Sheppard's correction

In statistics, Sheppard's corrections are approximate corrections to estimates of moments computed from binned data. The concept is named after William Fleetwood Sheppard.

Let $m_k$ be the measured k^th moment, $\hat{\mu}_k$ the corresponding corrected moment, and $c$ the class interval (bin width). No correction is necessary for the mean (first moment about zero). The first few measured and corrected moments about the mean are then related as follows:

\begin{align} \hat{\mu}_2 &= m_2 - \frac{1}{12} c^2 \\ \hat{\mu}_3 &= m_3 \\ \hat{\mu}_4 &= m_4 - \frac{1}{2} m_2c^2 + \frac{7}{240} c^4. \end{align}

Derivation

When data is grouped, moments are calculated using the mid-points of the groups. This tends to underestimate moments of even order. Sheppard's correction treats the data as spread evenly throughout each group. This is done by adding to the discrete distribution of the grouped variable, an independent random variable uniformly distributed between -0.5c and 0.5c, where c is the class width.

The cumulants of the sum of the grouped variable and the uniform variable are the sums of the cumulants. As odd cumulants of a uniform distribution are zero, only even moments are affected.

The second and fourth cumulants of the uniform distribution on (-0.5c, 0.5c) are respectively, c²/12 and -c⁴/120.

The correction to moments can be derived from the relation between cumulants and moments.

References

Weisstein, Eric W. "Sheppard's Correction". MathWorld—A Wolfram Web Resource. Retrieved March 2, 2014.
Weatherburn, C.E. (1949), A first course in mathematical statistics, Cambridge University Press

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