N – 1 = 119.

For the values given above, the limits of the 95% CI are as follows: Lower limit = 130 – 1.96 × 1.37 = 127.31. Upper limit = 130 + 1.96 × 1.37 = 132.69. The following exercises ask you to experiment to see how changing some of the values involved in compu ng the CI influences the width of the CI. Recalculate the CI for the emo onal IQ informa on in the preceding ques on to see how the lower and upper limits (and the width of the CI) change as you vary the N in the sample (and leave all the other values the same).

a) What are the upper and lower limits of the CI and the width of the 95% CI if all the other values remain the same (M = 130, SD = 15), but you change the value of N to 16? Note that when you change N, you need to change two things: the computed value of SEM and the degrees of freedom used to look up the cri cal values for t.

b) What are the upper and lower limits of the CI and the width of the 95% CI if all the other values remain the same, but you change the value of N to 25?

c) What are the upper and lower limits of the CI and the width of the 95% CI if all the other values remain the same (M = 130, SD = 15), but you change the value of N to 49?

d) On the basis of the numbers you reported for sample size N of 16, 25, and 49, how does the width of the CI change as N (the number of cases in the sample) increases?

e) What are the upper and lower limits and the width of this CI if you change the confidence level to 80% (and con nue to use M = 130, SD = 15, and N = 49)?

f) What are the upper and lower limits and the width of the CI if you change the confidence level to 99% (con nue to use M = 130, SD = 15, and N = 49)?

g) How does changing the level of confidence from 80% to 99% affect the width of the CI?

DIGITAL RESOURCES Find free study tools to support your learning, including eFlashcards, data sets, and web resources, on the accompanying website at edge.sagepub.com/warner3e. Descrip ons of Images and Figures Back to Figure The image shows a combina on of diagrams; a histogram as well as a sampling distribu on where the values have been drawn across many samples.

1. In the histogram, the M is equals to 73.8 and the SD is equal to 7.06. The distribu on has 130 individual scores.

The X axis denotes the HR for each person and ranges from 50 to 90, with the mean at 73.8. The Y axis denotes the frequency and ranges from 0 to 20. The SD has been depicted on the graph itself and a statement below the graph says that the varia on in individual scores is indicated by the SD. There are several bars on the graph corresponding to the scores.

2. In the second diagram, the sampling distribu on of M, a histogram of values of M has been drawn from 500 samples, where each sample has an N of 25 cases.

The mu or mean of the set of samples is equal to 73.8, and the sigma or standard devia on or varia on of the set of sample means is 1.38. The X axis ranges from 50 to 90 and the Y axis ranges from 0 to 40. The histogram is shaped like a normal distribu on. Back to Figure The X axis denotes the heart rates and ranges from 0 to 90. The Y axis denotes the frequency and ranges from 0 to 20. There are 17 bars that shows the scores. This popula on is not normally distributed. Back to Figure The image is a set of three histograms that show the result from different Monte Carlo simula ons. The sampling size varies; N is 9, 25 and 64. Details are below:

 First simula on N equals 9 in each of the many samples. N in the simula on is 500. Predicted mean using CLT equals 73.76. Mean from simula on equals 73.82. Predicted sigma subscript M from CLT equals 7.062 divided by 3 equals 2.354. Value of sigma subscript M from simula on equals 2.27. The X axis ranges from 0 to 90, rising in increments of 10. The Y axis, represen ng frequency, ranges from 0 to 60, rising in increments of 20. The histogram resembles a normal distribu on.

 Second simula on N equals 25 in each of the many samples. N in the simula on is 500. Predicted mean from CLT equals 73.76. Mean from simula on equals 73.85.

Predicted sigma subscript M from CLT equals 7.062 divided by 5 equals 1.41. Value of sigma subscript M from simula on equals 1.38. The X axis ranges from 0 to 90, rising in increments of 10. The Y axis, represen ng frequency, ranges from 0 to 60, rising in increments of 20. The histogram resembles a normal distribu on.

