Let E be an elliptic curve defined over Q and let N be a positive integer. Now, M_E(N) counts the number of primes p such that the group E_p(F_p) is of order N. In an earlier joint work with Balasubramanian, we showed that M_E(N) follows Poisson distribution when an average is taken over a large class of curves. In this talk we show how this result can be improved so that the same averaging result holds for a larger range of elliptic curves.