This article deals with outliers in statistics. For Polynesian outliers, see the article Polynesian outliers

In statistics, an outlier is a single observation far away from the rest of the data.

One definition of "far away" in this context is:

less than Q1 − 1.5 × IQR or greater than Q3 + 1.5 × IQR

where Q1 and Q3 are the first and third quartiles, respectively, and IQR is the interquartile range (equal to Q3 − Q1).

These values define the so-called inner fences, beyond which an observation would be labeled a mild outlier.

Extreme outliers are observations that are beyond the outer fences:

less than Q1 − 3 × IQR or greater than Q3 + 3 × IQR

In the case of normally distributed data, using the above definitions, only about 1 in 150 observations will be a mild outlier and only about 1 in 425,000 an extreme outlier. Because of this, outliers usually demand special attention since they may indicate problems in sampling or data collection or transcription. Alternatively, an outlier could be the result of, for example, a truly unusual response to a given treatment, calling for further investigation by the researcher.

Even when a normal model is appropriate to the data being analyzed, outliers are expected for large sample sizes and should not automatically be discarded if that is the case. Also, the possibility should be considered that the underlying distribution of the data is not approximately normal, having fat tails . For instance, when sampling from a Cauchy distribution, the sample variance increases with the sample size, the sample mean fails to converge as the sample size increases, and outliers are expected at far larger rates than for a normal distribution.

See also: box plot, Studentized residual, Chauvenet's criterion