On the number of primes for which a polynomial is Eisenstein

Previously Heyman and Shparlinski gave an asymptotic formula with error term for the number of Eisenstein polynomials of fixed degree and bounded height. Let $\psi (f)$ denote the number of primes for which a polynomial $f$ is Eisenstein. We give expressions for the mean and variance of the function $\psi$ for each fixed degree, where the polynomials are ordered according to their height.