Week 2 -- PPOL-505 Exercise 1

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2 COMPUTING AND UNDERSTANDING AVERAGES MEANS TO AN END

2: MEDIA LIBRARY

Premium Videos

Core Concepts in Stats Video

· Measures of Central Tendency

Lightboard Lecture Video

· Choosing a Measure of Central Tendency

Time to Practice Video

· Chapter 2: Problem 6

Difficulty Scale

(moderately easy)

WHAT YOU WILL LEARN IN THIS CHAPTER

· Understanding measures of central tendency

· Computing the mean for a sample of scores

· Computing the median for a sample of scores

· Computing the mode for a sample of scores

· Understanding and applying scales or levels of measurement

· Selecting a measure of central tendency

You’ve been very patient, and now it’s finally time to get started working with some real, live data. That’s exactly what you’ll do in this chapter. Once data are collected, a usual first step is to begin to understand all those bits of information using single numbers to describe the data. The easiest way to do this is through computing an average, of which there are several different types.

An average is the one value that best represents an entire group of scores. It doesn’t matter whether the group of scores represents the number correct on a spelling test for 30 fifth graders or the typical batting percentage for all the baseball players on the New York Yankees or how voters feel about a congressional candidate. In all of these examples, a big group of data can be summarized using an average. You can usually think of an average as the “middle” space or as a fulcrum on a seesaw. It’s the point in a range of values that seems to most fairly represent all the values.

Averages, also called measures of central tendency , come in three flavors: the mean, the median, and the mode. Each provides you with a different type of information about a distribution of scores and is simple to compute and interpret.

LIGHTBOARD LECTURE VIDEO

Choosing a Measure of Central Tendency

COMPUTING THE MEAN

The mean is the most common type of average that is computed. It is so popular that scientists sometimes sloppily treat the word average as if it means mean when it only sometimes means mean. The mean is simply the sum of all the values in a group, divided by the number of values in that group. So, if you had the spelling scores for 30 fifth graders, you would simply add up all the scores to get a total and then divide by the number of students, which is 30.

We are about to show a formula or equation for the first time in this book. Don’t panic. Equations are just statements or sentences that use symbols instead of words. We will always tell you what words the symbols stand for. The formula for computing the mean is shown in Formula 2.1:

(2.1)

¯¯¯X=∑Xn,X¯=∑Xn,

where

· the letter X with a line above it (also sometimes called “X bar”) is the mean value of the group of scores;

· the ∑, or the Greek letter sigma, is the summation sign, which tells you to “sum up” or add together whatever follows it;

· the X is each individual score in a group of scores; and

· the n is the size of the sample from which you are computing the mean, the number of scores.

To compute the mean, follow these steps:

1. List the entire set of values in one or more columns. These are all the Xs.

2. Compute the sum or total of all the values.

3. Divide the total or sum by the number of values.

compute a mean for that value.

Location	Number of Shoppers Last Year
Lanham Park Store	2,150
Williamsburg Store	1,534
Downtown Store	3,564

The mean or average number of shoppers in each store is 2,416. Formula 2.2 shows how this average was computed using the formula you saw in Formula 2.1:

(2.2)

¯¯¯X=∑Xn=2,150+1,534+3,5643=7,2483=2,416.X¯=∑Xn=2,150+1,534+3,5643=7,2483=2,416.

Or, if you needed to compute the average number of students in each grade in a school building, you would follow the same procedure.

Grade	Number of Students
Kindergarten	18
1	21
2	24
3	23
4	22
5	24
6	25

The mean or average number of students in each class is 22.43. Formula 2.3 shows how this average was computed using the formula you saw in Formula 2.1:

(2.3)

¯¯¯X=∑Xn=18+21+24+23+22+24+257=1577=22.43.X¯=∑Xn=18+21+24+23+22+24+257=1577=22.43.

See, we told you it was easy. No big deal. By the way, when you calculated that mean just now, you may have gotten a number with lots more digits in it: 22.42857143 or something like that. Statisticians are usually okay with you shortening numbers to just a couple digits past the decimal. So, we felt fine reporting the mean as 22.43 (rounding up for that last digit).

· The mean is sometimes represented by the letter M and is also called the typical, average, or most central score. If you are reading another statistics book or a research report and you see something like M = 45.87, it probably means that the mean is equal to 45.87. Technically, that capital letter M is used when you are talking about the mean of the larger population represented by the sample in front of you. Those sorts of distinctions aren’t important right now but might be interesting later on.

· In the formula, a small n represents the sample size for which the mean is being computed. A large N (← like this) would represent the population size. In some books and in some journal articles, no distinction is made between the two. Notice, as with the capital M when talking about the mean of a population, statistical types often capitalize a letter symbol to refer to a population and keep the letter as lowercase when talking about samples.

· The mean is like the fulcrum on a seesaw. It’s the centermost point where all the values on one side of the mean are equal in weight to all the values on the other side of the mean.

· Finally, for better or worse, the mean is very sensitive to extreme scores. An extreme score can pull the mean in one or the other direction and make it less representative of the set of scores and less useful as a measure of central tendency. This, of course, all depends on the values for which the mean is being computed. And, if you have extreme scores and the mean won’t work as well as you want, we have a solution! More about that later.

The mean is also referred to as the arithmetic mean , and there are other types of means that you may read about, such as the harmonic mean. Those are used in special circumstances and need not concern you here. And if you want to be technical about it, the arithmetic mean (which is the one that we have discussed up to now) is also defined as the point about which the sum of the deviations is equal to zero (whew!). Each score in a sample is some distance from the mean. If you add up all those distances, it will equal zero. Always. Every time. That’s why we like the mean. For instance, if you have scores like 3, 4, and 5 (whose mean is 4), the sum of the deviations about the mean (−1, 0, and +1) is 0.

Remember that the word average means only the one measure that best represents a set of scores and that there are many different types of averages. Which type of average you use depends on the question that you are asking and the type of data that you are trying to summarize. This is a levels of measurement issue that we will cover later in this chapter when we talk about when to use which statistic.

In basic statistics, an important distinction is made between those values associated with samples (a part of a population) and those associated with populations. To do this, statisticians use the following conventions. For a sample statistic (such as the mean of a sample), Roman letters are used. For a population parameter (such as the mean of a population), Greek letters are used. So, for example, the mean for the spelling score for a sample of 100 fifth graders is represented as ¯¯¯X=5,X¯=5, whereas the mean for the spelling score for the entire population of fifth graders is represented, using the Greek letter mu, as µ5.

COMPUTING THE MEDIAN

The median is also an average, but of a very different kind. The median is defined as the midpoint in a set of scores. It’s the point at which one half, or 50%, of the scores fall above and one half, or 50%, fall below. It’s got some special qualities that we will talk about later in this section, but for now, let’s concentrate on how it is computed. There’s not really a formula for computing the median but instead a set of steps.

To compute the median, follow these steps:

1. List the values in order, from either highest to lowest or lowest to highest.

2. Find the middle-most score. That’s the median.

For example, here are the annual incomes from five different households:

· $135,456

· $45,500

· $62,456

· $54,365

· $37,668

Here is the list ordered from highest to lowest:

· $135,456

· $62,456

· $54,365

· $45,500

· $37,668

There are five values. The middle-most value is $54,365, and that’s the median.

Now, what if the number of values is even? An even number of scores means there is no middle value. Let’s add a value ($64,500) to the list so there are six income levels. Here they are sorted with the largest value first:

· $135,456

· $64,500

· $62,456

· $54,365

· $45,500

· $37,668

When there is an even number of values, the median is simply the mean of the two middle values. In this case, the middle two cases are $54,365 and $62,456. The mean of those two values is $58,410.50. That’s the median for that set of six values.

What if the two middle-most values are the same, such as in the following set of data?

· $45,678

· $25,567

· $13,234

Then the median is same as both of those middle-most values. In this example, it’s $25,567.

If we had a series of values that was the number of days spent in rehabilitation for a sports-related injury for seven different patients, the numbers might look like this:

· 43

· 34

· 32

· 12

· 51

· 6

· 27

As we did before, we can order the values (51, 43, 34, 32, 27, 12, 6) and then select the middle value as the median, which in this case is 32. So, the median number of days spent rehabilitating an injury is 32.

If you know about medians, you should also know about percentile ranks . Percentile ranks are used to define the percentage of cases equal to or below a certain point in a distribution or set of scores. For example, if a score is “at the 75th percentile,” it means that the score is at or above 75% of the other scores in the distribution. The median is also known as the 50th percentile, because it’s the point at or below which 50% of the cases in the distribution fall. Other percentiles are useful as well, such as the 25th percentile, often called Q1, and the 75th percentile, referred to as Q3. So what’s Q2? The median, of course.

Here comes the answer to the question you’ve probably had in the back of your mind since we started talking about the median. Why use the median instead of the mean? For one very good reason. The median is insensitive to extreme scores, whereas the mean is not. When you have a set of scores in which one or more scores are extreme, the median better represents the centermost value of that set of scores than any other measure of central tendency. Yes, even better than the mean.

What do we mean by extreme? It’s probably easiest to think of an extreme score as one that is very different from the group to which it belongs. For example, consider the list of five incomes that we worked with earlier (shown again here):

· $135,456

· $54,365

· $37,668

· $32,456

· $25,500

The value $135,456 is more different from the other four than is any other value in the set. We would consider that an extreme score.

The best way to illustrate how useful the median is as a measure of central tendency is to compute both the mean and the median for a set of data that contains one or more extreme scores and then compare them to see which one best represents the group. Here goes.

The mean of the set of five scores you see above is the sum of the set of five divided by 5, which turns out to be $57,089. On the other hand, the median for this set of five scores is $37,668. Which is more representative of the group? The value $37,668, because it clearly lies closer to most of the scores, and we like to think about “the average” (in this case, we are using the median as a measure of average) as being representative or assuming a central position. In fact, the mean value of $57,089 falls above the fourth highest value ($54,365) and is not very central or representative of the distribution.

