A correction which must be applied to the measured moments  obtained from normally distributed data which have been binned in order to obtain correct estimators  for the population moments . The corrected versions of the second, third, and fourth moments are then

			(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. (1998).

REFERENCES:

Fisher, R. A. Statistical Methods for Research Workers, 14th ed., rev. and enl. Darien, CO: Hafner, 1970.

Kendall, W. S.; Barndorff-Nielson, O.; and van Lieshout, M. C. Current Trends in Stochastic Geometry: Likelihood and Computation. Boca Raton, FL: CRC Press, 1998.

Kenney, J. F. and Keeping, E. S. "Sheppard's Correction for Grouping Errors." §7.6 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 95-96, 1962.

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.

Stuart, A.; and Ord, J. K. Kendall's Advanced Theory of Statistics, Vol. 1: Distribution Theory, 6th ed. New York: Oxford University Press, 1998.

Whittaker, E. T. and Robinson, G. "Sheppard's Corrections." §99 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 194-196, 1967.