Definition:Sheppard's Correction

Definition

Let $S$ be a set of grouped data.

Let it be assumed that $S$ obeys a Gaussian (normal) distribution.

When calculating the moments of $S$ using the mid-interval values, it is in general necessary to apply an adjustment to the calculated values.

This is called a Sheppard's correction, and it is subtracted from the calculated moment thus:

\(\ds \hat \mu_2\)

\(=\)

\(\ds m_2 - \dfrac {h^2} {12}\)

\(\ds \hat \mu_3\)

\(=\)

\(\ds m_3\)

\(\ds \hat \mu_4\)

\(=\)

\(\ds m_4 - \dfrac {m_2} 2 + \dfrac {7 h^2} {240}\)

where:

$m_i$ is the $i$th moment calculated from the mid-interval values

$\hat \mu_i$ is the adjusted value of the $i$th moment

$h$ is the bin width.

Also see

• Results about Sheppard's corrections can be found here.

Source of Name

This entry was named for William Fleetwood Sheppard.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): grouped data
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): grouped data

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