Is there a good expository account of Sheppard's correction, written in a way that any ordinary mathematician can readily follow?
(I've thought of writing a Wikipedia article titled "Sheppard's correction", but I haven't done enough homework on it yet.)
Sheppard's correction can be done for various functionals of a probability distribution. The Wolfram article treats cumulants. Curiously, each correction is $-1$ times the correction I'd expect to do for a uniform distribution. I'm not sure whether to believe this.
Think of this: split the interval $(0,\theta)$ into bins of equal length. For a sample from the uniform distribution on the interval, suppose only the midpoint of the bin is reported. Then the variance of the reported data would be too small, since it would fail to report variability within bins. The variability within bins would be just $1/12$ of the square of the bin length. But Sheppard's correction for normally distributed data tells you to add $-1/12$ of the square of the bin length. With the normal distribution, the error from binning is negatively correlated with the observation; with the uniform distribution it is uncorrelated with the observation. If we believe Wolfram, this multiplication by $-1$ seems to apply to all cumulants, not just the second cumulant. If so, that seems like a mystery to be investigated. I hesitate to believe Wolfram on this.