If F(x) is a probability distribution with zero mean and ρ = integral_(-∞)^∞ ( left bracketing bar x right bracketing bar )^3 d F(x)<∞, where the above integral is a stieltjes integral, then for all x and n, left bracketing bar F_n(x) - Φ(x) - 1/2 right bracketing bar <33/4 ρ/(σ^3 sqrt(n)), where Φ(x) is the normal distribution function, Φ(x) + 1/2 = N(x) in Feller's notation, and F_n(x) = F^(n★)(x σ sqrt(n)) is the normalized n-fold convolution of F(x).