It’s for this reason that certain social and economic indicators (often involving income) are reported using a median as a measure of central tendency—“The median income of the average American family is …”—rather than using the mean to summarize the values. There are just too many extreme scores that would skew, or significantly distort, what is actually a central point in the set or distribution of scores. As an example, the median annual family income in the United States for 2014 (the most recent year we could find data for) was about $54,000, while the mean annual family income was about $73,000. Which is closer to your family’s income?

You learned earlier that sometimes the mean is represented by the capital letter M instead of ¯¯¯XX¯. Well, other symbols are used for the median as well. We like the letter M, but some people confuse it with the mean, so they use Med or Mdn for median. Don’t let that throw you—just remember what the median is and what it represents, and you’ll have no trouble adapting to different symbols.

Here are some interesting and important things to remember about the median:

· We use the word median to describe other middle things, like the median on a highway, that stripe down the middle of a road.

· Because the median is based on how many cases there are, and not the values of those cases, extreme scores (sometimes called outliers ) only count a little.

COMPUTING THE MODE

The third and last measure of central tendency that we’ll cover, the mode, is the most general and least precise measure of central tendency, but it plays a very important part in understanding the characteristics of a sample of scores. The mode is the value that occurs most frequently. Like the median, there is no formula for computing the mode.

To compute the mode, follow these steps:

1. List all the values in a distribution but list each value only once.

2. Tally the number of times that each value occurs.

3. The value that occurs most often is the mode.

For example, an examination of the political party affiliation of 300 people might result in the following distribution of scores:

Party Affiliation	Number or Frequency
Democrats	90
Republicans	70
Independents	140

The mode is the value that occurs most frequently, which in the preceding example is Independents. That’s the mode for this distribution.

If we were looking at the modal response on a 100-item multiple-choice test, we might find that Alternative A was chosen more frequently than any other. The data might look like this:

Item Alternative Selected	A	B	C	D
Number of times	57	20	12	11

On this 100-item multiple-choice test where each item has four choices (A, B, C, and D), A was the answer selected 57 times. It’s the modal response.

Want to know the easiest and most commonly made mistake made when computing the mode? It’s selecting the number of times a category occurs rather than the label of the category itself. Instead of the mode being Independents (in our first example), it’s easy for someone to conclude the mode is 140. Why? Because he or she is looking at the number of times the value occurred, not the value that occurred most often! This is a simple mistake to make, so be on your toes when you are asked about these things.

Apple Pie à la Bimodal

If every value in a distribution contains the same number of occurrences, then there really isn’t a single mode. But if more than one value appears with equal frequency, the distribution is multimodal. The set of scores can be bimodal (with two modes), as the following set of data using hair color illustrates:

Hair Color	Number or Frequency
Red	7
Blond	12
Black	45
Brown	45

In the previous example, the distribution is bimodal because the frequency of the values of black and brown hair occurs equally. You can even have a bimodal distribution when the modes are relatively close together but not exactly the same, such as 45 people with black hair and 44 with brown hair. The question becomes, How much does one class of occurrences stand apart from another?

Can you have a trimodal distribution? Sure—where three values have the same frequency. It’s unlikely, especially when you are dealing with a large set of data points , or observations, but certainly possible. The real answer to the above stand-apart question is that categories have to be mutually exclusive—you simply cannot have both black and red hair (although if you look around the classroom, you may think differently). Of course, you can have those two colors, but each person’s hair color is forced into only one category.

CORE CONCEPTS IN STATS VIDEO

Measures of Central Tendency

WHEN TO USE WHAT MEASURE OF CENTRAL TENDENCY (AND ALL YOU NEED TO KNOW ABOUT SCALES OF MEASUREMENT FOR NOW)

Which measure of central tendency you use depends on certain characteristics of the data you are working with—specifically the scale of measurement at which those data occur. And that scale or level dictates the specific measure of central tendency you will use.

But let’s step back for just a minute and make sure that we have some vocabulary straight, beginning with the idea of what measurement is.

Measurement is the assignment of values to outcomes following a set of rules—simple. The results are the different scales we’ll define in a moment, and an outcome is anything we are interested in measuring, such as hair color, gender, test score, or height.

These scales of measurement, or rules, are the particular levels at which outcomes are observed. Each level has a particular set of characteristics, and scales of measurement come in four flavors (there are four types): nominal, ordinal, interval, and ratio.

Let’s move on to a brief discussion and examples of the four scales of measurement and then discuss how these levels of scales fit with the different measures of central tendency discussed earlier.

A Rose by Any Other Name: The Nominal Level of Measurement

The nominal level of measurement is defined by the characteristics of an outcome that fit into one and only one class or category. For example, gender can

be a nominal variable (female and male), as can ethnicity (Caucasian or African American), as can political affiliation (Republican, Democrat, or Independent). Nominal-level variables are “names” (nominal in Latin), and the nominal level is the least precise level of measurement. Nominal levels of measurement have categories that are mutually exclusive; for example, political affiliation cannot be both Republican and Democrat.

Any Order Is Fine With Me: The Ordinal Level of Measurement

The ord in ordinal level of measurement stands for order, and the characteristic of things being measured here is that they are ordered. The perfect example is a rank of candidates for a job. If we know that Russ is ranked 1, Marquis is ranked 2, and Hannah is ranked 3, then this is an ordinal arrangement. We have no idea how much higher on this scale Russ is relative to Marquis than Marquis is relative to Hannah. We just know that it’s “better” to be 1 than 2 than 3 but not by how much.

1 + 1 = 2: The Interval Level of Measurement

Now we’re getting somewhere. When we talk about the interval level of measurement , a test or an assessment tool is based on some underlying continuum such that we can talk about how much more a higher performance is than a lesser one. For example, if you get 10 words correct on a vocabulary test, that is 5 more than getting 5 words correct. A distinguishing characteristic of interval-level scales is that the intervals or spaces or points along the scale are equal to one another. Ten words correct is 2 more than 8 correct, and 6 is 2 more than 4 correct. And it’s not just the actual math that determines whether there are equal intervals; it is assumptions about the underlying concept being measured. If two more spelling words correct always represents two more units of spelling ability (whatever that means), then the measurement is at the interval level.

Can Anyone Have Nothing of Anything? The Ratio Level of Measurement

Well, here’s a little conundrum for you. An assessment tool at the ratio level of measurement is characterized by the presence of an absolute zero on the scale. What that zero means is the absence of any of the trait that is being measured. The conundrum? Are there outcomes we measure where it is possible to have nothing of what is being measured? In some disciplines, that can be the case. For example, in the physical and biological sciences, you can have the absence of a characteristic, such as absolute zero (no molecular movement) or zero light. In the social and behavioral sciences, it’s a bit harder. Even if you score zero on that spelling test or miss every item of an intelligence test (written in Martian), that does not mean that you have no spelling ability or no intelligence, right?

In Sum …

These scales of measurement, or rules, represent particular levels at which outcomes are measured. And, in sum, we can say the following:

· Any outcome can be assigned to one of four levels of measurement.

· Levels of measurement have an order, from the least precise being nominal to the most precise being ratio.

· The “higher up” the scale of measurement, the more precise the data being collected, and the more detailed and informative the data are. It may be enough to know that some people are rich and some poor (and that’s a nominal or categorical distinction), but it’s much better to know exactly how much money they make (ratio). We can always make the “rich” versus “poor” distinction if we want to once we have all the information.

· Finally, the more precise scales contain all the qualities of the scales below them; for example, the interval scale includes the characteristics of the ordinal and nominal scales. If you know that the Cubs’ batting average is .350, you know it is better than that of the Tigers (who hit .250) by 100 points, but you also know that the Cubs are better than the Tigers (but not by how much) and that the Cubs are different from the Tigers (but there’s no direction to the difference).

Okay, we’ve defined levels of measurement and discussed three different measures of central tendency and given you good examples of each. But the most important question remains unanswered. That is, “When do you use which measure?”

In general, which measure of central tendency you use depends on the type of data that you are describing, which in turn means at what level of measurement the data occur. Unquestionably, a measure of central tendency for qualitative, categorical, or nominal data (such as racial group, eye color, income bracket, voting preference, and neighborhood location) can be described using only the mode. It’s not meaningful to talk about the mean eye color in a classroom.

For example, you can’t be interested in the most central measure that describes which political affiliation is most predominant in a group and use the mean—what in the world could you conclude, that everyone is half Republican? Rather, saying that out of 300 people, almost half (140) are Independent seems to be the best way to describe the value of this variable. In general, the median and the mean are best used with quantitative data, such as height, income level in dollars (not categories), age, test score, reaction time, and number of hours completed toward a degree.

It’s also fair to say that the mean is a more precise measure than the median and that the median is a more precise measure than the mode. The best strategy, then, if you are measuring at a high level, such as interval or ratio, is to use the mean, and indeed, the mean is the most often used measure of central tendency. However, we do have occasions when the mean would not be appropriate as a measure of central tendency—for example, when we have categorical or nominal data, such as hospitalized versus nonhospitalized people. Then we use the mode.

So, here is a set of three guidelines that may be of some help. And remember, there can always be exceptions.

1. Use the mode when the data are categorical (“nominal”) in nature and the people or things can fit into only one class, such as hair color, political affiliation, neighborhood location, and religion. When this is the case, these categories are called mutually exclusive.

2. Use the median when you have extreme scores and you don’t want an average that is misleading, such as when the variable of interest is income expressed in dollars.

3. Finally, use the mean when you have data that do not include extreme scores and are not categorical, such as the numerical score on a test or the number of seconds it takes to swim 50 meters.

