Cronbach's alpha measures how well a set of items (or variables) measures a single unidimensional latent construct. When data have a multidimensional structure, Cronbach's alpha will usually be low. 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 and r-bar is the average inter-item correlation among the items.
All this sounds somewhat difficult to understand. So what does this mean?
If the inter-item correlations are high, then there is evidence that the items are measuring the same underlying construct. This is really what is meant when someone says they have "high" or "good" reliability. They are referring to how well their items measure a single unidimensional latent construct.
Reliability coefficient of .70 or higher is considered "acceptable" in most Social Science research situations.
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