Let be an elliptic curve. For a prime p of good reduction, let be the set of rational points defined over the finite field . We denote by , the number of distinct prime divisors of . We prove that the quantity (assuming the GRH if E is non-CM) distributes normally. This result can be viewed as a “prime analogue” of the Erdős–Kac theorem. We also study the normal distribution of the number of distinct prime factors of the exponent of .