Sheppard's Correction
A correction which must be applied to the measured moments m_k obtained from normally distributed data which have been binned in order to obtain correct estimators mu^^_i for the population moments mu_i. The corrected versions of the second, third, and fourth moments are then
where c is the class interval.
If kappa_r^' is the rth cumulant of an ungrouped distribution and kappa_r the rth cumulant of the grouped distribution with class interval c, the corrected cumulants (under rather restrictive conditions) are
where B_r is the rth Bernoulli number, giving
For a proof, see Kendall et al. (1998).
See also
Bin, Class Interval, HistogramExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Sheppard's CorrectionCite this as:
Weisstein, Eric W. "Sheppard's Correction." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SheppardsCorrection.html