Lowest common correlation for seven variables

Seven random variables $X_1,\dots,X_7$ each have mean $0$ and variance $1$, and every pair of them has the same correlation $\rho$. What is the smallest value $\rho$ can take?

$-\tfrac16$. The variance of the sum is $7+7\cdot 6\,\rho$ and cannot be negative, so $\rho\ge-\tfrac16$. The bound is attained, e.g. by centring seven i.i.d. variables around their sample mean and rescaling, which makes the sum identically zero.