info prev up next book cdrom email home

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

$\displaystyle \mu_2$ $\textstyle =$ $\displaystyle {\mu_2}^{(0)}-{\textstyle{1\over 12}} c^2$ (1)
$\displaystyle \mu_3$ $\textstyle =$ $\displaystyle {\mu_3}^{(0)}$ (2)
$\displaystyle \mu_4$ $\textstyle =$ $\displaystyle {\mu_4}^{(0)}-{\textstyle{1\over 2}}{\mu_2}^{(0)}+{\textstyle{7\over 240}} c^2,$ (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 & for $r$\ odd\cr
\kappa_r-{B_r\over r} c^r & for $r$\ even,\cr}
\end{displaymath} (4)

where $B_r$ is the $r$th Bernoulli Number, giving
$\displaystyle \kappa_1'$ $\textstyle =$ $\displaystyle \kappa_1$ (5)
$\displaystyle \kappa_2'$ $\textstyle =$ $\displaystyle \kappa_2-{\textstyle{1\over 12}}c^2$ (6)
$\displaystyle \kappa_3'$ $\textstyle =$ $\displaystyle \kappa_3$ (7)
$\displaystyle \kappa_4'$ $\textstyle =$ $\displaystyle \kappa_4+{\textstyle{1\over 120}}c^4$ (8)
$\displaystyle \kappa_5'$ $\textstyle =$ $\displaystyle \kappa_5$ (9)
$\displaystyle \kappa_6'$ $\textstyle =$ $\displaystyle \kappa_6-{\textstyle{1\over 252}}c^6.$ (10)

For a proof, see Kendall et al. (1987).


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.

© 1996-9 Eric W. Weisstein