USING SPSS TO COMPUTE DESCRIPTIVE STATISTICS

If you haven’t already, now would be a good time to check out Appendix A so you can become familiar with the basics of using SPSS. Then come back here.

Let’s use SPSS to compute some descriptive statistics. The data set we are using is named Chapter 2 Data Set 1, and it is a set of 20 scores on a test of prejudice. All of the data sets are available in Appendix C and from the SAGE website edge.sagepub.com/salkindfrey7e . There is one variable in this data set:

Variable	Definition
Prejudice	The value on a test of prejudice as measured on a scale from 1 to 100

Here are the steps to compute the measures of central tendency that we discussed in this chapter. Follow along and do it yourself. With this and all exercises, including data that you enter or download, we’ll assume that the data set is already open in SPSS.

1. Click Analyze → Descriptive Statistics → Frequencies …

2. Double-click on the variable named Prejudice to move it to the Variable(s) box.

3. Click Statistics, and you will see the Frequencies: Statistics dialog box shown in Figure 2.1.

4. Under Central Tendency, click the Mean, Median, and Mode boxes.

5. Click Continue.

6. Click OK.

The SPSS Output

Figure 2.2 shows you selected output from the SPSS procedure for the variable named Prejudice.

In the Statistics part of the output, you can see how the mean, the median, and the mode are all computed along with the sample size and the fact that there were no missing data. SPSS does not use symbols such as X in its output. Also listed in the output are the frequency of each value and the percentage of times it occurs, all useful descriptive information.

It’s a bit strange, but if you select Analyze → Descriptive Statistics → Descriptives (instead of clicking Frequencies last) in SPSS and then click Options, there’s no median or mode option, which you might expect because they are basic descriptive statistics. The lesson here? Statistical analysis programs are usually quite different from one another, use different names for the same things, and make different assumptions about what’s where. If you can’t find what you want, it’s probably there. Just keep hunting. Also, be sure to use the Help feature to get help navigating through all this new information until you find what you need.

Figure 2.1 ⬢ The Frequencies: Statistics dialog box from SPSS

Figure 2.2 ⬢ Descriptive statistics from SPSS

Understanding the SPSS Output

This SPSS output is pretty straightforward and easy to interpret.

The average mean score for the 20 scores is 84.70 (and remember that the span of possible scores is from 0 to 100). The median, or the point at which 50% of the scores fall above and 50% fall below, is 87 (which is pretty close to the mean), and the most frequently occurring score, or the mode, is 87.

SPSS output can be full of information or just give you the basics. It all depends on the type of analysis that you are conducting. In the preceding example, we have just the basics and, frankly, just what we need. Throughout this book, you will be seeing output and then learning about what it means, but in some cases, discussing the entire collection of output information is far beyond the scope of the book. We focus on output that is directly related to what you learned in the chapter.

Real-World Stats

The Pew Research Center reported a recent study of who uses the big social media sites, like Facebook, Instagram, LinkedIn, and Twitter. About 70% of all Americans use one or more of these sites! Women tend to use Facebook and Instagram more than men do, but there wasn’t really a gender difference for LinkedIn or Twitter. The researchers also looked at the relationship between age and the percentage of people who used each site. In the following table, notice that, as you might suspect, younger adults use social networks more than older adults.

Age	Facebook	Instagram	LinkedIn	Twitter
18 to 29	81%	64%	29%	40%
30 to 49	78%	40%	33%	27%
50 to 64	65%	21%	24%	19%
65 and over	41%	10%	9%	8%

Want to know more? Go online and read about these and other findings at https://www.pewinternet.org/fact-sheet/social-media/.

Summary

No matter how fancy-schmancy your statistical techniques are, you will almost always start by simply describing what’s there—hence the importance of understanding the simple notion of central tendency. From here, we go to another important descriptive characteristic: variability, or how different scores are from one another. That’s what we’ll explore in Chapter 3!

Time to Practice

1. By hand (which means using a calculator if you’d like), compute the mean, the median, and the mode for the following set of 40 chemistry final scores.

93	85	99	77
94	99	86	76
95	99	97	84
91	89	77	87
97	83	80	98
75	94	81	85
78	92	89	94
76	94	96	94
90	79	80	92
77	86	83	81

2. Compute the mean, the median, and the mode for the following three sets of scores saved as Chapter 2 Data Set 2. Do it by hand or use a computer program such as SPSS. Show your work, and if you use SPSS, print out a copy of the output.

Score 1	Score 2	Score 3
3	34	154
7	54	167
5	17	132
4	26	145
5	34	154
6	25	145
7	14	113
8	24	156
6	25	154
5	23	123

3. Compute the means for the following set of scores saved as Chapter 2 Data Set 3 using SPSS. Print out a copy of the output.

Number of Beds (infection rate)	Infection Rate (per 1,000 admissions)
234	1.7
214	2.4
165	3.1
436	5.6
432	4.9
342	5.3
276	5.6
187	1.2
512	3.3
553	4.1

1. You are the manager of a fast-food restaurant. Part of your job is to report to the boss at the end of each day which item is selling best. Use your vast knowledge of descriptive statistics and write one paragraph to let the boss know what happened today. Here are the data. Don’t use SPSS to compute important values; rather, do it by hand. Be sure to include a copy of your work.

Special	Number Sold	Cost
Huge Burger	20	$2.95
Baby Burger	18	$1.49
Chicken Littles	25	$3.50
Porker Burger	19	$2.95
Yummy Burger	17	$1.99
Coney Dog	20	$1.99
Total specials sold	119

2. Imagine you are the CEO of a small group of clothing stores and you are planning an expansion. You’d like your new store to post similar numbers as the other three that are in your empire. By hand, provide some idea of what you want the store’s financial performance to look like. And remember that you have to select whether to use the mean, the median, or the mode as an average. Good luck, young Jedi.

Average	Store 1	Store 2	Store 3	New Store
Sales (in thousands of dollars)	323.6	234.6	308.3
Number of items purchased	3,454	5,645	4,565
Number of visitors	4,534	6,765	6,654

3. Here are ratings (on a scale from 1 through 5) for various Super Bowl party foods. You have to decide which food is rated highest (5 is a winner and 1 a loser). Decide what type of average you will use and why. Do this by hand or use SPSS.

Snack Food	North Fans	East Fans	South Fans	West Fans
Loaded Nachos	4	4	5	4
Fruit Cup	2	1	2	1
Spicy Wings	4	3	3	3
Gargantuan Overstuffed Pizza	3	4	4	5
Beer Chicken	5	5	5	4

4. Time to Practice Video

5. Chapter 2: Problem 6

6. Under what conditions would you use the median rather than the mean as a measure of central tendency? Why? Provide an example of two situations in which the median might be more useful than the mean as a measure of central tendency.

7. Suppose you are working with a data set that has some very “different” (much larger or much smaller than the rest of the data) scores. What measure of central tendency would you use and why?

Chapter 2 problem 6 is the best kind of problem to answer-- what is the best type of food to serve in your Super Bowl party? In this problem, there are five choices-- loaded nachos, fruit cups, spicy wings, gargantuan overstuffed pizza, and beer chicken. They asked fans from four regions to evaluate these choices on a 5 point scale, with 5 being the most preferred. When we look at our SPSS data set-- because that is the easiest way to do this, though it can be done by hand-- but when you look at the data set, you want to look under Variable view and make sure that it's been set up correctly. Each food choice is identified as a variable, and they are listed as the measurement of a scale, which shows the little ruler. So it's been set up correctly. Now, to assess which measure of central tendency is the best, you have to think about your goal. The goal here is to evaluate one thing against the other-- in this case, one type of food against other types of food. And since they're measured on a scale, the mean score would be the most logical choice and the one that gives us the best insight. So under Analyze, Descriptive Statistics, click on Frequencies. What you'll see here is the different types of food choices. Bring those all over to the right hand side. And then click on Statistics. Statistics, we see we have, for measures of central tendency, four options. Only three of them are really measures of central tendency-- mean, median, and mode. We want the mean score for a number of reasons. One, we only have one score for each food choice from each region, so the mode wouldn't be the right choice, because that gives us the most frequently occurring. Two, it's measured on a scale with fixed points, so the median, or middle score, would provide limited insight. Three, the mean, or the average, makes the most sense for this problem, because it gives us the most detailed evaluation. So click on the mean score. Hit Continue. Notice down here it says Display Frequency Tables. That's going to show us how often each score was given. Hit OK. And here is our output. You'll see that, when we look at loaded nachos, the average was a 4.25, and beer chicken a 4.75. In other words, beer chicken was most preferred, but people seemed to really like the loaded nachos as well. The mean score for the fruit cup, on the other hand, was a 1.5-- in other words, not terribly popular. When you look at the frequency table, you can see that, for loaded nachos, they got scores of 4 and 5, three 4s and one 5. The beer chicken got one 4 and three 5s. The fruit cup only got 1s and 2s. So this gives you a sense of how statistics can help you make decisions that affect a party. Presenting data so that it is helpful to the viewer requires that you select the best measure of central tendency. When something is measured on a scale, often the mean score is the best measure, unless-- and this is very important-- you are measuring something in which there is really an enormous range. The most common example is household income in the United States. A few people have super high income, and they would skew the average, because they would move it so much that it would no longer represent most of the people. In that particular case, a median score, or the household income that fell in the mid-range, would be better. In this problem, loaded nachos and beer and chicken are the best choice for your Super Bowl party. Good luck.

8. For this exercise, use the following set of 16 scores (ranked) that consists of income levels ranging from about $50,000 to about $200,000. What is the best measure of central tendency, and why?

