# Reliability and Item Analysis Introductory Overview - Cronbach's Alpha

To return to the prejudice example (see Basic ideas), if there are several subjects who respond to our items, then we can compute the variance for each item, and the variance for the sum scale. The variance of the sum scale will be smaller than the sum of item variances if the items measure the same variability between subjects, that is, if they measure some true score. Technically, the variance of the sum of two items is equal to the sum of the two variances minus (two times) the covariance, that is, the amount of true score variance common to the two items.

We can estimate the proportion of true score variance that is captured by the items by comparing the sum of item variances with the variance of the sum scale. Specifically, we can compute:

a = (k/(k-1)) * [1- S(s2i)/s2sum]

This is the formula for the most common index of reliability, namely, Cronbach's coefficient Alpha (a). In this formula, the s2i's denote the variances for the k individual items; s2sum denotes the variance for the sum of all items. If there is no true score but only error in the items (which is esoteric and unique, and, therefore, uncorrelated across subjects), then the variance of the sum will be the same as the sum of variances of the individual items. Therefore, coefficient Alpha will be equal to zero. If all items are perfectly reliable and measure the same thing (true score), then coefficient Alpha is equal to 1. (Specifically, 1-S(si2)/ssum2 will become equal to (k-1)/k; if we multiply this by k/(k-1) we obtain 1.)

Alternative terminology. Cronbach's Alpha, when computed for binary (e.g., true/false) items, is identical to the so-called Kuder-Richardson-20 formula of reliability for sum scales. In either case, because the reliability is actually estimated from the consistency of all items in the sum scales, the reliability coefficient computed in this manner is also referred to as the internal-consistency reliability.