Let’s say you have a test and it is composed of a bunch of different questions. Or you have a set of statistics–say statistics measuring how good a basketball player is–and you want to know how closely these statistics reliably measure the same thing. What you are looking for is a way to measure test (or statistic) reliability.
One common way to do this is with Cronbach’s alpha. If you have two measures, a simple way to measure reliability is with a correlation. However, consider the case where you have 3 measures. How would you measure relability? The correlation of measure 1 with 2? 2 with 3? 1 with 3?
Cronbach’s alpha basically is a scaled measured of the average correlation across questions. Mathematically, it can be calculated as:
where K is the number of questions and ř is the average correlation (not counting the correlations along the diagonal of a correlation matrix which are always 1). If all questions are perfectly correlated (i.e., ř=1), then α=1. If the average correlation is 0, then α=0 as well.
An equivalent way to calculate Cronbach’s Alpha is:
The last term is the mean squared residuals across individuals, while the first term is the sum of squared residuals not explained by variation in average scores across questions for each person or average person’s score across questions. If there is more variability in total scores across individuals, then small differences in the error residuals are determined to be more reliable. If individuals have more similar overall test scores, then smaller unexplain differences in residuals appear less reliable.
Hopefully this explanation was helpful. What is probably more helpful, however, is if you download an Excel sheet with my calculations here. All statistical packages (R, SAS, Stata, SPSS) calculate Cronbach’s Alpha, but it’s good to know what this is doing when you type in that command.