1. $199,999

2. $98,789

3. $90,878

4. $87,678

5. $87,245

6. $83,675

7. $77,876

8. $77,743

9. $76,564

10. $76,465

11. $75,643

12. $66,768

13. $65,654

14. $58,768

15. $54,678

16. $51,354

9. Use the data in Chapter 2 Data Set 4 and, manually, compute the average attitude scores (with a score of 10 meaning a very positive attitude and a 1 meaning a very negative attitude) for three groups of individuals’ attitudes reflecting their experience with urban transportation. Choose the mean as the average to use.

10. Take a look at the following number of pie orders from the Lady Bird diner and determine the average number of orders for each week.

Week	Chocolate Silk	Apple	Douglas County Pie
1	12	21	7
2	14	15	12
3	18	14	21
4	27	12	15

3 UNDERSTANDING VARIABILITY VIVE LA DIFFÉRENCE

3: MEDIA LIBRARY

Premium Videos

Difficulty Scale

(moderately easy but not a cinch)

WHAT YOU WILL LEARN IN THIS CHAPTER

· Understanding the value of variability as a descriptive tool

· Computing the range

· Computing the standard deviation

· Computing the variance

· Understanding what the standard deviation and variance have in common—and how they are different

WHY UNDERSTANDING VARIABILITY IS IMPORTANT

In Chapter 2, you learned about different types of averages, what they mean, how they are computed, and when to use them. But when it comes to descriptive statistics and describing the characteristics of a distribution, averages are only half the story. The other half is measures of variability.

In the simplest of terms, variability reflects how much scores differ from one another. For example, the following set of scores shows some variability:

7, 6, 3, 3, 1

The following set of scores has the same mean (4) but has less variability than the previous set:

3, 4, 4, 5, 4

The next set has no variability at all—the scores do not differ from one another—but it also has the same mean as the other two sets we just showed you.

4, 4, 4, 4, 4

Variability (also called spread or dispersion) can be thought of as a measure of how different scores are from one another. It’s even more accurate (and maybe even easier) to think of variability as how different scores are from one particular score. And what “score” do you think that might be? Well, instead of comparing each score to every other score in a distribution, the one score that could be used as a comparison is—that’s right—the average. So, variability becomes a measure of how much each score in a group of scores differs from the average, usually the mean. More about this in a moment.

Remember what you already know about computing averages—that an average (whether it is the mean, the median, or the mode) is a representative score of a set of scores. Now, add your new knowledge about variability—that it reflects how different scores are from one another. Each is an important descriptive statistic. Together, these two (average and variability) can be used to describe the characteristics of a distribution and show how distributions differ from one another.

Three measures of variability are commonly used to reflect the degree of variability, spread, or dispersion in a group of scores. These are the range, the standard deviation, and the variance. Let’s take a closer look at each one and how each one is used.

Actually, how data points differ from one another is a central part of understanding and using basic statistics. But when it comes to differences between individuals and groups (a mainstay of most social and behavioral sciences), the whole concept of variability becomes really important. Sometimes it’s called fluctuation, or error, or one of many other terms, but the fact is, variety is the spice of life, and what makes people different from one another also makes understanding them and their behavior all the more challenging (and interesting). Without variability either in a set of data or between individuals and groups, things are just boring.

COMPUTING THE RANGE

The range is the simplest measure of variability and kind of intuitive. It is the distance of the biggest score from the smallest score. The range is computed simply by subtracting the lowest score in a distribution from the highest score in the distribution.

In general, the formula for the range is

(3.1)

r=h−l,r=h−l,

where

· r is the range,

· h is the highest score in the data set, and

· l is the lowest score in the data set.

Take the following set of scores, for example (shown here in descending order):

98, 86, 77, 56, 48

In this example, 98 − 48 = 50. The range is 50. Even if there was a set of 500 scores, where the largest is 98 and the smallest was 48, the range would still be 50.

There really are two kinds of ranges. One is the exclusive range, which is the highest score minus the lowest score (or h − l) and the one we just defined. The second kind of range is the inclusive range, which is the highest score minus the lowest score plus 1 (or h − l + 1). The difference is that the inclusive range counts both the high and low scores as being among the values in the range. You most commonly see the exclusive range in research articles, but the inclusive range is also used on occasion if the researcher prefers it.

The range tells you how different the highest and lowest values in a data set are from one another—that is, the range shows how much spread there is from the lowest to the highest point in a distribution. So, although the range is fine as a general indicator of variability, it should not be used to reach any conclusions regarding how individual scores differ from one another. And you will usually never see it reported as the only measure of variability but as one of several—which brings us to….

COMPUTING THE STANDARD DEVIATION

Now we get to the most frequently used measure of variability, the standard deviation. Just think about what the term implies—it’s a deviation from something (guess what?) that has been standardized. Actually, the standard deviation (sometimes abbreviated as SD, sometimes s) represents the average amount of variability in a set of scores. In practical terms, it’s the average distance of each score from the mean. The larger the standard deviation, the larger the average distance each data point is from the mean of the distribution and the more variable the set of scores is.

So, what’s the logic behind computing the standard deviation? Your initial thoughts may be to compute the mean of a set of scores and then subtract each individual score from the mean. Then, compute the average of that distance. That’s a good idea—you’ll end up with the average distance of each score from the mean. But it won’t work. (For fun, play with a small set of scores and see if you can figure out why this simple formula won’t work. We’ll show you why in a moment.)

Here’s the formula for computing the standard deviation:

(3.2)

s=

⎷∑(X−¯¯¯X)2n−1,s=∑(X−X¯)2n−1,

where

· s is the standard deviation;

· ∑ is sigma, which tells you to find the sum of what follows;

· X is each individual score;

· ¯¯¯XX¯ is the mean of all the scores; and

· n is the sample size.

This formula finds the difference between each individual score and the mean (X−¯¯¯XX−X¯ ), squares each difference, and sums them all together. Then, it divides the sum by the size of the sample (minus 1) and takes the square root of the result. As you can see, and as we mentioned earlier, the standard deviation is an average deviation from the mean.

1. List each score. It doesn’t matter whether the scores are in any particular order.

2. Compute the mean of the group.

3. Subtract the mean from each score.

4. Square each individual difference. The result is the column marked (X−¯¯¯X)2(X−X¯)2 .

5. Sum all the squared deviations about the mean. As you can see, the total is 28.

6. Divide the sum by n − 1, or 10 − 1 = 9, so then 28/9 = 3.11.

7. Compute the square root of 3.11, which is 1.76 (after rounding). That is the standard deviation for this set of 10 scores.

Here are the data we’ll use in the following step-by-step explanation of how to compute the standard deviation:

5, 8, 5, 4, 6, 7, 8, 8, 3, 6

Here’s what we’ve done so far, where (X−¯¯¯X)(X−X¯) represents the difference between the actual score and the mean of all the scores, which is 6.

X	¯¯¯XX¯	((X−¯¯¯X)(X−X¯)
8	6	8 − 6 = +2
8	6	8 − 6 = +2
8	6	8 − 6 = +2
7	6	7 − 6 = +1
6	6	6 − 6 = 0
6	6	6 − 6 = 0
5	6	5 − 6 = −1
5	6	5 − 6 = −1
4	6	4 − 6 = −2
3	6	3 − 6 = −3
X	(X−¯¯¯X)(X−X¯)	(X−¯¯¯X)2(X−X¯)2
8	+2	4
8	+2	4
8	+2	4
7	+1	1
6	0	0
6	0	0
5	−1	1
5	−1	1
4	−2	4
3	−3	9
Sum	0	28

What we now know from these results is that each score in this distribution differs from the mean by an average of 1.76 points.

They’re important to review and will increase your understanding of what the standard deviation is.

First, why didn’t we just add up the deviations from the mean? Because the sum of the deviations from the mean is always equal to zero. Try it by summing the deviations (2 + 2 + 2 + 1 + 0 + 0 − 1 − 1 − 2 − 3). In fact, that’s the best way to check whether you computed the mean correctly.

There’s another type of deviation that you may read about, and you should know what it means. The mean deviation (also called the mean absolute deviation) is the sum of the absolute value of the deviations from the mean divided by the number of scores. You already know that the sum of the deviations from the mean must equal zero (otherwise, the mean is computed incorrectly). That’s why we square the deviations before summing them. Another option, though, would be to take the absolute value of each deviation (which is the value regardless of the sign). Sum the absolute values and divide by the number of data points, and you have the mean deviation. So, if you have a set of scores such as 3, 4, 5, 5, 8 and the arithmetic mean is 5, the mean deviation is the sum of 2 (the absolute value of 5 − 3), 1, 0, 0, and 3, for a total of 6. Then divide this by 5 to get the result of 1.2. (Note: The absolute value of a number is usually represented as that number with a vertical line on each side of it, such as |5|. For example, the absolute value of −6, or |−6|, is 6.)

Second, why do we square the deviations? Because we want to get rid of the negative sign so that when we do eventually sum them, they don’t add up to 0.

And finally, why do we eventually end up taking the square root of the entire value in Step 7? Because we want to return to the same units with which we originally started. We squared the deviations from the mean in Step 4 (to get rid of negative values) and then took the square root of their total in Step 7 to get back where we started. Pretty tidy.

LIGHTBOARD LECTURE VIDEO

In statistics, there are all these complicated formulas, and they look like they're more complicated than they need to be. But they are exactly as complicated as they need to be and a good example is the standard deviation. Here's the definition, right? Average distance of each score from the mean. That should be a pretty easy equation, right? You would sum up all the different scores. And then you'd figure out how far they are away from the mean. And then you'd average that by dividing by the number of scores. So that should be a good equation to figure out the standard deviation. The problem is that when you do that-- and let me show you what would happen when you do that. We're going to subtract the mean from each of these four scores. The mean happens to be 2. So when we do that, we're going to get negative 1, OK? We're going to add those up. And that adds up to We divide by 4, and we're going to get It turns out that with the mean, you'll always get So that formula won't work for standard deviation. Instead, you've got to get rid of negative numbers because they will counteract all the positive numbers. Now, you could do that in a variety of ways. We're actually going to square those. That's what statisticians do. So if you square 1, it becomes 1. But now when we add that up, we get a 2. That's like a real number. So you divide that by the number of scores to get your average. And you're going to get a standard deviation of We're still not done, though, I'm afraid. Because we squared all these distances, that's like cheating. So you've got to go back and un-square everything. So the actual standard deviation now has got to be the square root of this, which is something like And for this set of numbers. So the formula for standard deviation is super complicated, but it has to be.

