In this paper we conduct a Monte Carlo study to determine the power of Pearson’s overall goodness-of-fit test as well as the “Pearson analog” tests (see Anderson (1994)) to detect rejections due to shifts in variance, skewness and kurtosis, as we vary the number and location of the partition points. Simulations are conducted for small and moderate sample sizes. While it is generally recommended that to improve the power of the goodness-of-fit test the partition points are equiprobable, we find that power can be improved by the use of non-equiprobable partitions.