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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. One thing to keep in mind is that alpha is heavily dependent on the number of items composing the scale. Even using items with poor internal consistency you can get a reliable scale if your scale is long enough. For example, 10 items that have an average interitem 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.

 Avg Corr # of Vars Corr 0.5 1 0.500 0.5 2 0.667 0.5 3 0.750 0.5 4 0.800 0.5 5 0.833 0.5 6 0.857 0.5 7 0.875 0.5 8 0.889 0.5 9 0.900 0.5 10 0.909 0.5 11 0.917 0.5 12 0.923 0.5 13 0.929 0.5 14 0.933 0.5 15 0.938

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.

#### 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.
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