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
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
For a proof, see Kendall et al. (1998).
See also
Bin, Class Interval, Histogram
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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.
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Cite this as:
Weisstein, Eric W. "Sheppard's Correction." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SheppardsCorrection.html