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.

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.

See also: box plot, Studentized residual