Why n − 1? What’s Wrong With Just n?

It might make sense to square the deviations about the mean and then later go back and take the square root of their sum. But how about subtracting the value of 1 from the denominator of the formula? Why do we divide by n − 1 rather than just plain ol’ n, like we usually do when we calculate the mean? Good question.

The answer is that s (the standard deviation) is an estimate of the population standard deviation, and it is an unbiased estimate . Unbiased means that your sample estimate of the mean is just as likely to be a little higher than the population mean as it is to be a little lower. It is unbiased, though, only when we subtract 1 from n. By subtracting 1 from the denominator, we artificially force the standard deviation to be a tiny bit larger than it would be otherwise. Why would we want to do that? Because, as good scientists, we are conservative. Being conservative means that if we have to err (and there is always a pretty good chance that we will), we will do so on the side of overestimating what the standard deviation of the population is. Dividing by a smaller denominator lets us do so. Thus, instead of dividing by 10, we divide by 9. Or instead of dividing by 100, we divide by 99.

Notice that with larger sample sizes, the adjustment of dividing by n − 1 doesn’t make as much difference as it does with a smaller set of scores. This pattern, that larger sample sizes lead to more accurate estimates of population statistics (like the standard deviation and the mean), is reflected throughout all statistics and throughout this book.

Biased estimates are appropriate if your intent is only to describe the characteristics of the sample. But if you intend to use the sample as an estimate of a population parameter, then it’s best to calculate the unbiased statistic. That’s why spreadsheets and calculators sometimes offer two equations for computing the standard deviation, one that divides the sum of deviations by n − 1 and one that divides the sum of deviations by just n.

Take a look in the following table and see what happens as the size of the sample gets larger (and moves closer to the population in size). The n − 1 adjustment has far less impact on the difference between the biased and the unbiased estimates of the standard deviation (the bold column in the table) as the sample size increases.

All other things being equal, then, the larger the size of the sample, the less difference there is between the biased and the unbiased estimates of the standard deviation.

Sample Size	Value of Numerator in Standard Deviation Formula	Biased Estimate of the Population Standard Deviation (dividing by n)	Unbiased Estimate of the Population Standard Deviation (dividing by n − 1)	Difference Between Biased and Unbiased Estimates
10	500	7.07	7.45	0.38
100	500	2.24	2.25	0.01
1,000	500	0.7071	0.7075	0.0004

The moral of the story? When you compute the standard deviation for a sample, which is an estimate of the population, the larger the sample is, and the more accurate the estimate will be.

What’s the Big Deal?

The computation of the standard deviation is very straightforward. But what does it mean? As a measure of variability, all it tells us is how much each score in a set of scores, on the average, varies from the mean. But it has some very practical applications, as you will find out in Chapter 4. Just to whet your appetite, consider this: The standard deviation can be used to help us compare scores from different distributions, even when the means and standard deviations are different. Amazing!

· The standard deviation is computed as the average distance from the mean. So, you will need to first compute the mean as a measure of central tendency. Don’t fool around with the median or the mode in trying to compute the standard deviation.

· The larger the standard deviation, the more spread out the values are, and the more different they are from one another.

· Just like the mean, the standard deviation is sensitive to extreme scores. When you are computing the standard deviation of a sample and you have extreme scores, note that fact somewhere in your written report and in your interpretation of what the data mean.

· If s = 0, there is absolutely no variability in the set of scores, and the scores are essentially identical in value. This will rarely happen and actually makes it hard to compute statistics about the relationship between scores. Without variability, we can’t see connections between variables. And, when you think about it, a variable is only a variable when it varies.

So, what’s the big deal about variability and its importance? As a statistical concept, you learned above that it is a measure of dispersion or how scores differ from one another. But, “shakin’ it up” really is the spice of life, and as you may have learned in your biology or psychology classes or from your own personal reading, variability is the key component of evolution, for without change (which variability always accompanies), organisms cannot adapt. All very cool.

CORE CONCEPTS IN STATS VIDEO

Measures of Variability

Measures of central tendency focus on what scores have in common with each other. But in order to describe a distribution of data, we also have to think about how scores or different. Measures of variability allow us to do that. The first measure of variability is the range, which is the difference between the highest and lowest score in a set of data. With these five scores, the highest score is seven, and the lowest is one. To describe the variability, we would subtract the lowest score from the highest score, and say that these five scores have a range of six. The problem is that it isn't based on all of the scores. It's only based on the highest and lowest. So it may not accurately portray the variability among scores. Take a look at the two sets of five scores. Both sets have the same range, 6. But if you compare the two sets of scores with each other, they're clearly different. We need a measure of variability that's based on all the scores. But when you have more and more scores, it gets to be a lot of comparisons to make. This is when we look to compare each score against the mean of the scores. Then we can answer the question, on average, how different is a score from the mean? In this table, at the bottom of the first column, you'll see that the mean is 4. In the second column, we're going to calculate the difference between each score and the mean. For the first score of 3, the difference between this score and the mean is 3 minus 4, of negative 1. We can do this for all the scores. Now, we can calculate the average difference, known as the deviation, of a score and the mean. So in the numerator in this formula, you see x minus x bar. That's a deviation. That's the difference between a score and the mean, and we have five of them. So to calculate the average deviation, we're simply going to add up the deviations. As you can see, the average deviations add up to to calculate the average deviation of a score from the mean, we would come up with To say that there's zero variability implies that all the scores are the same, and they're clearly not. In any set of data, the average deviation will always be equal to of the deviations, the numerator of the formula, will always be And it doesn't matter what the scores are. In any set of scores, the sum of the deviations will be the positive deviations. In this example, the negative deviations of negative 1 and 3 balance out the positive deviations of 3 and 1. How do we get rid of the negative deviations? The easiest way is to square them, because any squared number is a positive number. This takes us to our second measure of variability, the variance. In the denominator, you expect to see n because we're talking about the average squared deviation, but we actually divide by n minus 1. To calculate the variance, we calculate the squared deviation of each score from the mean. For the first score, the squared deviation is negative 1 squared, which is 1. If we were to calculate the average squared deviation, we would provide the formula for the variance. And the numerator of this formula is the sum of the squared deviations. The average squared deviation here is 5, so we would conclude that the variance in this set of scores is equal to 5. Now we've represented the variability with a statistic. We want the average deviation of a score from the mean, but remember, what we've calculated is the average squared deviation. So undo the squaring by taking the square root of the variance to get the standard deviation. In this example, we calculated the standard deviation by taking the square root of the variance, which in this case is the square root of 5, and we ended up with a value of 2.24. That represents the average deviation of a score from the mean. It shows on average in this set of five scores how different each score is from the mean of 4. Researchers need to be able to describe the amount of differences among the scores of a variable, which makes variability an important aspect of descriptive statistics.

COMPUTING THE VARIANCE

Here comes another measure of variability and a nice surprise. If you know the standard deviation of a set of scores and you can square a number, you can easily compute the variance of that same set of scores. This third measure of variability, the variance, is simply the standard deviation squared.

In other words, it’s the same formula you saw earlier but without the square root bracket, like the one shown in Formula 3.3:

(3.3)

s2=∑(X−¯¯¯X)2n−1s2=∑(X−X¯)2n−1

If you take the standard deviation and never complete the last step (taking the square root), you have the variance. In other words, s 2 = s × s, or the variance equals the standard deviation times itself (which is what squaring does). In our earlier example, where the standard deviation was equal to 1.76 (before rounding), the variance is equal to 1.762, or 3.10. As another example, let’s say that the standard deviation of a set of 150 scores is 2.34. Then, the variance would be 2.342, or 5.48.

You are not likely to see the variance mentioned by itself in a journal article or see it used as a descriptive statistic. This is because the variance is a difficult number to interpret and apply to a set of data. After all, it is based on squared deviation scores, and its size is only a function of the number of scores that happen to be in a distribution.

But the variance is important because it is used both as a concept and as a practical measure of variability in many statistical formulas and techniques. You will learn about these later in Statistics for People Who (Think They) Hate Statistics.

People Who Loved Statistics

You may know about the famous English nurse Florence Nightingale (1820–1910), who managed and trained nurses in the Crimean War and developed standards and procedures to make nursing a profession. She also, though, was a gifted mathematician and was influential in promoting the collection and analysis of statistical data. She was one of the first medical researchers to use graphs and charts to display the variability of data.

She used these visual forms of communication to examine causes of death for soldiers and to convince administrators that environmental conditions, like unsanitary water, affected death rates. Her application of statistics to real-world problems of life and death like sanitation and other issues is believed to have dramatically increased life expectancy throughout English towns in the latter parts of the 19th century.

The Standard Deviation Versus the Variance

How are standard deviation and the variance the same, and how are they different?

Well, they are both measures of variability, dispersion, or spread. The formulas used to compute them are very similar. You see them (but mostly the standard deviation) reported all over the place in the “Results” sections of journals.

They are also quite different.

First, and most important, the standard deviation (because we take the square root of the average summed squared deviation) is stated in the original units from which it was derived. The variance is stated in units that are squared (the square root of the final value is never taken).

