Week 3 - Assignment: Analyze Numerical Summaries of Data Using Excel and SPSS

Using Descriptive Statistics In This Chapter

▶ Defining levels of measurement

▶ Looking at frequencies for categorical and continuous variables

▶ Using the descriptives procedure to summarize continuous variables

S ummaries of individual variables provide the basis for more complex analysis (as you see in the next few chapters). They also help establish

base rates, answer important questions (for example, the percent of satisfied customers), allow users to check sample size and the data for unusual cases or errors, and provide insights into ways in which you may combine differ- ent groups. Ideally, you want to obtain as much information as possible from your data. In practice, however, given the measurement level of the variables, only some information is meaningful.

In this chapter, we begin by discussing level of measurement. Next, we run the frequencies procedure to obtain summary statistics for both categorical and continuous variables. Finally, we use the descriptives procedure to sum- marize continuous variables.

Looking at Levels of Measurement The level of measurement of a variable determines the appropriate statistics, and graphs that can be used to describe the data. For example, if we have a variable like marital status, it wouldn’t make sense to ask for the mean of this variable; instead, we may ask for the percentages associated with the different categories. In addition, level of measurement also determines the kind of research questions we can answer, so it’s a critical step in the research process.

The term levels of measurement refers to the coding scheme or the meaning of the numbers associated with each variable. Many statistical techniques

Chapter 13

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202 Part V: Analyzing Data are appropriate only for data measured at particular levels or combinations of levels. Different statistical measures are appropriate for different types of variables, and the statistical summaries depend on the level of measurement.

Defining the four levels of measurement Introductory statistics textbooks present four levels of measurement, each defined by certain properties. Each successive level can be said to contain the properties of the preceding types and records information at a higher level. The four levels of measurement are as follows:

✓ Nominal: For nominal data, each value represents a category. There is no inherent order to the categories. For example, the variable gender may be coded as 0 (male) and 1 (female), but all these values tell us is that we have two distinct categories, not that one category has more or less or is better or worse than the other.

✓ Ordinal: For ordinal data, each value is a category, but there is a mean- ingful order or rank to the categories. However with ordinal data, there is not a measurable distance between categories. For example, if we’re measuring the outcome of a foot race, we can determine which con- testant came in first, second, third, and so on. However, based on the ranking, we can’t tell how much faster each competitor was compared to the others, nor can we say that the difference between first and second place is the same as the difference between second and third place. Other examples of ordinal variables are attitudinal questions with cat- egories, such as Strongly Disagree (1), Disagree (2), Neutral (3), Agree (4), and Strongly Agree (5), or variables such as income coded into cat- egories representing ranges of values.

✓ Interval: For interval data, a one‐unit change in numeric value repre- sents the same change in quantity regardless of where it occurs on the scale. For example, for a variable like temperature measured in Fahrenheit, the difference between 20 degrees and 21 degrees (1 unit) is equal to the difference between 50 degrees and 51 degrees. In other words, they have equal intervals between points on the scale.

✓ Ratio: For ratio data, you have all the properties of interval variables with the addition of a true zero point, representing the complete absence of the property being measured. For example, temperature measured in Fahrenheit is measured on an interval scale, because zero degrees does not represent the absence of temperature. However, a variable like number of purchases is a ratio variable because zero indi- cates no purchases. Ratios can then be calculated (for example, eight purchases represents twice as many purchases as four purchases).

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203 Chapter 13: Using Descriptive Statistics These four levels of measurement are often combined into two main types:

✓ Categorical: Nominal and ordinal measurement levels

✓ Continuous (or scale): Interval and ratio measurement levels

Defining summary statistics The most common way to summarize variables is to use measures of central tendency and variability:

✓ Central tendency: One number that is often used to summarize the distri- bution of a variable. Typically, we think of central tendency as referring to the “average” value. There are three main measures of central tendency:

• Mode: The category or value that contains the most cases. This measure is typically used on nominal or ordinal data and can easily be determined by examining a frequency table.

• Median: The midpoint of a distribution; it is the 50th percentile. If all the cases for a variable are arranged in order according to their value, the median is the value that splits the data into two equally sized groups.