 Third simula on N equals 64 in each of the many samples. N in the simula on is 500. Predicted mean from CLT equals 73.76. Mean from simula on equals 73.76. Predicted sigma subscript M from CLT equals 7.062 divided by 8 equals .88. Value of sigma subscript M from simula on equals.85. The X axis ranges from 0 to 90, rising in increments of 10. The Y axis, represen ng frequency, ranges from 0 to 60, rising in increments of 20. The histogram resembles a normal distribu on. As the N rises in each simula on, the histogram narrows, and the value of sigma subscript M drops. Back to Figure The image is a table that shows both the popula on and sample descrip ve sta s cs for heart rate data. Popula on;

 N – Valid: 130  N – Missing: 0  Mean or Mu: 73.76  Std Devia on or sigma: 7.062

Sample;

 N – Valid: 9  N – Missing: 0  Mean or Mu: 76.22

Back to Figure The image is a diagram of the normal distribu on that shows the loca on of z equals 1 on the curve.

The X axis denotes the z score and ranges from minus 3 to plus 3, with 0 as the center. The mean is 0. Plus 1 on the X axis has been circled. The area on either side of the mean, that is, between 1 and 0 on the right as well as between minus 1 and 0 on the le , is equal to 34.13 percent. The area between plus 1 and plus 2 on the right and minus 1 and minus 2 on the le is equal to 13.59 percent each. The area between plus 2 and plus 3 on the right and minus 2 and minus 3 on the le is equal to 2.14 percent each. The area beyond plus 3 and minus 3 on either side is 13 percent. At the top of the figure, three lines show the area under the curve. The area under minus 1 to plus 1 is 68 percent. The area under minus 2 to plus 2 is 95 percent. The area under minus 3 and plus 3 is around 99 percent. Back to Figure The image shows different examples of a t distribu on where the curves are fla er or higher based on differences in df and how the resultant range of t values will change accordingly. The first df equals infinity and this curve resembles a normal distribu on. The t values extend from plus and minus 1.96 on either side of the mean. The second df equals 6. This curve is much fla er in the middle, and has thicker tails, compared with the normal distribu on. The t values extend from plus and minus 2.45 on either side of the mean. The last df equals 3. This is the fla est curve and its tail is much thicker than the others. The t values extend from plus and minus 3.18 on either side of the mean. The image has been sourced from a website with the following link: h p://www.psychstat.missouristate.edu/introbook/sbk24.htm. Back to Figure The image is a diagram of the normal distribu on that shows the lower tail, middle 95 percent, and upper tail. The X axis has 0 as the center. Minus 1.96 and plus 1.96 have also been marked. The area between plus 1.96 on the right and minus 1.96 on the le is equal to 95 percent.

The area beyond plus 1.96 and minus 1.96 on either side is 2.5 percent each. A statement below the diagram men ons that Most extreme 5 percent is sum of areas in lower and upper tails beyond z equals 1.96. Back to Figure The image is an extract from the cri cal values for T distribu on and has been adapted from the table by Fisher and Yates. The table lists different confidence intervals, and levels of significance for one tailed and two tailed tests. It also shows the df ranges that result in the cri cal values. Details are below:

The eighth row of the df and the 95 percent confidence interval column have been circled. Back to Figure The image is a diagram of a t distribu on that shows the percentage of area under the curve. The area between plus 2.034 on the right and minus 2.034 on the le is equal to the middle 95 percent. The area beyond plus 2.034 is the upper 2.5 percent and beyond minus 2.034 is lower 2.5 percent. Back to Figure The image shows a hypothe cal outcome for 18 confidence intervals. The X axis is the ver cal axis with its center as mu, and the outcomes are indicated by circles in the center with ver cal lines on either side. Each ver cal line represents the lower and upper bounds of the confidence intervals for each of the 18 samples.