What does this mean? Let’s say that we need to know the variability of a group of production workers assembling circuit boards. Let’s say that they average 8.6 boards per hour and the standard deviation is 1.59. The value 1.59 means that the difference in the average number of boards assembled per hour by each worker is about 1 1/2 circuit boards. This kind of information is quite valuable when trying to understand the overall performance of groups.

Let’s look at an interpretation of the variance, which is 1.592, or 2.53. This would be interpreted as meaning that the average difference between the workers is about 2.53 circuit boards squared from the mean. Which of these two makes more sense?

USING SPSS TO COMPUTE MEASURES OF VARIABILITY

Let’s use SPSS to compute some measures of variability. We are using the file named Chapter 3 Data Set 1.

There is one variable in this data set:

Variable	Definition
ReactionTime	Reaction time on a tapping task in seconds

Here are the steps to compute the measures of variability that we discussed in this chapter:

1. Open the file named Chapter 3 Data Set 1.

2. Click Analyze → Descriptive Statistics → Frequencies.

3. Double-click on the ReactionTime variable to move it to the Variable(s) box. If it’s highlighted, you can also click on that left arrow between the two boxes to move a variable.

4. Click on the Statistics button, and you will see the Frequencies: Statistics dialog box. This dialog box is used to select the variables and procedures you want to perform.

5. Under Dispersion, click Std. deviation.

6. Under Dispersion, click Variance.

7. Under Dispersion, click Range.

8. Click Continue.

9. Click OK.

The SPSS Output

Figure 3.1 shows selected output from the SPSS procedure for ReactionTime.

Figure 3.1 ⬢ SPSS output for the variable reaction time

Source: IBM.

Understanding the SPSS Output

There are 30 valid cases with no missing cases, and the standard deviation is 0.70255. The variance equals 0.494 (or s2), and the range is 2.60. As you already know, the standard deviation, variance, and range are measures of dispersion or variability. In the case of this set of 30 observations, the standard deviation (the most commonly used of all these measures) is equal to 0.703 or about 0.70.

We will go into much greater detail as to how this value is used in Chapter 8, but for now, the value of 0.70 represents the average amount each score is from the centermost point in the set of scores, or the mean. For these reaction time data, it means that the average reaction time varies from the mean by about 0.70 seconds.

Let’s try another one, using the data titled Chapter 3 Data Set 2. There are two variables in this data set:

Variable	Definition
Math_Score	Score on a mathematics test
Reading_Score	Score on a reading test

Follow the same set of instructions as given previously, only in Step 3, select both variables (either by double-clicking on each one or by selecting it and clicking the “move” arrow).

More SPSS Output

The SPSS output is shown in Figure 3.2, where you can see selected output from the SPSS procedure for these two variables. There are 30 valid cases with no missing cases, and the standard deviation for math scores is 12.36 with a variance of 152.70 and a range of 43. For reading scores, the standard deviation is 18.70, the variance is a whopping 349.69 (that’s pretty big), and the range is 76 (which is large as well, reflecting the similarly large variance).

Figure 3.2 ⬢ Output for the variables Math_Score and Reading_Score

Understanding the SPSS Output

As with the example shown in Figure 3.1, the interpretation of the SPSS output is relatively straightforward.

On average, the amount of deviation from the mean for math sores is 12.4, and on average, the amount of deviation for reading scores is 18.70. Whether these measures of dispersion are considered “large” or “small” is a question that can only be answered relative to a variety of factors, including what’s being measured, the number of observations, and the range of possible scores. Context is everything.

Real-World Stats

If you were like a mega stats person, then you might be interested in the properties of measures of variability for the sake of those properties. That’s what mainline statisticians spend lots of their time doing—looking at the characteristics and performance and assumptions (and the violation thereof) of certain statistics.

But we’re more interested in how these tools are used, so let’s take a look at one such study that actually focused on variability as an outcome. And, as you read earlier, variability among scores is interesting for sure, but when it comes to understanding the reasons for variability among substantive performances and people, then the topic becomes really interesting.

This is exactly what Nicholas Stapelberg and his colleagues in Australia did when they looked at variability in heart rate as it related to coronary heart disease. Now, they did not look at this phenomenon directly, but they entered the search terms heart rate variability, depression, and heart disease into multiple electronic databases and found that decreased heart rate variability is found in conjunction with both major depressive disorders and coronary heart disease.

Why might this be the case? The researchers think that both diseases disrupt control feedback loops that help the heart function efficiently. This is a terrific example of how looking at variability can be the focal point of a study rather than an accompanying descriptive statistic.

Want to know more? Go online or to the library and read …

Stapelberg, N. J., Hamilton-Craig, I., Neumann, D. L., Shum, D. H., & McConnell, H. (2012). Mind and heart: Heart rate variability in major depressive disorder and coronary heart disease—a review and recommendations. The Australian and New Zealand Journal of Psychiatry, 46, 946–957.

Summary

Measures of variability help us even more fully understand what a distribution of data points looks like. Along with a measure of central tendency, we can use these values to distinguish distributions from one another and effectively describe what a collection of test scores, heights, or measures of personality looks like and what those individual scores represent. Now that we can think and talk about distributions, let’s explore ways we can look at them.

Time to Practice

1. Why is the range the most convenient measure of dispersion, yet the most imprecise measure of variability? When would you use the range?

2. Fill in the exclusive and inclusive ranges for the following items.

High Score	Low Score	Inclusive Range	Exclusive Range
12.1	3
92	51
42	42
7.5	6
27	26

3. Why would you expect more variability on a measure of personality in college freshmen than you would on a measure of age?

4. Why does the standard deviation get smaller as the individuals in a group score more similarly on a test? And why would you expect the amount of variability on a measure to be relatively less with a larger number of observations than with a smaller one?

5. For the following set of scores, compute the range, the unbiased and the biased standard deviations, and the variance. Do the exercise by hand.

94, 86, 72, 69, 93, 79, 55, 88, 70, 93

6. In Question 5, why is the unbiased estimate greater than the biased estimate?

7. Use SPSS to compute all the descriptive statistics for the following set of three test scores over the course of a semester. Which test had the highest average score? Which test had the smallest amount of variability?

Test 1	Test 2	Test 3
50	50	49
48	49	47
51	51	51
46	46	55
49	48	55
48	53	45
49	49	47
49	52	45
50	48	46
50	55	53

Problem 7 in Chapter 3, asks you to compute the descriptive statistics for three different tests. As you see here, there are scores for each test, and this would assume that every student has a score. So, 5 You want to make a spreadsheet that's going to look like this. So in SPSS, you look here-- and you see I've already set it up. Under Variable View, you would want to set it up by identifying Test 1, Test 2, and Test 3, and then making sure the level of measurement is indicated as scale, meaning that this is continuous data. And test scores are continuous data. You know that when you look back here on the top of the column and you see a little ruler. Computing descriptive statistics is really straightforward. On the Menu bar, you're going to see the tab called Analyze, you want to click on it. And you're going to see the Descriptive Statistics Tab, when you hover over that, you're going to see have some options. Let's click on Frequencies. And you see this big table, or big screen, open up. And in the left hand column, we see the three variables. Highlight those and bring them to the right hand side, which is the side that tells us, or tells the computer, what test we're going to do. Under Statistics, when you click on that, it gives you some options. We want Central Tendency. So let's look at the Mean, Median, and Mode. Also click down here under Dispersion, Standard Deviation and Range. Click Continue. Down here, in the bottom left hand side, unclick Display Frequency Tables, because that just tells you how many times a given score occurred. Click OK. And here comes our output. What you see here is, we had 1 what saying was valid, and then it's showing us the mean score, or this is the average, a 5 As that will show you, the student scored really closely here. The median, or middle score, is almost exactly the same, 49.5, 49.5, and 48, slightly lower. The mode, or the most commonly occurring score, 49, 48, 45. Where you see the difference is in the standard deviation. And this is the range of scores within the average, so how many scores fall within one standard deviation of the mean. Here, the lowest standard deviation is under Test 2. So, the answer to the question, which test had the highest score? As you see, Test 2 had the highest average, but only by very little bit. Which test had the smallest variability? Again, it's Test 2. What this means is that the student scored slightly higher on Test 2 and there was less deviation, meaning their scores were closer together.

1. For the following set of scores, compute by hand the unbiased estimates of the standard deviation and variance.

· 58

· 56

· 48

· 76

· 69

· 76

· 78

· 45

· 66

2. The variance for a set of scores is 36. What is the standard deviation, and what is the range?

3. Find the inclusive range, the sample standard deviation, and the sample variance of each of the following sets of scores:

c. 5, 7, 9, 11

c. 0.3, 0.5, 0.6, 0.9

c. 6.1, 7.3, 4.5, 3.8

c. 435, 456, 423, 546, 465

1. This practice problem uses the data contained in the file named Chapter 3 Data Set 3. There are two variables in this data set:

Variable	Definition
Height	Height in inches
Weight	Weight in pounds

1. Using SPSS, compute all of the measures of variability you can for height and weight.

1. How can you tell whether SPSS produces a biased or an unbiased estimate of the standard deviation?

1. Compute the biased and unbiased values of the standard deviation and variance for the set of accuracy scores shown in Chapter 3 Data Set 4. Use SPSS if you can; otherwise, do it by hand. Which one is smaller, and why?

1. On a spelling test, the standard deviation is equal to 0.94. What does this mean?

APPENDIX A SPSS STATISTICS IN LESS THAN 30 MINUTES

This appendix will teach you enough about IBM SPSS to complete the exercises in Statistics for People Who (Think They) Hate Statistics. Learning SPSS is not rocket science—just take your time, work as slowly as you need to, and ask a fellow student or your instructor for help if necessary.