• Mean: The mathematical average of all the values in the distribu- tion (that is, the sum of the values of all cases divided by the total number of cases).

✓ Variability: The amount of spread or dispersion around the measure of central tendency. There are a number of measures of variability:

• Maximum: The highest value for a variable.

• Minimum: The lowest value in the distribution.

• Range: The difference between the maximum and minimum values.

• Variance: Provides information about the amount of spread around the mean value. It’s an overall measure of how clustered data values are around the mean. The variance is calculated by summing the square of the difference between each value and the mean and dividing this quantity by the number of cases minus one. In general terms, the larger the variance, the more spread there is in the data; the smaller the variance, the more the data values are clustered around the mean.

• Standard deviation: The square root of the variance. The variance measure is expressed in the units of the variable squared. This can cause difficulty in interpretation, so more often, the standard devi- ation is used. The standard deviation restores the value of variabil- ity to the units of measurement of the original variable.

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204 Part V: Analyzing Data We care about level of measurement because it determines appropriate sum- mary statistics and graphs to describe the data. Table 13-1 summarizes the most common summary statistics and graphs for each of the measurement levels used by SPSS.

Focusing on Frequencies for Categorical Variables

The most common technique for describing categorical data — nominal and ordinal levels of measurement — is to request a frequency table, which provides a summary showing the number and percentage of cases falling into each category of a variable. Users can also request additional summary statistics like the mode or median, among others.

Here’s how to run the frequencies procedure so you can create a frequency table that will allow you to obtain summary statistics for categorical variables:

1. From the main menu, choose File ➪ Open ➪ Data and load the merchandise.sav data file.

The file is not in the SPSS installation directory. You have to download it from this book’s companion website.

It contains the customer’s purchase history and has 16 variables and 3,338 cases.

Table 13-1 Level of Measurement and Descriptive Statistics Nominal Ordinal Scale

Definition Unordered categories

Ordered categories

Numeric values

Examples Gender, geo­ graphic location, job category

Satisfaction ratings, income groups, rank­ ing of prefer­ ences

Number of purchases, cholesterol level, age

Measures of central tendency

Mode Mode, median Mode, median, mean

Measures of dispersion

None Min/max/range Min/max/ range, standard deviation/ variance

Graph Pie or bar Pie or bar Histogram

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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205 Chapter 13: Using Descriptive Statistics 2. Choose Analyze ➪ Descriptive Statistics ➪ Frequencies.

The Frequencies dialog box, shown in Figure 13-1, appears.

In this example, we want to study the distribution of the variables Payment_Method (Auto Pay, Check, or Credit Card), Premier (Yes or No), and Status (Current or Churned). You can place these variables in the Variable(s) box and each will be analyzed separately.

3. Select the variables Payment_Method, Premier, and Status, and place them in the Variable(s) box, as shown in Figure 13-2.

If you were to run the Frequencies procedure now, you would get three tables, each showing the distribution of one variable. It’s customary though to request additional summary statistics.

4. Click the Statistics button.

The Frequencies: Statistics dialog box, shown in Figure 13-3, appears.

This dialog box provides many statistics, but it’s critical that you request only those appropriate for the level of measurement of the vari- ables you placed in the Variable(s) box. For nominal variables, only the mode is suitable.

5. In the Central Tendency section, select the Mode check box, as shown in Figure 13-4.

6. Click Continue.

Requesting a graph, so you can have a visual display of the data, can be useful. That’s what we’ll do now.

Figure 13-1: The

Frequencies dialog box.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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206 Part V: Analyzing Data

7. Click the Charts button.

The Frequencies: Charts dialog box, shown in Figure 13-5, appears.

This dialog box has options for pie charts and bar charts. Either type of chart is acceptable for a nominal variable. Charts can be built using either counts or percentages, but normally percentages are a better choice.

Figure 13-3: The

Frequen­ cies:

Statistics dialog box.

