Sheppard's Correction
A correction which must be applied to the Moments computed from Normally Distributed data which have been binned. The corrected versions of the second, third, and fourth moments are
 |  |  | (1) |
 | |  | (2) |
 | |  | (3) |
where
is the Class Interval. If 
is the 
th Cumulant of an ungrouped distribution and
the th Cumulant of the grouped distribution with Class Interval , the corrected cumulants(under rather restrictive conditions) are | (4) |
is the th Bernoulli Number, giving | |  | (5) |
 | |  | (6) |
 | |  | (7) |
 | |  | (8) |
 | |  | (9) |
 | |  | (10) |
For a proof, see Kendall et al. (1987).
References
Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall's Advanced Theory of Statistics, Vol. 1: Distribution Theory, 6th ed. New York: Oxford University Press, 1987. Kenney, J. F. and Keeping, E. S. ``Sheppard's Correction.'' §4.12 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 80-82, 1951.
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