You are probably familiar with other Windows applications, and you will find that many SPSS features operate exactly the same way. We assume you know about dragging, clicking, double-clicking, and working with Windows or the Mac (the two versions of SPSS for these operating systems are very similar). If you do not, you can refer to one of the many popular trade computer books for help. SPSS works with Microsoft Windows XP, Vista, Versions 7 and 8, and later versions of the operating system. For the Mac, it works on Mountain Lion 10.8 and later. This appendix focuses almost exclusively on the Windows version because it is much more popular than the Mac version, but if you are a Mac user, you should have no difficulty at all following the instructions and examples.

This appendix is an introduction to SPSS (Version 25) and shows you just some of the things it can do. Almost all of the information in this appendix also can be applied to earlier versions of SPSS, from Versions 11 through 24.

For many of the examples in this appendix, we will use the sample data set shown in Appendix C named Sample Data Set.sav. You are welcome to enter those data manually or download the set from the SAGE website at edge.sagepub.com/salkindfrey7e .

Starting SPSS

Like other Windows-based applications, SPSS is organized as a group and is available on the Start menu. This group was created when you first installed SPSS. To start SPSS, follow these steps:

1. Click Start and then point to Programs.

2. Find and click the SPSS icon. When you do this, you will see the SPSS opening screen, as shown in Figure A.1. You should note that some computers are set up differently, and your SPSS icon might be located on the desktop. In that case, to open SPSS, just double-click on the icon.

Figure A.1 ⬢ The Data View window

The SPSS Opening Window

As you can see in Figure A.1, the opening window shows the Data View (also called the Data Editor). This is where you enter data you want to use with SPSS once those data have been defined. If you think the Data Editor is similar to a spreadsheet in form and function, you are right. In form, it certainly is, because the Data Editor consists of rows and columns just like an Excel spreadsheet. Values can be entered and then manipulated. In function as well, the Data Editor is much like a spreadsheet. Values that are entered can be transformed, sorted, rearranged, and more. The Data Editor, though, differs from a spreadsheet in that you cannot put equations and functions in a cell. The cells only hold data, and those data do not change once you enter them, unless you reenter or recode them.

Although you cannot see it when SPSS first opens, there is another open (but not active) window as well. This is the Variable View, where variables are defined and the parameters for those variables are set.

The Viewer displays statistical results and charts that you create. An example of the Viewer section of the Output window is shown in Figure A.2. A data set is created using the Data Editor, and once the set is analyzed or graphed, you examine the results of the analysis in the Viewer.

Figure A.2 ⬢ The Viewer

The SPSS Toolbar and Status Bar

The use of the Toolbar—the set of icons below the menus—can greatly facilitate your SPSS activities. If you want to know what an icon on the Toolbar does, just roll the mouse pointer over it, and you will see a tip telling you what the tool does. Some of the buttons on the Toolbar are dimmed, meaning they are not active.

The Status Bar, located at the bottom of the SPSS window, is another useful on-screen tool. Here, you can see a one-line report telling you which activity SPSS is currently involved in. The message IBM SPSS for Windows Processor is Ready tells you that SPSS is ready for your directions or input of data. As another example, Running Means … tells you that SPSS is in the middle of the procedure named Means.

Using SPSS Help

SPSS Help is only a few mouse clicks away, and it is especially useful when you are in the middle of a data file and need information about an SPSS feature. SPSS Help is so comprehensive that even if you are a new SPSS user, it can show you the way.

You can get help in SPSS by using the Help menu you see in Figure A.3.

Figure A.3 ⬢ The various help options

The SPSS Toolbar and Status Bar

Using SPSS Help

You can get help in SPSS by using the Help menu you see in Figure A.3.

Figure A.3 ⬢ The various help options

There are 12 options on the Help menu (which is greatly expanded from earlier versions of SPSS), and 6 of these are directly relevant to helping you:

· Topics gives you a list of topics for which you can get help.

· Tutorial offers you a short tutorial on all aspects of using SPSS.

· Case Studies gives you real live examples of how SPSS can be applied.

· Working with R provides you with information about how to work with the open source statistics package named R.

· Statistics Coach walks you through procedures step by step.

· Command Syntax Reference helps you to learn and use SPSS’s programming language.

· SPSS Community gives you access to other SPSS users and information.

· About … gives you some technical information about SPSS, including the version you are using.

· Algorithms focuses on the calculations that are used to produce the results you see in SPSS.

· IBM SPSS Products Home takes you to the home page for SPSS on the Internet.

· Programmability provides you with information about creating add-ons and other program enhancements for SPSS.

· Diagnose helps you to diagnose why SPSS may not be running properly.

A Brief Tour of SPSS

Now, sit back and enjoy a brief tour of what SPSS can do. Nothing fancy here. Just some simple descriptions of data, a test of significance, and a graph or two. What we are trying to show you is how easy it is to use SPSS.

Opening a File

You can enter your own data to create a new SPSS data file, use an existing file, or even import data from such applications as Microsoft Excel into SPSS. Any way you do it, you need to have data to work with. In Figure A.4, the data contained in Appendix C (named Sample Data Set.sav) are shown. This file is also available on the book’s companion website or directly from the author.

A Simple Table and Graph

Now it is time to get to the reason why we are using SPSS in the first place—the various analytical tools that are available.

First, let’s say we want to know the general distribution of males and females. That’s all—just a count of how many males and how many females are in our total sample. We also want to create a simple bar graph of the distribution.

Figure A.4 ⬢ An open SPSS file

In Figure A.5, you see the output that provides exactly the information we asked for, which was the frequency of the number of males and females. We used the “Frequencies” option on the Descriptive Statistics menu (under the main menu Analyze) to compute these values, and then we created a simple bar graph.

Figure A.5 ⬢ The results of a simple descriptive analysis

A Simple Analysis

Let’s see if males and females differ in their average Test1 scores. This is a simple analysis requiring a t test for independent samples. The procedure is a comparison between males and females for the mean of Test1 for each group.

In Figure A.6, you can see a summary of the results of the t test. Notice that the listing in the left pane (the outline view) of the SPSS Viewer now shows the Frequencies and t test procedures listed. To see any part of the output, all we need do is click on that element. Almost always, when SPSS produces output in the Viewer, you will have to scroll to see the entire output.

Figure A.6 ⬢ The results of an independent-samples t test

Creating and Editing a Data File

As a hands-on exercise, let’s create the beginning of the sample data file you see in Appendix C. The first step is to define the variables in your data set, and the second step is to enter the data. You should have a new Data Editor window open (Click File → New → Data).

Defining Variables

SPSS cannot work unless variables are defined. You can have SPSS define the variables for you, or you can do the defining yourself, thereby having much more control over the way things look and work. SPSS will automatically name the first variable VAR00001. If you defined a variable in row 1, column 5, then SPSS would name the variable VAR00005 and number the other columns sequentially.

Custom Defining Variables: Using the Variable View Window

To define a variable yourself, you must first go to the Variable View window by clicking the Variable View tab at the bottom of the SPSS screen. Once that is done, you will see the Variable View window, as shown in Figure A.7, and be able to define any one variable as you see fit.

Figure A.7 ⬢ The Variable View window

Once in the Variable View window, you can define variables along the following parameters:

· Name allows you to give a variable a name of up to eight characters.

· Type defines the type of variable, such as text, numeric, string, scientific notation, and so on.

· Width defines the number of characters of the column housing the variable.

· Decimals defines the number of decimals that will appear in the Data View.

· Label defines a label of up to 256 characters for the variable.

· Values defines the labels that correspond to certain numerical values (such as 1 for male and 2 for female).

· Missing indicates how any missing data will be dealt with.

· Columns defines the number of spaces allocated for the variable in the Data View window.

· Align defines how the data are to appear in the cell (right, left, or center aligned).

· Measure defines the scale of measurement that best characterizes the variable (nominal, ordinal, or interval).

· Role defines the part that the variable plays in the overall analysis (input, target, etc.).

If you place the cursor in the first cell under the “Name” column, enter any name, and press the Enter key, then SPSS will automatically provide you with the default values for all the variable characteristics. Even if you are not in the Data View screen (click the tab on the bottom of the window), SPSS will automatically name the variables var0001, var0002, and so on.

In the Variable View, enter the names of the variables as you see in Figure A.8.

Figure A.8 ⬢ Defining variables in the Variable View window

Now, if you wanted, you could switch to the Data View (see Figure A.9) and just enter the data as you see in Figure A.4. But first, let’s look at just one of the cool SPSS bells and whistles.

Figure A.9 ⬢ The Data View window ready for data entry

Defining Value Labels

You can leave your data as numerical values in the SPSS Data Editor, or you can have labels represent the numerical values (as you saw in Figure A.4).

Why would you want to change the label of a variable? You probably already know that, in general, it makes more sense to work with numbers (like 1 or 2) than with string or alphanumeric variables (such as male or female). An often-made error is entering data as a string (such as male or female) rather than as the number representing the variable named gender. When it comes time for an analysis, it is very difficult to work with nonnumerical entries (such as “male”).

But it sure is a lot easier to look at a data file and see words rather than numbers. Just think about the difference between data files with numbers representing various levels (such as 1 and 2) of a variable and with the actual values (such as male and female). The Values option in the Variable View screen allows you to enter values in the cell, but what you will see are value labels.

If you click the ellipsis button in the Values column (see Figure A.10), you will see the Value Labels dialog box, as shown in Figure A.11.

Figure A.10 ⬢ The Values column in the Variable View screen

Figure A.11 ⬢ The Value Labels dialog box

Changing Value Labels

To assign or change a variable label, follow these steps. Here, we will label males as 1 and females as 2.

1. For the variable gender, click on the ellipsis (see Figure A.10) to open the Value Labels dialog box.

2. Enter a value for the variable—in this case, 1 for males.

3. Enter the value label for the value, which is male.

4. Click Add.

5. Do the same for female and value 2. When you finish your business in the Define Labels dialog box (see Figure A.12), click OK, and the new labels will take effect.

