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

$$\mu_2 = {\mu_2}^{(0)}-{\textstyle{1\over 12}} c^2$$ (1)

$$\mu_3 = {\mu_3}^{(0)}$$ (2)

$$\mu_4 = {\mu_4}^{(0)}-{\textstyle{1\over 2}}{\mu_2}^{(0)}+{\textstyle{7\over 240}} c^2,$$ (3)

where $c$ is the Class Interval. If $\kappa_r'$ is the $r$th Cumulant of an ungrouped distribution and $\kappa_r$ the $r$th Cumulant of the grouped distribution with Class Interval $c$, the corrected cumulants (under rather restrictive conditions) are

$$\kappa_r'=\cases{ \kappa_r & for $r$\ odd\cr \kappa_r-{B_r\over r} c^r & for $r$\ even,\cr}$$ (4)

where $B_r$ is the $r$th Bernoulli Number, giving

$$\kappa_1' = \kappa_1$$ (5)

$$\kappa_2' = \kappa_2-{\textstyle{1\over 12}}c^2$$ (6)

$$\kappa_3' = \kappa_3$$ (7)

$$\kappa_4' = \kappa_4+{\textstyle{1\over 120}}c^4$$ (8)

$$\kappa_5' = \kappa_5$$ (9)

$$\kappa_6' = \kappa_6-{\textstyle{1\over 252}}c^6.$$ (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.