In
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, an outlier is a
data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the
data set
A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more table (database), database tables, where every column (database), column of a table represents a particular Variable (computer sci ...
. An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses.
Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in the data-set,
measurement error, or that the population has a
heavy-tailed distribution. In the case of measurement error, one wishes to discard them or use statistics that are
robust to outliers, while in the case of heavy-tailed distributions, they indicate that the distribution has high
skewness and that one should be very cautious in using tools or intuitions that assume a
normal distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
f(x) = \frac ...
. A frequent cause of outliers is a mixture of two distributions, which may be two distinct sub-populations, or may indicate 'correct trial' versus 'measurement error'; this is modeled by a
mixture model.
In most larger samplings of data, some data points will be further away from the
sample mean than what is deemed reasonable. This can be due to incidental
systematic error
Observational error (or measurement error) is the difference between a measurement, measured value of a physical quantity, quantity and its unknown true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. Such errors are ...
or flaws in the
theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
that generated an assumed family of
probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
s, or it may be that some observations are far 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, in large samples, a small number of outliers is to be expected (and not due to any anomalous condition).
Outliers, being the most extreme observations, may include the
sample maximum or
sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum are not always outliers because they may not be unusually far from other observations.
Naive interpretation of statistics derived from data sets that include outliers may be misleading. For example, if one is calculating the
average
In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
temperature of 10 objects in a room, and nine of them are between 20 and 25
degrees Celsius, but an oven is at 175 °C, the
median
The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
of the data will be between 20 and 25 °C but the mean temperature will be between 35.5 and 40 °C. In this case, the median better reflects the temperature of a randomly sampled object (but not the temperature in the room) than the mean; naively interpreting the mean as "a typical sample", equivalent to the median, is incorrect. As illustrated in this case, outliers may indicate data points that belong to a different
population
Population is a set of humans or other organisms in a given region or area. Governments conduct a census to quantify the resident population size within a given jurisdiction. The term is also applied to non-human animals, microorganisms, and pl ...
than the rest of the
sample set.
Estimators capable of coping with outliers are said to be robust: the median is a robust statistic of
central tendency, while the mean is not.
Occurrence and causes

In the case of
normally distributed data, the
three sigma rule means that roughly 1 in 22 observations will differ by twice the
standard deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
or more from the mean, and 1 in 370 will deviate by three times the standard deviation. In a sample of 1000 observations, the presence of up to five observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice the expected number and hence within 1 standard deviation of the expected number – see
Poisson distribution
In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
– and not indicate an anomaly. If the sample size is only 100, however, just three such outliers are already reason for concern, being more than 11 times the expected number.
In general, if the nature of the population distribution is known ''a priori'', it is possible to test if the number of outliers deviate
significantly from what can be expected: for a given cutoff (so samples fall beyond the cutoff with probability ''p'') of a given distribution, the number of outliers will follow a
binomial distribution
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
with parameter ''p'', which can generally be well-approximated by the
Poisson distribution
In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
with λ = ''pn''. Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, ''p'' is approximately 0.3%, and thus for 1000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a Poisson distribution with λ = 3.
Causes
Outliers can have many anomalous causes. A physical apparatus for taking measurements may have suffered a transient malfunction. There may have been an error in data transmission or transcription. Outliers arise due to changes in system behaviour, fraudulent behaviour, human error, instrument error or simply through natural deviations in populations. A sample may have been contaminated with elements from outside the population being examined. Alternatively, an outlier could be the result of a flaw in the assumed theory, calling for further investigation by the researcher. Additionally, the pathological appearance of outliers of a certain form appears in a variety of datasets, indicating that the causative mechanism for the data might differ at the extreme end (
King effect).
Definitions and detection
There is no rigid mathematical definition of what constitutes an outlier; determining whether or not an observation is an outlier is ultimately a subjective exercise.
There are various methods of outlier detection, some of which are treated as synonymous with novelty detection.
Some are graphical such as
normal probability plots. Others are model-based.
Box plot
In descriptive statistics, a box plot or boxplot is a method for demonstrating graphically the locality, spread and skewness groups of numerical data through their quartiles.
In addition to the box on a box plot, there can be lines (which are ca ...
s are a hybrid.
Model-based methods which are commonly used for identification assume that the data are from a normal distribution, and identify observations which are deemed "unlikely" based on mean and standard deviation:
*
Chauvenet's criterion
*
Grubbs's test for outliers
*
Dixon's ''Q'' test
*
ASTM
ASTM International, formerly known as American Society for Testing and Materials, is a standards organization that develops and publishes voluntary consensus technical international standards for a wide range of materials, products, systems and s ...
E178: Standard Practice for Dealing With Outlying Observations
*
Mahalanobis distance
The Mahalanobis distance is a distance measure, measure of the distance between a point P and a probability distribution D, introduced by Prasanta Chandra Mahalanobis, P. C. Mahalanobis in 1936. The mathematical details of Mahalanobis distance ...
and
leverage are often used to detect outliers, especially in the development of linear regression models.
* Subspace and correlation based techniques for high-dimensional numerical data
Peirce's criterion
It is proposed to determine in a series of observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as 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. (Quoted in the editorial note on page 516 to Peirce (1982 edition) from ''A Manual of Astronomy'' 2:558 by Chauvenet.)
Tukey's fences
Other methods flag observations based on measures such as the
interquartile range
In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the differen ...
. For example, if
and
are the lower and upper
quartiles respectively, then one could define an outlier to be any observation outside the range:
: