The limiting distribution of the chi-square goodness-of-fit statisticTnunder alternatives is noncentral chi-square if the alternative probabilities approach the null probabilities at an appropriate rate asn→ ∞. It is shown that for fixed alternative probabilities, the limiting distribution of (Tn− μn)/σnis standard normal. Both of these asymptotic results can be used to approximate the power of the goodness-of-fit test. Numerical comparisons between these two approximations indicate that for large values of the true power, the normal approximation is best, but for moderate values of power, the chi-square approximation is best.