Cronbach's Alpha

We use Cronbach's alpha to evaluate the unidimensionality of a set of scale items. It's a measure of the extent to which all the variables in your scale are positively related to each other. In fact, it is really just an adjustment to the average correlation between every variable and every other.

The formula for alpha is this:

In the formula, K is the number of variables, and r-bar is the average correlation among all pairs of variables.

People always want to know what's an acceptable alpha. Nunnally (1978) offered a rule of thumb of 0.7. More recently, one tends to see 0.8 cited as a minimum alpha. 

Another way to think about alpha is that it is the average split-half reliability for all possible splits. A split half reliability is obtained by taking, at random, half of the variables in your scale, averaging them into a single variable and then averaging the remaining half, and correlating the two composite variables. The expected value for the random split-half reliability is alpha.

Some issues with Cronbach's Alpha

First, while unidimensional scales should have a high alpha, having a high alpha doesn't guarantee unidimensionality. If you have a set of 20 variables, and 10 are very highly with each other, and other 10 are also highly correlated among themselves, but there is low correlation between the two sets, the Alpha can still be quite high. So if alpha is low, then you don't have a unidimensional scale. If alpha high, you may or may not. It's a weak test. 

Second, Alpha gets bigger with the number of variables. Even using items with poor internal consistency you can get an apparently "reliable" scale if your scale has enough items. For example, 10 items that have an average inter-item correlation of only .2 will produce a scale with a reliability of .714. Similarly, if the average correlation among 5 variables is .5, the alpha coefficient will be 0.833. But if the number of variables is 10 (with the same average correlation), the alpha coefficient will be 0.909.

References

• Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334.

• Nunnally, J. C. (1978). Psychometric theory (2nd ed.). New York: McGraw-Hill.