Measures of Variability

CHAPTER 4 Measures of Variability Chapter 3 introduced descriptive statistics, the purpose of which is to numerically describe or summarize data for a Chapter Outline variable. As we learned in Chapter 3, a set of data is often 4.1 An Example From the described in terms of the most typical, common, or frequently Research: How Many “Sometimes” occurring score by using a measure of central tendency such in an “Always”? as the mode, median, or mean. Measures of central ten- 4.2 Understanding Variability dency numerically describe two aspects of a distribution of scores, modality and symmetry, based ondistribute what the scores 4.3 The Range have in common with each other. However, researchers • Strengths and weaknesses of also describe the amount of differences among the scores of a the range variable in analyzing and reportingor their data. This chapter 4.4 The Interquartile Range introduces measures of variability, which are designed to rep- • Strengths and weaknesses of resent the amount of differences in a distribution of data. In the interquartile range this chapter, we will present and evaluate different measures 4.5 The Variance (s2) of variability in terms of their relative strengths and weak- • Definitional formula for the nesses. As in the previous chapters, our presentation will variance depend heavily on the findings and analysis from a published • Computational formula for the research study.post, variance • Why not use the absolute value of the deviation in 4.1 An Example From the calculating the variance? Research: How Many • Why divide by N – 1 rather than “Sometimes” in an “Always”? N in calculating the variance? 4.6 The Standard Deviation (s) copy,Throughout our daily lives, we are constantly asked to fill out • Definitional formula for the questionnaires and surveys. Restaurants ask us to evaluate the standard deviation quality of the food and service they provide; companies want • Computational formula for the to know how often we shop online; political organizations are standard deviation interested in knowing what we think about our elected offi- 4.7 Measures of Variabilitynot for cials. In responding to these types of scenarios, we are often Populations asked to describe our feelings, behaviors, or beliefs by choos- • The population variance (σ2) ing among comparative terms such as excellent, good, or poor. • The population standard To what degree do words such as these have the same mean- deviationDo (σ) ing for different people? Researchers who collect data using questionnaires are concerned about the clarity of the words ww 104 Copyright ©2016 by SAGE Publications, Inc. This work may not be reproduced or distributed in any form or by any means without express written permission of the publisher. they use. If research participants do not agree about the mean- ing of key terms and phrases, it is difficult to combine or com- • Measures of variability for pare their responses. The following example demonstrates how samples vs. populations important it is to understand and address these differences when 4.8 Measures of Variability: research is conducted with people of different backgrounds, cul- Drawing Conclusions tures, or languages. As members of an international research team, two British 4.9 Looking Ahead researchers, Suzanne Skevington and Christine Tucker, con- 4.10 Summary ducted a study assessing differences in interpretation of labels for 4.11 Important Terms questionnaire items measuring the frequency of health-related 4.12 Formulas Introduced in This behaviors, such as, “How often do you suffer pain?” (Skevington Chapter & Tucker, 1999). As a part of their study, they examined how British people define frequency-related terms such as seldom, 4.13 Using SPSS usually, and rarely. To this end, the researchers assigned numeric 4.14 Exercises values to participants’ evaluation of these words in order to measure the degree to which people differ in their interpretation of these words. distribute In their study, Skevington and Tucker (1999) presented 20 British adults a series of frequency-related words. For each word, they included a line 100 millimeters (about 4 inches) in length. At opposite ends of the line were attached the orlabels Never and Always, representing the lowest and highest possible values for frequency. After reading each word, participants were asked to place an X on the point in the line that in their opinion best represented the frequency of the word, relative to the end points of Never (0%) and Always (100%). An example of their methodology is provided in Figure 4.1. After the participants had completed their responses, the researchers used a ruler to measure, for each word, the distance from the word Never to the “X” provided by the participant. This distance, which was measured in millimeters, representedpost, the perceived “frequency” of the word. For example, in Figure 4.1, the X for the word sometimes is in the exact middle of the line; consequently, this word is given a frequency rating of 50% because it is 50 mm from the left end. In this study, the variable, which we will call “frequency rating,” had possible values ranging from 0 to 100. In this example, we will focus on the study’s findings regarding the word sometimes. The frequency ratings for the word sometimes for the 20 participants are listed in Table 4.1(a). Table 4.1(b) organizes the raw data into a grouped frequency distribution table, and Figure 4.2 illustrates the distribution using a frequency polygon. A frequency polygon is used for this variable (rather than a histogram or pie chart) because copy,the variable “frequency rating” is measured at the ratio level of measurement, with the value of 0 representing the complete absence of distance from the left end of the 100-mm line. As was discussed in Chapter 3, the data for a variable can be summarized using a measure of central tendency. Using Formula 3-3 from Chapter 3, the mean frequency rating of the word some- timesnot for the 20 participants is calculated as follows: ΣX X = N ++++ ++ = 46 15 21... 22 66 26 = 666 Do 20 20 = 33.30 Chapter 4 | Measures of Variability 105 Copyright ©2016 by SAGE Publications, Inc. This work may not be reproduced or distributed in any form or by any means without express written permission of the publisher. Figure 4.1 Example of Ratings of Frequency-Related Words Along a 100-mm Line Sometimes Never Always Seldom Never Always Usually Never Always Rarely Never Alwaysdistribute From this calculation, we conclude that the participants in the study believed, on average, the word sometimes represents a 33.30% frequency. However, the grouped frequency distribution table in Table 4.1(b) reveals that participants provided a wide range of ratings,or some of which were far from the average rating of 33.30%. For example, the ratings provided by 11 of the 20 participants were lower than 30%, while 6 of the 20 participants provided ratings of 40% or higher. Given the variety of ratings these participants gave the word sometimes, in order to accurately describe the distribution of responses for this variable, one must not only include a measure of central tendency (which focuses on what the scores have in common with each other) but also represent the degree to which the scores differ, or vary. The next section introduces the concept of variability. post, 4.2 Understanding Variability We begin our discussion of variability by examining the three distributions presented in Figure 4.3. Chapter 2 discussed three aspects of distributions: modality, symmetry, and variability. The modality and symmetry of a distribution can be described using measures of central tendency such as those introduced in copy,Chapter 3. However, because all three distributions in Figure 4.3 are unimodal and symmetric, they would be described as having the same mode, median, and mean. It is obvious, however, that the shapes of the three distributions are very different. As the remainder of this chapter will illustrate, these differences can be portrayed using a statistic that describes the third aspect of distributions:not variability. The word variability typically evokes words such as differences, dispersion, or changes. In a statis- tical sense, variability refers to the amount of spread or scatter of scores in a distribution. The concept of variability is a critical issue in the behavioral sciences, where research frequently examines differences in such things as characteristics, attitudes, and cognitive abilities. Different people, for example,Do express different levels of extroversion, different attitudes toward capital punishment, and 106 Fundamental Statistics for the Social and Behavioral Sciences Copyright ©2016 by SAGE Publications, Inc. This work may not be reproduced or distributed in any form or by any means without express written permission of the publisher. Table 4.1 Frequency Rating of the Word Sometimes by 20 Participants (a) Raw Data Frequency Frequency Frequency Participant Rating Participant Rating Participant Rating 1 46 8 27 15 49 2 15 9 20 16 32 3 21 10 23 17 45 4 49 11 58 18 22 5 23 12 24 19 66 6 39 13 36 20 26 7 25 14 20 distribute (b) Grouped Frequency Distribution Table or Frequency Rating f % 70–100 0 0% 60–69 1 5% 50–59 post,1 5% 40–49 4 20% 30–39 3 15% 20–29 10 50% 10–19 1 5% 0–9 copy, 0 0% Total 20 100% different learning styles. Ultimately, the primary goal of a science such as psychology is to describe, understand,not explain, and predict variability. Researchers have developed statistics designed to measure variability. A measure of variability is a descriptive statistic of the amount of differences in a set of data for a variable.

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