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In statistics, an outlier is a data point that differs significantly from other observations.[1][2] 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.[3][4] 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. 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, 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 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 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 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.[5] However, the mean is generally a more precise estimator.[6]

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 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.

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).

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.[8] There are various methods of outlier detection, some of which are treated as synonymous with novelty detection.[9][10][11][12][13] Some are graphical such as normal probability plots. Others are model-based. Box plots are a hybrid.

Other methods flag observations based on measures such as the interquartile range. For example, if Q 1 \displaystyle Q_1 and Q 3 \displaystyle Q_3 are the lower and upper quartiles respectively, then one could define an outlier to be any observation outside the range:

In various domains such as, but not limited to, statistics, signal processing, finance, econometrics, manufacturing, networking and data mining, the task of anomaly detection may take other approaches. Some of these may be distance-based[20][21] and density-based such as Local Outlier Factor (LOF).[22] Some approaches may use the distance to the k-nearest neighbors to label observations as outliers or non-outliers.[23]

The modified Thompson Tau test[citation needed] is a method used to determine if an outlier exists in a data set. The strength of this method lies in the fact that it takes into account a data set's standard deviation, average and provides a statistically determined rejection zone; thus providing an objective method to determine if a data point is an outlier.[citation needed][24]How it works:First, a data set's average is determined. Next the absolute deviation between each data point and the average are determined. Thirdly, a rejection region is determined using the formula:

The modified Thompson Tau test is used to find one outlier at a time (largest value of δ is removed if it is an outlier). Meaning, if a data point is found to be an outlier, it is removed from the data set and the test is applied again with a new average and rejection region. This process is continued until no outliers remain in a data set.

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. The application should use a classification algorithm that is robust to outliers to model data with naturally occurring outlier points.

Deletion of outlier data is a controversial practice frowned upon 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. An outlier resulting from an instrument reading error may be excluded but it is desirable that the reading is at least verified.

The two common approaches to exclude outliers are truncation (or trimming) and Winsorising. Trimming discards the outliers whereas Winsorising replaces the outliers with the nearest "nonsuspect" data.[26] Exclusion can also be a consequence of the measurement process, such as when an experiment is not entirely capable of measuring such extreme values, resulting in censored data.[27]

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,[29] 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. Even a slight difference in the fatness of the tails can make a large difference in the expected number of extreme values.

A set membership approach considers that the uncertainty corresponding to the ith measurement of an unknown random vector x is represented by a set Xi (instead of a probability density function). If no outliers occur, x should belong to the intersection of all Xi's. When outliers occur, this intersection could be empty, and we should relax a small number of the sets Xi (as small as possible) in order to avoid any inconsistency.[30] This can be done using the notion of q-relaxed intersection. As illustrated by the figure, the q-relaxed intersection corresponds to the set of all x which belong to all sets except q of them. Sets Xi that do not intersect the q-relaxed intersection could be suspected to be outliers.

Outlier Jets Inc arranges flights on behalf of clients with FAR Part 135 air carriers that exercise full operational control of charter flights at all times. Flights will be operated by FAR Part 135 air carriers that have been certified to provide service for Outlier Jets and that meet all FAA safety standards and additional safety standards established by Outlier Jets Inc. (Refer to for details.)

Despite its great potential in studying brain anatomy and structure, diffusion magnetic resonance imaging (dMRI) is marred by artefacts more than any other commonly used MRI technique. In this paper we present a non-parametric framework for detecting and correcting dMRI outliers (signal loss) caused by subject motion. Signal loss (dropout) affecting a whole slice, or a large connected region of a slice, is frequently observed in diffusion weighted images, leading to a set of unusable measurements. This is caused by bulk (subject or physiological) motion during the diffusion encoding part of the imaging sequence. We suggest a method to detect slices affected by signal loss and replace them by a non-parametric prediction, in order to minimise their impact on subsequent analysis. The outlier detection and replacement, as well as correction of other dMRI distortions (susceptibility-induced distortions, eddy currents (EC) and subject motion) are performed within a single framework, allowing the use of an integrated approach for distortion correction. Highly realistic simulations have been used to evaluate the method with respect to its ability to detect outliers (types 1 and 2 errors), the impact of outliers on retrospective correction of movement and distortion and the impact on estimation of commonly used diffusion tensor metrics, such as fractional anisotropy (FA) and mean diffusivity (MD). Data from a large imaging project studying older adults (the Whitehall Imaging sub-study) was used to demonstrate the utility of the method when applied to datasets with severe subject movement. The results indicate high sensitivity and specificity for detecting outliers and that their deleterious effects on FA and MD can be almost completely corrected. 041b061a72

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