Estimating true standard deviations

A basic statistic that students learn about in their classes is the standard deviation. Like any statistic, standard deviations are influenced by systematic factors and randomness. I propose that researchers should report “corrected” or “true” standard deviations and I show how to calculate them. The notion of “true” statistics comes out of classical test theory (see Lord and Novick, 1968; Gulliksen, 1987 for reviews). This theory commences with the definition of a “true score”—as the expectation across an infinite set of independent responses—and with an assumption that an observed score equals the true score plus error (X = T + E). Thus, measures of constructs necessarily include random variance, as well as non-random variance. Many statistics—possibly the most famous of which is the correlation coefficient— are influenced by random measurement error (e.g., Spearman, 1904). The deleterious effects of random measurement error are well known, and many statistical packages contain provisions for correcting correlation coefficients, so these corrected correlations can be used in complex path analyses and structural equation analyses, thereby increasing their accuracy (Skrondal and Rabe-Hesketh, 2004). In addition, Baguley (2009) has addressed the correction of effect sizes. Given that it is widely accepted that “corrected” or “true” statistics, uncontaminated by random measurement error, are necessary for complex analyses such as those mentioned above, why not obtain them even for simple cases such as standard deviations? According to the classical theory, the reliability of a measure (ρXX′) equals the ratio of the true score variance (symbolized as σ 2 T) to observed score variance (symbolized as σ 2 X) or ρXX′ = σ 2 T σ 2 X .