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# A note on Cronbach's alpha

The above article suggests rules of thumb for Cronbach's $$\alpha$$ and examples of its use. In particular a value of 0.70 is deemed to be 'satisfactory'.

Cronbach's alpha is defined as

$$\frac{k}{k-1} (1 - \frac{\mbox{Sum of k item variances}}{\mbox{Variance of total scores}} )$$

In particular Bland and Altman (1997) note that

Cronbach's alpha has a direct interpretation. The items in our test are only some of the many possible items which could be used to make the total score. If we were to choose two random samples of k of these possible items, we would have two different scores each made up of k items. The expected correlation between these scores is $$\alpha$$.

R code is available from here for obtaining confidence intervals for Cronbach's alpha. You just need to run this function. An EXCEL spreadsheet also computes in the same way a confidence interval for alpha. This method corresponds to the exact method of Koning AJ and Franses PH (2003). This paper also gives R code in how to compare a pair of Cronbach alphas from two different studies.

Liu and Weng (2009) propose a simple effect size for comparing two Cronbach's alpha based on the same number of tests (items) which is analogous to Cohen's d. It can be computed by this spreadsheet.

* For another interpretation of Cronbach's alpha involving the average correlation and details of a suggested inproved alternative, Raykov's rho, see here.

References

Koning AJ and Franses PH (2003) Confidence intervals for Cronbach's coefficient alpha values. ERIM Report Series Research in Management. ERIM Report Series reference number ERS-2003-041-MKT.

Liu H-Y and Weng L-J (2009) An effect size index for comparing two independent alpha coefficients. British Journal of Mathematical and Statistical Psychology 62 385-400.

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