Tag Info

Cronbach's $\alpha$ is an estimate of the reliability (specifically, of one its aspect called internal consistency or item-item homogeneity) of a unidimensional test, a construct. It is used mostly in psychometrics and occasionally in other disciplines.

Assume that a total test score $X$ is a sum of scores on $K$ separate items (which could be "actual" test items, or subscales, or different raters - anything that purports to measure the same underlying construct):

$$X=Y_1+\dots+Y_K.$$

Then Cronbach's $\alpha$ is defined as follows:

$$\alpha:=\frac{K}{K-1}\left(1-\frac{1}{\sigma_X^2}\sum_{i=1}^K\sigma_{Y_i}^2\right),$$

where $\sigma_{Y_i}^2$ and $\sigma_X^2$ denote the variance of the $i$-th item and the total score, respectively. Theoretically, $0\leq\alpha\leq 1$, but estimated $\alpha$ is only constrained by $\alpha<1$ and can take negative values.

This post gives more motivation behind Cronbach's $\alpha$.