Sheppard's Correction
Sheppard's correction is used to adjust the variance in a frequency distribution when the data is grouped into classes: $$ \sigma^2_R = \sigma^2 - \frac{ \alpha^2 }{12} $$ where σ2 represents the variance and α is the class width.
You can only apply Sheppard's correction when all the classes in the distribution have the same width (α) and the statistical phenomenon is continuous.
Why is it used?
Grouping data into classes in frequency distributions introduces some approximation in the calculation of statistical measures.
In certain cases, these approximations cancel each other out (for example, with the arithmetic mean).
However, when it comes to variance, the approximation is more significant because variance is a squared measure relative to the unit of the phenomenon.
To correct this, Sheppard's correction can be applied.
A practical example
Consider a frequency distribution that is divided into classes, where each class has a width of α = 10.
$$ \alpha = 10 $$
The variance of the phenomenon is σ2 = 12.
$$ \sigma^2 = 12 $$
By using Sheppard's correction, we can minimize the approximation caused by class grouping:
$$ \sigma^2_R = \sigma^2 - \frac{ \alpha^2 }{12} $$
$$ \sigma^2_R = 12 - \frac{ 10^2 }{12} $$
$$ \sigma^2_R = 12 - 8.33 $$
$$ \sigma^2_R = 3.67 $$
And so forth.
