Central Tendency and Variability

This chapter begins by revisiting the idea of a “normal” distribution and uses a relevant example of exam grades within a college class to illustrate the importance of understanding central tendency. We then describe the three measures of central tendency—mean, median, and mode—explaining the relevant scientific notation and using a small data set to demonstrate the calculations of each measure. The utility of the mean, median, and mode is discussed in the context of levels of measurement (nominal, ordinal, interval, and ratio) that were learned in Chap. 1 . Histograms are included to illustrate unimodal, bimodal, and uniform distributions within the context of explaining the mode. The mean, median, and mode are also discussed within the context of normal and skewed distributions, with illustrations showing how positive and negative skew impact these measures by pulling the mean in the direction of the skew, leaving the median falling somewhere between the mean and mode. The chapter then moves to a discussion of variability, focusing on the range, variance, and standard deviation. We review leptokurtic and platykurtic distributions and discuss the utility of understanding variability in the context of teaching a course: when all students’ grades are clustered tightly around the mean, we would teach differently than if students’ scores were spread out. In the former scenario, all students have a similar understanding of the material, so teaching can occur at one level for the entire class. In the latter scenario, some students need remediation and others need to be challenged. We then demonstrate calculations of variability using the same small data set that we used to calculate measures of central tendency. Deviation scores, sums of squares, variance, and standard deviation are calculated step-by-step. Using an additional example, we review all concepts and calculations covered in the chapter, and finally we put these concepts back into the context of a real-world example of understanding where one’s own exam grade fits within the distribution of a class. Sidebar notes throughout the text highlight ways to check one’s math and errors to avoid. The chapter concludes with a brief summary of important concepts and a step-by-step illustrated guide to computing measures of central tendency and variability using Excel’s Data Analysis Toolpak. We also provide critical thinking questions and practice problems.