Figure 13-2: Place the variables

in the Variable(s)

box.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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207 Chapter 13: Using Descriptive Statistics

8. In the Chart Type section, click the Bar Charts radio button; in the Chart Values section, click the Percentages radio button (see Figure 13-6).

Figure 13-4: Select the

Mode check box.

Figure 13-5: The

Frequen­ cies: Charts dialog box.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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208 Part V: Analyzing Data

9. Click Continue.

10. Click OK.

SPSS runs the frequencies procedure and calculates the summary statis- tics, frequency table, and bar chart you requested.

The Statistics table (shown in Figure 13-7) displays the number of valid and missing cases for each variable requested in the Frequencies procedure.

Be sure to review this table to check the number of missing cases. In this example, we have 3,338 valid cases and we don’t have any missing data.

The Statistics table also displays any additional statistics that were requested. In our case, we asked only for the mode, the category that has the

Figure 13-6: Click Bar

Charts and Percent­

ages.

Figure 13-7: The

Statistics table.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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209 Chapter 13: Using Descriptive Statistics highest frequency, so only the mode is shown for each of the variables. In this example, the mode is represented by values of 3, 1, and 2, respectively, and represents the category of “Credit Card” for Payment_Method, “No” for Premier, and the “Current” group for Status.

We could’ve asked for additional summary statistics like the mean, and the frequencies procedure would’ve produced it. This is why it’s important to understand measurement level and what statistics are relevant.

The Frequency table (shown in Figure 13-8) shows the distribution of the variable Payment_Method (in this case, we focus on the variable Payment_Method because all the other Frequency tables will have similar information). The information in the Frequency table is comprised of counts and percentages. The Frequency column contains counts, or the number of occurrences of each data value. So, for the variable Payment_Method, it’s easy to see why the category “Credit Card” was the mode — 1,926 custom- ers made purchases this way. The Percent column shows the percentage of cases in each category relative to the number of cases in the entire dataset, including those with missing values. In our current example, those 1,926 cus- tomers who paid via credit card account for 57.7% of all customers. The Valid Percent column contains the percentage of cases in each category relative to the number of valid (nonmissing) cases. Because there is no missing data, the percentages in the Percent column and in the Valid Percent column are identi- cal. The Cumulative Percent column contains the percentage of cases whose values are less than or equal to the indicated value. Cumulative percent is useful only for variables that are ordinal.

Bar charts (like the one in Figure 13-9) summarize the distribution that was observed in the Frequency table and allow you to see the distribution. For the variable Payment_Method, more than half of the people fall into the Credit Card category.

Figure 13-8: The

Frequency table for the Payment_

Method variable.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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210 Part V: Analyzing Data

Understanding Frequencies for Continuous Variables

As we have seen, frequency tables show counts and percentages, which are extremely useful when working with categorical variables. However, for continuous variables that have many values, frequency tables become less useful. For example, if we were working with a variable like income, it wouldn’t be very useful to know that there was only one person in the data- set who made $22,222 last year. In this case, it’s likely that each response would have a different value, so the frequency table would be very large and not particularly useful as a summary of the variable.

Instead, if the variables of interest are continuous, the frequencies procedure can be useful because of the summary statistics it can produce. To run the frequencies for continuous variables, follow these steps:

1. From the main menu, choose File ➪ Open ➪ Data and load the merchandise.sav data file.

The file is not in the SPSS installation directory. You have to download it from this book’s companion website.

2. Choose Analyze ➪ Descriptive Statistics ➪ Frequencies.

Figure 13-9: A bar chart

for the Payment_ Method variable.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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211 Chapter 13: Using Descriptive Statistics 3. Select the variables Stereos, TVs, Speakers, Delivery_Problems, Years_as_customer, and Estimated_Revenue, and place them in the Variable(s) box.

4. Deselect the Display Frequency Tables check box, as shown in Figure 13-10.

A warning dialog box appears saying, “You have turned off all output. Unless you select any Output Options this procedure will not be run.” We receive this warning because at the moment nothing is selected. This is okay because we will now select the summary statistics we want to display.

5. Click the Statistics button.

The Frequencies: Statistics dialog box appears.

