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Peirce's criterion

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Outliers removed by Peirce's criterion

The problem of outliers

In data sets containing real-numbered measurements, the suspected outliers are the measured values that appear to lie outside the cluster of most of the other data values. The outliers would greatly change the estimate of location if the arithmetic average were to be used as a summary statistic of location. The problem is that the arithmetic mean is very sensitive to the inclusion of any outliers; in statistical terminology, the arithmetic mean is not robust.

In the presence of outliers, the statistician has two options. First, the statistician may remove the suspected outliers from the data set and then use the arithmetic mean to estimate the location parameter. Second, the statistician may use a robust statistic, such as the median statistic.

Peirce's criterion is a statistical procedure for eliminating outliers.

Uses of Peirce's criterion

The statistician and historian of statistics Stephen M. Stigler wrote the following about Benjamin Peirce:

"In 1852 he published the first significance test designed to tell an investigator whether an outlier should be rejected (Peirce 1852, 1878). The test, based on a likelihood ratio type of argument, had the distinction of producing an international debate on the wisdom of such actions (Anscombe, 1960, Rider, 1933, Stigler, 1973a)."

Peirce's criterion is derived from a statistical analysis of the Gaussian distribution. Unlike some other criteria for removing outliers, Peirce's method can be applied to identify two or more outliers.

"It is proposed to determine in a series of m observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as n such observations. The principle upon which it is proposed to solve this problem is, that the proposed observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many, and no more, abnormal observations."

Hawkins provides a formula for the criterion.

Peirce's criterion was used for decades at the United States Coast Survey, which was renamed the United States Coast and Geodetic Survey in 1878:

"From 1852 to 1867 he served as the director of the longitude determinations of the U. S. Coast Survey and from 1867 to 1874 as superintendent of the Survey. During these years his test was consistently employed by all the clerks of this, the most active and mathematically inclined statistical organization of the era."

Peirce's criterion was discussed in William Chauvenet's book.

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