A STATISTICAL ODDITY

The standard deviation is almost always smaller than the variance, as the former is equal to the square root of the latter. For example, if the variance is 25, SD = 5. But consider what happens if the variance is between 0 and 1. In these fully legitimate situations, SD > variance.

Do real data ever produce variances and SDs that are smaller than 1.00? Most certainly! Here are 5 examples:

  •  Statements in an attitude inventory typically are set up in a Likert format, with response-options extending from “strongly agree” to “strongly disagree” and scored 1 through 5. The variance and SD of the responses to any given item usually are < 1.00.
  • The correlation between 2 variables is often computed separately for subgroups, with the mean r reported along with the SD or variance of the correlation coefficients. This SD or variance must be < 1.00.
  • Both the SD and variance of a set of proportions necessarily will turn out to be < 1.00
  • In test-development efforts, the mean item discrimination for subgroups of items (or for all items) is often reported, along with the SD or variance of these item indices. Because item discrimination can range from 0-to-1, the variability of a set of these values must be < 1.00.
  • In meta-analysis investigations that combine the results of different factor analytic studies, the mean factor loading is sometimes reported for each factor along with the SD or variance (that must be < 1.00) of the given factor’s loadings.
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