# Ancillary Statistic - Example

Example

Suppose X1, ..., Xn are independent and identically distributed, and are normally distributed with unknown expected value μ and known variance 1. Let

be the sample mean.

The following statistical measures of dispersion of the sample

• Range: max(X1, ..., Xn) − min(X1, ..., Xn)
• Interquartile range: Q3Q1
• Sample variance:

are all ancillary statistics, because their sampling distributions do not change as μ changes. Computationally, this is because in the formulas, the μ terms cancel – adding a constant number to a distribution (and all samples) changes its sample maximum and minimum by the same amount, so it does not change their difference, and likewise for others: these measures of dispersion do not depend on location.

Conversely, given i.i.d. normal variables with known mean 1 and unknown variance σ2, the sample mean is not an ancillary statistic of the variance, as the sampling distribution of the sample mean is N(1, σ2/n), which does depend on σ 2 – this measure of location (specifically, its standard error) depends on dispersion.