What does Cronbach's alpha mean? 

Cronbach's alpha is a measure of internal consistency, that is, how closely related a set of items are as a group.  A "high" value of alpha is often used (along with substantive arguments and possibly other statistical measures) as evidence that the items measure an underlying (or latent) construct. However, a high alpha does not imply that the measure is unidimensional. If, in addition to measuring internal consistency, you wish to provide evidence that the scale in question is unidimensional, additional analyses can be performed. Exploratory factor analysis is one method of checking dimensionality. Technically speaking, Cronbach's alpha is not a statistical test - it is a coefficient of reliability (or consistency). 

Cronbach's alpha can be written as a function of the number of test items and the average inter-correlation among the items.  Below, for conceptual purposes, we show the formula for the standardized Cronbach's alpha:

Here N  is equal to the number of items, c-bar is the average inter-item covariance among the items and v-bar equals the average variance. 

One can see from this formula that if you increase the number of items, you increase Cronbach's alpha.  Additionally, if the average inter-item correlation is low, alpha will be low.  As the average inter-item correlation increases, Cronbach's alpha increases as well (holding the number of items constant).

An example

Let's work through an example of how to compute Cronbach's alpha using SPSS, and how to check the dimensionality of the scale using factor analysis. For this example, we will use a dataset that contains four test items - q1, q2, q3 and q4. You can download the dataset by clicking on alpha.sav. To compute Cronbach's alpha for all four items - q1, q2, q3, q4 - use the reliability command:

  /VARIABLES=q1 q2 q3 q4.

Here is the resulting output from the above syntax:

The alpha coefficient for the four items is .839, suggesting that the items have relatively high internal consistency.  (Note that a reliability coefficient of .70 or higher is considered  "acceptable" in most social science research situations.) 

In addition to computing the alpha coefficient of reliability, we might also want to investigate the dimensionality of the scale. We can use the factor command to do this:

 /VARIABLES q1 q2 q3 q4
Here is the resulting output from the above syntax:

Looking at the table labeled Total Variance Explained, we see that the eigen value for the first factor is quite a bit larger than the eigen value for the next factor (2.7 versus 0.54). Additionally, the first factor accounts for 67% of the total variance. This suggests that the scale items are unidimensional.

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