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
(1)
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(2)
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(3)
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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)
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where is the th Bernoulli number, giving
(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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For a proof, see Kendall et al. (1998).