Table 1.

Natural Exponential Family Distributions With Quadratic Variance Functions and Their Corresponding KR20 and KR21 Estimators, Given by Equations (10) and (11), Respectively.

$Y_{\mathrm{ij}}$ distribution	$V(\theta )$	$\mathrm{KR}20$	$\mathrm{KR}21$
Normal	${\sigma }^2$	$(1-\frac{\sum _{j=1}^k{\sigma }^2}{s_x^2})$	$(1-\frac{k{\sigma }^2}{s_x^2})$
Bernoulli	$\theta -{\theta }^2$	$\frac{k}{k-1}(1-\frac{\sum _{j=1}^k{\overline{y}}_j(1-{\overline{y}}_j)}{s_x^2})$	$\frac{k}{k-1}(1-\frac{k\left(\frac{1}{k}\overline{x}\right)(1-\frac{1}{k}\overline{x})}{s_x^2})$
Poisson	$\theta$	$(1-\frac{\sum _{j=1}^k{\overline{y}}_j}{s_x^2})$	$(1-\frac{\overline{x}}{s_x^2})$
Exponential	${\theta }^2$	$\frac{k}{k+1}(1-\frac{\sum _{j=1}^k{{\overline{y}}_j}^2}{s_x^2})$	$\frac{k}{k+1}(1-\frac{\frac{1}{k}{\overline{x}}^2}{s_x^2})$
Geometric	$\theta +{\theta }^2$	$\frac{k}{k+1}(1-\frac{\sum _{j=1}^k({\overline{y}}_j+{{\overline{y}}_j}^2)}{s_x^2})$	$\frac{k}{k+1}(1-\frac{\overline{x}+\frac{1}{k}{\overline{x}}^2}{s_x^2})$
GHS	$1+{\theta }^2$	$\frac{k}{k+1}(1-\frac{\sum _{j=1}^k(1+{{\overline{y}}_j}^2)}{s_x^2})$	$\frac{k}{k+1}(1-\frac{k(1+\frac{1}{k^2}{\overline{x}}^2)}{s_x^2})$

Note. The sum scores $x_i$ for subject $i$ have sample mean $\overline{x}$ and sample variance $s_x^2$, where sample mean and variance are calculated over subjects. Each test item has response $y_{\mathrm{ij}}$, with $i$ indexing subject and $j$ indexing item, and where ${\overline{y}}_j$ is the mean response for item $j$, again averaging over subjects. The test length is $k$ items and the number of subjects is $n$. Note that the use of the normal density assumes that the noise variance ${\sigma }^2$ around each test item is known, which is unlikely to be the case in practice. Cronbach’s alpha is recommended when the data are believed to be normal. Where possible, terms are canceled to simplify formulas, except in the case of the binomial so as to preserve KR20 and KR21 in their original forms. Also note that because $\sum _{j=1}^k{\overline{y}}_j=\overline{x}$ as shown in Equation (9), the Poisson estimator is identical for both KR20 and KR21. The binomial estimators were first derived in Kuder and Richardson (1937) and the Poisson estimator was first derived in Allison (1978). GHS = generalized hyperbolic secant function.