Suppose Xj,i = 1,..., n are independent and identically distributed with E\X { \ r < oo, r = 1, 2 If Cov ((X -/*)', S 2 ) = 0 for r = 1, 2 , . . . , where /x = EX,, S 2 = £" = 1 (X, -X) 2 /(n -1), and X = Y11=i Xi/ n ' t n e n w e show that X, ~ ,yK(^,<7 2 ), where a 2 = Var(X,). This covariance zero condition characterizes the normal distribution. It is a moment analogue, by an elementary approach, of the classical characterization of the normal distribution by independence of X and S 2 using semiinvariants. 2000 Mathematics subject classification: primary 60E05, 62E10.