Suppose there is a probability distribution $p$, over money outcomes. Suppose one draws $N$ times from $p$, with replacement, and takes the average money outcome over these $N$ draws. Call this probabilistic process process "$(N; p)$".
For any $N' > N''$, it is clear that the variance of $(N'; p)$ is smaller than the variance of $(N''; p)$, and that both have the same expected value.
But what happens to skewness? Is the skewness of $(N'; p)$ also always smaller than the skewness of $(N''; p)$?