A Note on Sheppard's Corrections
C. C. Craig · The Annals of Mathematical Statistics · 1941
As far as the author is aware, H. C. Carver was the first to point out that while the formulae ordinarily given for Sheppard's corrections for central moments are valid for moments computed about the population mean, there are still systematic errors present when they are applied to central moments calculated from any particular grouped frequency distribution [1]. This is due, of course to the fact that the mean of a grouped frequency distribution is in general different from that of the distribution before grouping. For a fixed class interval k, Sheppard's corrections give the average value of a moment about a fixed point of a given order for all the groupings of this class width possible and will fail to do so if the moment in question is calculated for each position of the class limits about a point which varies as the class limits shift. Thus Carver [1] pointed that the commonly used formula (for a continuous variate),