Two different Cronbachs Alpha On the wikipedia artikle you have some randomvariables $Y_1, \dots, Y_K$ and $X = \sum_{i=1}^K Y_i$. Then Cronbach' $\alpha$ is definied as
$$\alpha = \frac{K}{K-1} \left(1 - \frac{\sum_{i=1}^K \sigma^2_{Y_i}}{\sigma^2_X} \right)$$
Alternativly its defined as
$$ \alpha = \frac{K \cdot r}{v + (K-1) r}$$
where $r$ is the average of all covariances between $Y_i$ and $Y_j$ with $i\neq j$ and $v$ is the average variance from all $Y_i's$. 
I have no idea why these two formulars are thought to be similiar at all. Can someone explain it to me?
 A: The equivalence uses the fact that the variance $V(\sum_i Y_i)$ is equal to $\sum_i V(Y_i) + \sum _{i \neq j} \operatorname{Cov}(Y_i,Y_j)$.  In detail, the first definition is equivalent to 
$$ \frac{K}{K-1} \frac{ V(X) - \sum_i V(Y_i)}{V(X)} $$
Replacing $V(X)$ with $\sum _iV(Y_i) + \sum _{i \neq j} \operatorname{Cov}(Y_i,Y_j)$ and simplifying gives
$$ \frac{K}{K-1} \frac{\sum_{i\neq j} \operatorname{Cov}(Y_i,Y_j)}{\sum_i V(Y_i) + \sum _{i\neq j} \operatorname{Cov}(Y_i,Y_j) }$$
There are $K$ different $Y_i$s, so $v= \frac{1}{K}\sum_i V(Y_i)$ and the first sum in the denominator of the second fraction is $Kv$.  Similarly there are $K(K-1)$ covariances $\operatorname{Cov}(Y_i,Y_j)$ (you can choose $i$ to be any number from 1 to $K$, and $j$ can then be anything except $i$, so there are $K-1$ possibilities for $j$, giving $K(K-1)$ possibilities in total), so $r= \frac{1}{K(K-1)} \sum_{i\neq j} \operatorname{Cov}(Y_i,Y_j)$ and the numerator of the second fraction is $K(K-1)r$.  Therefore we have
$$ \frac{K}{K-1} \frac{K(K-1)r}{Kv+K(K-1)r}$$
Cancelling a $K-1$ from the top and a $K$ from the bottom, you get the second definition.
