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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

mu^^_2=m_2-1/(12)c^2
(1)
mu^^_3=m_3
(2)
mu^^_4=m_4-1/2m_2+7/(240)c^2,
(3)

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

 kappa_r^'={kappa_r   for r odd; kappa_r-(B_r)/rc^r   for r even,
(4)

where B_r is the rth Bernoulli number, giving

kappa_1^'=kappa_1
(5)
kappa_2^'=kappa_2-1/(12)c^2
(6)
kappa_3^'=kappa_3
(7)
kappa_4^'=kappa_4+1/(120)c^4
(8)
kappa_5^'=kappa_5
(9)
kappa_6^'=kappa_6-1/(252)c^6.
(10)

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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Sheppard's Correction

Cite this as:

Weisstein, Eric W. "Sheppard's Correction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SheppardsCorrection.html

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