Outlier

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

In most samplings of data, some data points will be further away from their expected values than what is deemed reasonable. This can be due to systematic error, faults in the theory that generated the expected values, or it can simply be the case that some observations happen to be a long way from the center of the data. Outlier points can therefore indicate faulty data, erroneous procedures, or areas where a certain theory might not be valid. However, a small number of outliers is expected in normal distributions.

Estimators not sensitive to outliers are said to be robust.

Mathematical definitions

Mild outliers

Defining $Q_{1}$ and $Q_{3}$ to be first and third quartiles, and $IQR$ to be the interquartile range ( $Q_{3}-Q_{1}$ ), one possible definition of being "far away" in this context is:

<Q_{1}-1.5*IQR,

or

>Q_{3}+1.5*IQR

Q_{1}

and

Q_{3}

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

Extreme outliers

Extreme outliers are observations that are beyond the outer fences:

<Q_{1}-3\cdot IQR,

or

>Q_{3}+3\cdot IQR.

Occurrence and causes

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 a flaw in the assumed theory, calling for further investigation by the researcher.

Non-normal distributions

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.

External link

Definition of outlier from MathWorld.