Most of the circles and lines for the confidence intervals included the mu, except for two samples, which have been circled. While one lies above the mu, the other falls below the mu. The image has been adapted from Cumming and Finch. Back to Figure The image is a table that shows the following descrip ve sta s cs data: Sta s cs Temp underscore Fahrenheit

 N – Valid: 130  N – Missing: 0  Mean: 98.254  Std error of mean: .0667  Std Devia on: .7603

Back to Figure The image is a screenshot of the procedure to use SPSS One-Sample t Test. At the top of the spreadsheet are the following menu bu ons; analyze, graphs, u li es, extensions, window and help. Below these bu ons are icon bu ons for table edi ng op ons. On the clicking of the Analyze bu on, a drop-down menu with the following op ons has opened; reports, descrip ve sta s cs, Bayesian sta s cs, tables, compare means, general linear model, generalized linear models, mixed models, correlate, regression, loglinear, classify, dimension reduc on, scale, non-parametric tests, forecas ng, survival, mul ple response, simula on, quality control, ROC curve, and Spa al and temporal modelling. The compare means menu has been clicked and the following menu op ons are visible; means, one sample T test, independent samples T test, summary independent samples T test, Paired samples T test, and one-way ANOVA. The one sample T test dialog box is also open. This has a set of variables on the le , sex, hr, temp underscore Celsius, zscore open bracket temp underscore Fahrenheit close bracket and zscore open bracket temp underscore Celsius close bracket. There is an op on to move the required variable to the box on the right for test variables. Temp underscore Fahrenheit is in this box.

At the right is a bu on to control Op ons. The test value can be changed. At present, it has been set to 0. At the bo om of the dialog box are op ons bu ons for the following; OK, Paste, Reset, Cancel and Help. Back to Figure The image is the output for the one sample T test procedure. The sta s cs are in one table and the output is in another table. Both of them have been provided below:

 One sample sta s cs

Temp underscore Fahrenheit N: 130 Mean: 98.254 Std Devia on: .7603 Std Error Mean: .0667 One sample test Test value equals 0 Temp underscore Fahrenheit

o T: 1473.387 o Df: 129 o Sig – 2 tailed: .000 o Mean difference: 98.2542 o 95 percent confidence interval of the difference

o Lower: 98.122 o Upper: 98.386

The 95 percent confidence interval of the difference has been circled. Back to Figure The image is a dialog box for the error bar graph that allows for choosing the type of graph as well as how the data is summarized. The dialog box has two chart types; simple and clustered. The Simple op on has been chosen. The data in the chart can be arranged in two ways; summaries for groups of cases and summaries of separate variables. The second op on has been chosen.

At the bo om of the dialog box, there are three radio bu ons; define, cancel and help. The Define bu on has been selected. Back to Figure The image is a second dialog box to define error bar and shows how to represent the error bars in the graph. On the le are a set of variables, which can be chosen and moved to the box on the right. The variables available are sex, hr, temp underscore Celsius, zscore open bracket temp underscore Fahrenheit close bracket, zscore pen bracket temp underscore Celsius close bracket. The variable temp underscore Fahrenheit has been moved to the error bar variable box. There are two radio bu ons on the side; tles and op ons. There is a drop-down menu that allows a choice of what the bars represent. Here the bars represent confidence interval for mean. The level can also be chosen and 95 percent is the level currently. Back to Figure The image is an output graph of the 95 percent confidence interval for temperature data. In the image, which is the 95 percent confidence interval for temperature in Fahrenheit, the temperature is the Y axis and ranges from 98.1 to 98.4. The mean has been indicated as 98.25, and the upper and lower level of the line have also been shown. The upper limit is close to 98.38 and the lower limit is around 98.12. Back to Figure The image shows two bars in a graph that represent group means with 95 percent CI error bars. The bars represent student agreement with the statement – I feel guilty when I eat foods I know are unhealthy. The X axis denotes whether the bar is male or female. The Y axis denotes a range of guilt levels from 0 to 6. The height of the male bar reached 3.2 while the female bar height reached 4.3. The error bars upper value was 5.7 for females and 4.8 for males. The lower value was 2.8 and 1.7 recep vely for men and women.