When you select View → Variable Labels from the main menu (in the Data View), you will see the labels in the Data Editor. Notice that the value of the entry in Figure A.13 is actually 2, even though the label in the cell reads Female.

Figure A.12 ⬢ The completed Value Labels dialog box

Opening a Data File

Once a file is saved, you have to open or retrieve it when you want to use it again. The steps are simple:

1. Click File → Open. You will see the Open Data File dialog box.

2. Find the data file you want to open and highlight it.

3. Click OK.

A quick way to find and open an SPSS file is to click on the Recently Used Data option on the File menu. SPSS lists the most recently used files there.

Printing With SPSS

Here comes information on the last thing you will do once a data file is created. Once you have created the data file you want or completed any type of analysis or chart, you probably will want to print out a hard copy for safekeeping or for inclusion in a report or paper. Then, when your SPSS document is printed and you want to stop working, it will be time to exit SPSS.

Printing is almost as important a process as editing and saving data files. If you cannot print, you have nothing to take away from your work session. You can export data from an SPSS file to another application, but getting a hard copy directly from SPSS is often more timely and more important.

Printing an SPSS Data File

It is simple to print either an entire data file or a selection from one:

1. Be sure that the data file you want to print is the active window.

2. Click File → Print. When you do this, you will see the Print dialog box.

3. Click OK, and whatever is active will print.

As you can see, you can choose to print the entire document or a specific selection (which you will have already made in the Data Editor window), and you can increase the number of copies from 1 to 99 (the maximum number of copies you can print). You can also configure the Print dialog box so that a PDF file is produced.

Printing a Selection From an SPSS Data File

Printing a selection from a data file follows exactly the steps that we listed above for printing a data file, except that in the Data Editor window, you select what you want to print and click on the Selection option in the Print dialog box. The steps go like this:

1. Be sure that the data you want to print are selected.

2. Click File → Print.

3. Click Selection in the Print dialog box.

4. Click OK, and whatever you selected will be printed.

Creating an SPSS Chart

A picture is worth a thousand words, and SPSS offers you just the features to create charts that bring the results of your analyses to life. We’ll go through the steps to create several different types of charts and provide examples of different charts. Then, we will show you how to modify a chart, including adding a chart title, adding labels to axes, modifying scales, and working with patterns, fonts, and more. For whatever reason, SPSS uses the words graphs and charts interchangeably.

Creating a Simple Chart

The one thing that all charts have in common is that they are based on data. Although you may import data to create a chart, in this example, we will use the data from Appendix C to create a bar chart (like the one you saw in Figure A.5) of the number of males and females in each group.

Creating a Bar Chart.

The steps for creating any chart are basically the same. You first enter the data you want to use in the chart, select the type of chart you want from the Graphs menu, define how the chart should appear, and then click OK. Here are the steps we followed to create the chart you see in Figure A.5:

1. Enter the data you want to use to create the chart.

2. Click Graphs → Legacy Dialogs → Bar. When you do this, you will see the Bar Charts dialog box you see in Figure A.14.

3. Click Simple.

4. Click Summaries for groups of cases.

5. Click Define. When you do this, you will see the Define Simple Bar: Summaries for Groups of Cases dialog box.

6. Click Cum n of cases.

7. Click gender. Then move the variable to the Category Axis area by dragging it.

8. Click OK, and you see the results of the chart in Figure A.15.

That’s just the beginning of making the chart. To make any changes, you have to use the chart editor tools.

Figure A.14 ⬢ The Bar Charts dialog box

Saving a Chart

A chart is only one component of the Viewer window. A chart is part of the output generated when you perform some type of analysis. The chart is not a separate entity that stands by itself, and it cannot be saved as such. To save a chart, you need to save the contents of the entire Viewer. Follow these steps to do that:

1. Click File → Save.

2. Provide a name for the Viewer window.

3. Click OK. The output is saved under the name that you provide with an .spo extension.

Enhancing SPSS Charts

Once you create a chart as we showed you in the previous section, you can edit the chart to reflect exactly what you want to say. Colors, shapes, scales, fonts, and more can be changed. We will be working with the bar chart that was first shown to you in Figure A.15.

Editing a Chart

The first step in editing a chart is to double-click on the chart and then click the maximize button. You will see the entire chart in the Chart Editor window, as shown in Figure A.16.

Figure A.16 ⬢ The Chart Editor window

Working With Titles and Subtitles.

Our first task is to enter a title and subtitle on the chart you saw in Figure A.15:

1. Click the Insert a Title icon on the Toolbar. When you do this, as you see in Figure A.17, you can edit the element Title right on the screen and enter what you wish.

2. To insert a subtitle (or, in fact, as many titles as you like), just keep clicking the Insert a Title icon.

Figure A.17 ⬢ Inserting a title

Working With Fonts.

Once you have created a title or titles, you can work with fonts by double-clicking on the text you want to modify. You will see the Properties dialog box, as shown in Figure A.18. Click on the Text Style tab, and you can make whatever changes you wish.

Figure A.18 ⬢ Working with fonts

Working With Axes.

The x- and y-axes provide the calibration for the independent (usually the x-axis) variable and the dependent (usually the y-axis) variable. SPSS names the y-axis the Scale axis and the x-axis the Category axis. Each of these axes can be modified in a variety of ways. To modify either axis, double-click on the title of the axis.

How to Modify the Scale (y) Axis.

To modify the y-axis, follow these steps:

1. Still in the chart editor? We hope so. Double-click on the axis (not the axis label).

2. Click the Scale tab in the Properties dialog box. When you do this, you will see the Scale Axis dialog box, as shown in Figure A.19.

3. Select the options you want from the Scale Axis dialog box.

How to Modify the Category (x) Axis.

Working with the x-axis is no more difficult than working with the y-axis. Here is how the x-axis was modified:

1. Double-click on the x-axis. The Category Axis dialog box opens. It is very similar to the Scale Axis dialog box that you see in Figure A.19.

2. Select the options you want from the Category Axis dialog box.

When you are done, close the Chart Editor by double-clicking on the window icon or selecting File → Close.

Figure A.19 ⬢ The Scale Axis dialog box

Describing Data

Now you have some idea about how data files are created in SPSS. Let’s move on to some examples of simple analysis.

Frequencies and Crosstab Tables

Frequencies simply compute the number of times that a particular value occurs. Crosstabs allow you to compute the number of times that a value occurs when categorized by one or more dimensions, such as gender and age. Both frequencies and crosstabs are often reported first in research reports because they give the reader an overview of what the data look like. To compute frequencies, follow these steps. You should be in the Data Editor window.

1. Click Analyze → Descriptive Statistics → Frequencies. When you do this, you will see the Frequencies dialog box shown in Figure A.20.

2. Double-click the variables for which you want frequencies computed. In this case, they are Test1 and Test2.

3. Click Statistics. You will see the Frequencies: Statistics dialog box shown in Figure A.21.

4. Click the Statistics button.

5. In the Dispersion area, click Std. deviation.

6. Under the Central Tendency area, click Mean.

7. Click Continue.

8. Uncheck the Display frequency tables.

9. Click OK.

The output consists of a listing of the frequency of each value for Test1 and Test2, plus summary statistics (mean and standard deviation) for each, as you see in Figure A.22.

Figure A.20 ⬢ The Frequencies dialog box

Figure A.21 ⬢ The Frequencies: Statistics dialog box

Figure A.22 ⬢ Summary statistics for Test1 and Test2

Applying the Independent-Samples t Test

Independent-samples t tests are used to analyze data from a number of types of studies, including experimental, quasi-experimental, and field studies such as those shown in the following example, where we test the hypothesis that there are differences between males and females in reading.

How to Conduct an Independent-Samples t Test.

To conduct an independent-samples t test, follow these steps:

1. Click Analyze → Compare Means → Independent-Samples T Test. When you do this, you will see the Independent-Samples T Test dialog box, as shown in Figure A.23.

Figure A.23 ⬢ The Independent-Samples T Test frequencies dialog box

The Independent-Samples T Test Dialog Box.

On the left-hand side of the dialog box, you see a listing of all the variables that can be used in the analysis. What you now need to do is define the test and the grouping variable:

· 2. Click Test1 and drag it to the Test Variable(s) area.

· 3. Click Gender and drag it to the Grouping Variable area.

· 4. Click on Gender under the Grouping Variable box.

· 5. Click Define Groups.

· 6. In the Group 1 box, type 1.

· 7. In the Group 2 box, type 2.

· 8. Click Continue.

· 9. Click OK.

The output contains the means and standard deviations for each variable, plus the results of the t test, as shown in Figure A.24.

Figure A.24 ⬢ Results of the simple t test

Exiting SPSS

To exit SPSS, click File → Exit. SPSS will be sure that you get the chance to save any unsaved or edited windows and will then close.

We have just given you the briefest of introductions to SPSS, and certainly none of these skills means anything if you don’t know the value or meaningfulness of the data you originally entered. So, don’t be impressed by your or others’ skills at using programs like SPSS. Be impressed when those other people can tell you what the output means and how it reflects on your original question. And be really impressed if you can do it!

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94	99	86	76
95	99	97	84
91	89	77	87
97	83	80	98
75	94	81	85
78	92	89	94
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90	79	80	92
77	86	83	81

93	85	99	77
94	99	86	76
95	99	97	84
91	89	77	87
97	83	80	98
75	94	81	85
78	92	89	94
76	94	96	94
90	79	80	92
77	86	83	81

93	85	99	77
94	99	86	76
95	99	97	84
91	89	77	87
97	83	80	98
75	94	81	85
78	92	89	94
76	94	96	94
90	79	80	92
77	86	83	81