Cronbach's alpha

Cronbach's $$\alpha$$ (alpha) has an important use as a measure of the reliability of a psychometric instrument. It indicates the extent to which a set of test items can be treated as measuring a single latent variable. It was first named as alpha by Cronbach (1951), although an earlier version is the Kuder-Richardson Formula 20 (often shortened to KR-20), which is the equivalent for dichotomous items, and Guttman (1945) developed the same quantity under the name lambda-2.

Given that a sample was measured on a set of k items, Cronbach's $$\alpha$$ is defined as the mean correlation across the items, adjusted upward by the Spearman-Brown prediction formula by k. It is related to the outcome of an analysis of variance of the item data into variance due to the individuals in the sample and variance due to the items. The higher the proportion of variance due to individuals, the higher Cronbach's $$\alpha$$.

$$\alpha$$ can take values between minus infinity and 1 (although only positive values make sense). As a rule of thumb, a proposed psychometric instrument should only be used if an $$\alpha$$ value of 0.70 or higher is obtained on a substantial sample. However the standard of reliability required varies between fields of psychology: cognitive tests (tests of intelligence or achievement) tend to be more reliable than tests of attitudes or personality. There is also variation within fields: it is easier to construct a reliable test of a specific attitude than of a general one, for example.

Alpha is most appropriately used when the items measure different substantive areas within a single construct (conversely, alpha can be artificially inflated by making superficial changes to the wording within a set of items). Although this description of the use of $$\alpha$$ is given in terms of psychology, the statistic can be used in any discipline.

Formula
Below the formula for the standardized Cronbach's $$\alpha$$:

$$\alpha = {N\cdot\bar r \over (1 + (N-1)\cdot\bar r)}$$

Here N is equal to the number of items and $$\bar r$$ is the average inter-item correlation among the items.