Ancillary statistic

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In statistics, an ancillary statistic is a statistic whose probability distribution does not depend on which of the probability distributions among those being considered is the distribution of the statistical population from which the data were taken. This concept was introduced by the great statistical geneticist Sir Ronald Fisher.

[edit] Example

Suppose X₁, ..., X_n are independent and identically distributed, and are normally distributed with expected value μ and variance 1. (The use as an example, of this particular parametrized family of probability distributions, all having the same variance, is unrealistic, in that it amounts to a situation in which the statistician somehow knows the exact value of the population variance, but can only estimate the population mean by using the observed values of the data X₁, ..., X_n.) Let

$\overline{X}_n=(X_1+\,\cdots\,+X_n)/n$

be the sample mean.

The random variable

$\overline{X}_n-\mu$

is not an ancillary statistic, even though its probability distribution does not depend on μ. That is because it is not a statistic, since its value depends on the unobservable population mean μ.

The following measures of dispersion of the sample
- Range = $\max\{\,X_1,\dots,X_n\,\}-\min\{\,X_1,\dots,X_n\,\}\,\!$
- Interquartile range = $Q_3-Q_1\,$
- Sample variance = $\hat{\sigma}^2:=\,\frac{\sum \left(X_i-\overline{X}\right)^2}{n}$

are all ancillary statistics, because

their probability distributions do not change as μ changes, and
they depend only on the data X₁, ..., X_n and not on the unobservable parameter μ, i.e., it is a statistic.

[edit] Ancillary complement example

In baseball, suppose a scout observes a batter in N at-bats. Suppose (unrealistically) that the number N is chosen by some random process that is independent of the batter's ability -- say a coin is tossed after each at-bat and the result determines whether the scout will stay to watch the batter's next at-bat. The eventual data are the number N of at-bats and the number X of hits. The observed batting average X/N fails to convey all of the information available in the data because it fails to report the number N of at-bats (e.g., a batting average of 0.400, which is very high, based on only five at-bats does not inspire anywhere near as much confidence in the player's ability than a 0.400 average based on 100 at-bats). The number N of at-bats is an ancillary statistic because

It is a part of the observable data (it is a statistic), and
Its probability distribution does not depend on the batter's ability, since it was chosen by a random process independent of the batter's ability.

This ancillary statistic is an ancillary complement to the observed batting average X/N, i.e., the batting average X/N is not a sufficient statistic, in that it conveys less than all of the relevant information in the data, but conjoined with N, it becomes sufficient.