Cronbach's alpha

Cronbach's alpha (Cronbach's ${\displaystyle \alpha }$) or coefficient alpha (coefficient ${\displaystyle \alpha }$), is a reliability coefficient and a measure of the internal consistency of tests and measures. It was devised by the American psychometrician Lee Cronbach. Today it enjoys such wide-spread usage that numerous studies warn against using Cronbach's alpha uncritically.

In his initial 1951 publication, Lee Cronbach described the coefficient as Coefficient alpha and included an additional derivation. Coefficient alpha had been used implicitly in previous studies, but his interpretation was thought to be more intuitively attractive relative to previous studies and it became quite popular.

To use Cronbach's alpha as an accurate estimate of reliability, the following conditions must be met:

However, under the definition of CTT, the errors are defined to be independent.

This is often a source of confusion for users who might consider some aspect of the testing process to be an "error" (rater biases, examinee collusion, self-report faking). Anything that increases the covariance among the parts will contribute to greater true score variance. Under such circumstances, alpha is likely to over-estimate the reliability intended by the user.

Reliability can be defined as one minus the error score variance divided by the observed score variance:

$${\displaystyle \rho _{XX'}=\left(1-{\sigma _{E}^{2} \over \sigma _{X}^{2}}\right)}$$

Cronbach's alpha is best understood as a direct estimate of this definitional formula with error score variance estimated as the sum of the variances of each "part" (e.g., items or testlets):