Linear statistics, a random variable built out of the sum of the evaluation of functions at the eigenvalues of a N ×N random matrix, Σ f(x) or tr f(M), is an ubiquitous statistical characteristics in random matrix theory. Hermitian random matrix ensembles, under the eigenvalue-eigenvector decompositions give rise to the joint probability density functions of N random variables. We show that if f(·) is a polynomial of degree K, then the variance of tr f(M) is of the form of Σ n(d) and d is related to the expansion coefficients c of the polynomial f(x) = Σ c P(x), where P(x) are polynomials of degree n, orthogonal with respect to the weights [equation presented here], (0 < a < x < b < 1), respectively.