outlier

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Figure 1. Box plot of data from the Michelson-Morley Experiment displaying outliers in the middle column.

In statistics, an outlier^[1] is an observation that is numerically distant from the rest of the data. Grubbs^[2] defined an outlier as:

An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs.

Outliers can occur by chance in any distribution, but they are often indicative either of measurement error or that the population has a heavy-tailed distribution. In the former case one wishes to discard them or use statistics that are robust to outliers, while in the latter case they indicate that the distribution has high kurtosis and that one should be very cautious in using tool or intuitions that assume a normal distribution. 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 or flaws in the theory that generated an assumed family of probability distributions, 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, will include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum need not be outliers, if they are not 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 temperature of 10 objects in a room, and most are between 20 and 25 degrees Celsius, but an oven is at 175 °C, the median of the data may be 23 °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 than the mean; however, naively interpreting the mean as "a typical sample", equivalent to the median, is incorrect. As illustrated in this case, outliers may be indicative of data points that belong to a different population than the rest of the sample set.

Estimators capable of coping with outliers are said to be robust: the median is a robust statistic, while the mean is not.

Occurrence and causes

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In the case of normally distributed data, roughly 1 in 22 observations will differ by twice the standard deviation or more from the mean, and 1 in 370 will deviate by three times the standard deviation; see three sigma rule for details. 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, and not indicative of 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 with parameter p, which can generally be well-approximated by the Poisson distribution with λ = pn. Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, p is approximately .3%, and thus for 1,000 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.

Caution

Outliers may be mere measurement error, or may be actual events of consequence, such as the 1755 Lisbon earthquake.

Unless it can be ascertained that the deviation is not significant, it is ill-advised to ignore the presence of outliers. Outliers that cannot be readily explained demand special attention – see kurtosis risk and black swan theory.

Identifying outliers

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.

Outlier detection^[3][4] has been used for centuries to detect and, where appropriate, remove anomalous observations from data. Outlier detection can identify system faults and fraud before they escalate with potentially catastrophic consequences. The original outlier detection methods were arbitrary but now, principled and systematic techniques are used, drawn from the full gamut of computer science and statistics.

There are three fundamental approaches^[4] to the problem of outlier detection:

• Type 1 - Determine the outliers with no prior knowledge of the data. This is essentially a learning approach analogous to unsupervised clustering. The approach processes the data as a static distribution, pinpoints the most remote points, and flags them as potential outliers.
• Type 2 - Model both normality and abnormality. This approach is analogous to supervised classification and requires pre-labelled data, tagged as normal or abnormal.
• Type 3 - Model only normality (or in a few cases model abnormality). This is analogous to a semi-supervised recognition or detection task. It may be considered semi-supervised as the normal class is taught but the algorithm learns to recognise abnormality.

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' test for outliers
• Peirce's criterion
• ASTM E178 Standard Practice for Dealing With Outlying Observations

Other methods flag observations based on measures such as the interquartile range. For example, if Q₁ and Q₃ are the lower and upper quartiles respectively, then one could define an outlier to be any observation outside the range:

for some constant k.

Other approaches are distance-based^[5][6] and frequently use the distance to the k-nearest neighbors to label observations as outliers or non-outliers.

Working with outliers

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The choice of how to deal with an outlier should depend on the cause.

Retention

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.

Exclusion

Deletion of outlier data is a controversial practice frowned on by many scientists and science instructors; while mathematical criteria provide an objective and quantitative method for data rejection, they do 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.

In regression problems, an alternative approach may be to only exclude points which exhibit a large degree of influence on the parameters, using a measure such as Cook's distance.

If a data point (or points) is excluded from the data analysis, this should be clearly stated on any subsequent report.

Non-normal distributions

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.

Alternative models

In cases where the cause of the outliers is known, it may be possible to incorporate this effect into the model structure, for example by using a hierarchical Bayes model or a mixture model.

See also

• Robust statistics
• Robust regression
• Box plot
• Q test
• Sample maximum and minimum
• Studentized residual
• Winsorising

External links

• Renze, John, "Outlier" from MathWorld.
• Balakrishnan, N.; Childs, A. (2001), "Outlier", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Grubbs test described by NIST manual
• how to detect univariate outliers, how to detect multivariate outliers and how to deal with outliers

References

1. ^ Barnett, V. and Lewis, T.: 1994, Outliers in Statistical Data. John Wiley & Sons., 3^rd edition.
2. ^ Grubbs, F. E.: 1969, Procedures for detecting outlying observations in samples. Technometrics 11, 1–21.
3. ^ Rousseeuw, P. and Leroy, A.: 1996, Robust Regression and Outlier Detection. John Wiley & Sons., 3^rd edition.
4. ^ ^{a b} Hodge, V.J. and Austin, J. A Survey of Outlier Detection Methodologies. Artificial Intelligence Review, 22: pp. 85–126, Kluwer Academic Publishers, 2004
5. ^ Knorr, E. M. and Ng, R. T.: 1998, Algorithms for Mining Distance-Based Outliers in Large Datasets. In: Proceedings of the VLDB Conference. New York, USA, pp. 392–403
6. ^ Ramaswamy, S., Rastogi, R., and Shim, K.: 2000, ‘Efficient Algorithms for Mining Outliers from Large Data Sets’. In: Proceedings of the ACM SIGMOD Conference on Management of Data. Dallas, TX, pp.427–438.