  We prove that if $p$ is an odd prime, $G$ is a solvable group, and the
average value of the irreducible characters of $G$ whose degrees are not
divisible by $p$ is strictly less than $2(p+1)/(p+3)$, then $G$ is
$p$-nilpotent. We show that there are examples that are not $p$-nilpotent where
this bound is met for every prime $p$. We then prove a number of variations of
this result.