Several summary statistics are appropriate for scale variables. The statistics can be divided into those summarizing the central tendency, those measuring the amount of variation (dispersion) in the data, differ- ent percentile values you can request, and those statistics assessing the shape of the distribution.

6. In the Central Tendency section, select the Mean, Median, and Mode check boxes; in the Dispersion section, select the Std. Deviation, Minimum, and Maximum check boxes (see Figure 13-11).

Figure 13-10: The

Frequencies dialog box.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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212 Part V: Analyzing Data

7. Click Continue.

8. Click the Charts button.

The Frequencies: Charts dialog box appears.

Figure 13-12: The

Frequen­ cies: Charts dialog box.

Figure 13-11: The

Frequencies: Statistics

dialog box.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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213 Chapter 13: Using Descriptive Statistics 9. Click the Histograms radio button and select the Show Normal Curve

on Histogram check box, as shown in Figure 13-12.

10. Click Continue.

11. Click OK.

SPSS runs the frequencies procedure and calculates the summary statistics and the histogram you requested.

The Statistics table (shown in Figure 13-13) shows that we have 3,338 valid cases and we don’t have any missing data. The Statistics table contains the requested statistics. For example, for the variable Speakers, we can see that the minimum value is 0 and the maximum value is 451. This seems like a very large range of values, so it would be useful to double‐check the data to make sure there are no errors. Likewise, in an ideal world, we would like the mean, median, and mode to be similar, because they’re all measures of central tendency. In this case, note that for the variable Speakers, the mean (51.3), median (36), and mode (4) are very different from each other, which is an indication that this variable is probably not normally distributed (you see why this is important in later chapters).

You can visually check the distribution of these variables with a histogram (see Figure 13-14). A histogram has bars, but, unlike the bar chart, they’re plotted along an equal interval scale. The height of each bar is the count of values falling within the interval. Notice that the lower range of values is trun- cated at 0 and the number of speakers is greatest down toward the lower end of the distribution, although there are some extreme values. The distribution is not normal.

Figure 13-13: The

Statistics table.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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214 Part V: Analyzing Data

Summarizing Continuous Variables with the Descriptives Procedure

The descriptive procedure is an alternative to the frequencies procedure (see the preceding section) when the objective is to summarize continuous variables. The descriptives procedure provides a succinct summary of vari- ous statistics and the number of cases with valid values for each variable included in the table. To use the descriptives procedure, follow these steps:

1. From the main menu, choose File ➪ Open ➪ Data and load the merchandise.sav data file.

The file is not in the SPSS installation directory. You have to download it from this book’s companion website.

2. Choose Analyze ➪ Descriptive Statistics ➪ Descriptives.

The Descriptives dialog box, shown in Figure 13-15, appears.

3. Select the variables Stereos, TVs, Speakers, Delivery_ Problems, Years_as_customer, and Estimated_Revenue, and place them in the Variable(s) box, as shown in Figure 13-16.

Figure 13-14: A histogram

for the Speakers

variable.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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215 Chapter 13: Using Descriptive Statistics

4. Click OK.

SPSS runs the descriptives procedure and calculates the summary statistics.

The minimum and maximum values provide an efficient way to check for values outside the expected range(see Figure 13-17). Likewise, it’s always important to investigate the mean and determine if the value makes sense. Sometimes a mean may be lower or higher than expected, which can indi- cate a problem relating to how the data was coded or maybe even collected. Finally, the last row in the table, Valid N (listwise), gives the number of cases that have a valid value on all the variables appearing in the table. In this example, we have no missing data, so this number isn’t particularly useful for this set of variables. However, it would be useful for a set of variables that you intended to use for a multivariate analysis (an analysis looking at the relationships between many variables).

Figure 13-15: The

Descriptives dialog box.

Figure 13-16: Place the variables

in the Variable(s)

box.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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216 Part V: Analyzing Data

Figure 13-17: The

Descriptive Statistics

table.

McCormick, K., & Salcedo, J. (2015). Spss statistics for dummies. John Wiley & Sons, Incorporated. Created from ncent-ebooks on 2022-06-17 19:12:11.

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