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) |

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) |

For a proof, see Kendall

**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.

© 1996-9

1999-05-26