# Cronbach's Alpha

*Cronbach's alpha* is an estimate of the squared correlation of the estimated values of a sample obtained using a Multi-Item Scale and their true values (e.g., the squared correlation between the average IQ as measured in an IQ test and the true intelligence). (Note that the 'squared correlation' is equivalent to the proportion of variance explained.)

## Calculation

[math]\displaystyle{ \alpha = {K \over K-1 } \left(1 - {\sum_{i=1}^K \sigma^2_{Y_i}\over \sigma^2_X}\right) }[/math]

where [math]\displaystyle{ K }[/math] is the number of items, [math]\displaystyle{ \sigma^2_X }[/math] is the variance of the sum of all of the items, and [math]\displaystyle{ \sigma^2_{Y_i} }[/math] is the variance of the *i*th item.

## Interpretation

A score of 0.95 or greater implies that a measurement is acccurate enough to be regarded as the truth. In practice measurements of around 0.7 are often regarded as acceptable for real-world applications. Values of substantially less than this can be useful if all that is being done is comparing the multi-item scale's estimates by sub-groups (e.g., comparing differences by gender).