The original formulation for Cronbach’s alpha takes this form:
\[\alpha = \frac{N}{N-1}\left(1 - \frac{\sum_{i=1}^{N} \sigma_{y_{i}}^{2}}{\sigma_{T}^{2}}\right)\], where $N$ represents the number of items in the measure, $\sigma_{y_{i}}^{2}$ is the variance associated with each item $i$, and $\sigma_{T}^{2}$ is the variance associated with the total scores.
It’s often written in this form:
\[\alpha = \frac{N\bar{c}}{\bar{v} + (N-1)\bar{c}}\], where $\bar{c}$ is the average of the inter-item covariance, and $\bar{v}$ is the average variance.
To get the above, note that:
\[\bar{c} = \frac{\sum_{i < j} COV(y_{i}, y_{j})}{{N \choose 2}} = \frac{2\sum_{i < j} \sigma_{y_{i}y_{j}}}{N(N-1)}\]Tune in next time for Generalizability theory and SEM as alternatives to Cronbach alpha.