# Outlier

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## Overview

In statistics such as stratified samples, an outlier is an observation that is numerically distant from the rest of the data. Statistics derived from data sets that include outliers will often be misleading. For example, if one is calculating the average temperature of 10 objects in a room, and most are between 20-25° Celsius, but an oven is at 350° C, the median of the data may be 23 but the mean temperature will be 55. In this case, the median better reflects the temperature of a randomly sampled object than the mean. Outliers may be indicative of data points that belong to a different population than the rest of the sample set.

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.

Deletion of outlier data is a controversial practice frowned on by many scientists and science instructors; while mathematical criteria provides an objective and quantitiative method for data rejection, it does not make the practice more scientifically or methodologically sound, especially in small sets or where a normal distribution cannot be assumed. Rejection of outliers is more acceptable in areas of practice where the underlying model of the process being measured and the usual distribution of measurement error are confidently known. When practiced, rejection of outliers usually is based on some rule such as the quartile rules given below, Chauvenet's Criterion, or Grubbs Test.

## 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:

$< Q_1 - 1.5\cdot \mathrm{IQR},$

or

$> Q_3 + 1.5\cdot \mathrm{IQR}.$

$Q_1$ 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:

$< Q_1 - 3\cdot \mathrm{IQR},$

or

$> Q_3 + 3\cdot \mathrm{IQR}.$

### 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 distribution 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. Outliers play an important role in statistics.