## 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)

where 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.