Differences Between a Distribution of Scores and Dispersion of Scores

R.W. asked me to distinguish between a distribution and dispersion. When we speak of distributions, we normally refer to its shape (e.g., normal, skewed, bimodal, leptokurtic, etc.) to describe the nature of the respondents’ scores. Using one of the examples I just gave (and, for PS 205, covered in class), a leptokurtic distribution indicates uniformity or homogeneity of scores with respect to the variable under study (such as test scores in a class). Nothing exact here – we’re just eyeballing the shape of the distribution. However, if we want to specify exactly how much variability around a mean there is in any distribution of scores, we speak of dispersion as measured or quantified by the range or, better yet, standard deviation.

Example: Three different classes took my statistics exam. Each improbably earned the same mean, say 70, but were shaped much differently according to the following distributions:

While their respective shapes tell us that one class performed more uniformly than the other two (as evinced by a leptokurtic distribution), and another class revealed a lack of uniformity in their scores (a platykurtic distribution), we need a measure of variability to quantify precisely how much dispersion exists in each distribution. That’s where the standard deviation (s) comes in. The smaller the s, the smaller the dispersion around the mean and the more likely its shape will tend towards being leptokurtic (or homogeneous). Naturally, the opposite holds true. The larger the s, the larger the dispersion around the mean and the more likely its shape will tend towards being platykurtic (or heterogeneous).

In short, three classes with the same mean will reveal three different distributions (shapes) only when their respective s values (i.e., dispersion measured by the standard deviation) differ sizably in the ways I just described.