Dear Muneeb,

there is a very sophisticated book for this problem from D.M. Hawkins, Book Identification of Outliers

Introduction

The problem of outliers is one of the oldest in statistics, and during the last century and a half interest in it has waxed and waned several times. Currently it is once again an active research area after some years of relative neglect, and recent work has solved a number of old problems in outlier theory, and identified new ones. The major results are, however, scattered amongst many journal articles, and for some time there has been a clear need to bring them together in one place. That was the original intention of this monograph: but during execution it became clear that the existing theory of outliers was deficient in several areas, and so the monograph also contains a number of new results and conjectures. In view of the enormous volume ofliterature on the outlier problem and its cousins, no attempt has been made to make the coverage exhaustive. The material is concerned almost entirely with the use of outlier tests that are known (or may reasonably be expected) to be optimal in some way. Such topics as robust estimation are largely ignored, being covered more adequately in other sources. The numerous ad hoc statistics proposed in the early work on the grounds of intuitive appeal or computational simplicity also are not discussed in any detail.

A very good book!

Best

Tanja

You may have use of the attachment.

Article Resistant outlier rules and the non-Gaussian case

If you were willing to change your mindset from Gaussian thinking to Paretian thinking, there would be no outliers. Conventionally, an outlier is considered to be a value that is far from the the average - what if there is no such an average, by which I meant data with a heavy tailed distribution.

Under Gaussian thinking, a small probability event is an impossible event (negative), BUT, under Paretian thinking, a small probability event is highly improbable (positive, literally meaning an event with a high impact):

Article Geospatial Analysis Requires a Different Way of Thinking: Th...

Given such a data with a heavy tailed distribution, head/tail breaks (rather than conventional clustering methods) should be adopted to get insights into the underlying scaling hierarchy:

Article Head/tail Breaks: A New Classification Scheme for Data with ...

I would want to make a point re comment by Bin and Wulf. The standard book on outliers is by Barnett & Lewis: Outliers in statistical data. One point raised by them and simultaneously by John Tukey in his work on Exploratory Data Analysis was that: sure, an outlier is an abberant observation. However, its detection must not necessarily imply discarding it. One should consider statistical models that accomodate outliers. For instance and the simplest one, if you suspect outliers due to questionable observations - compute and report the median instead of the mean. As the median is resistant to outliers (while somewhat less efficient) you trade between the risk of a contaminated sample and the cost of a less efficient centrality measure. As the resistant statistical methods often are marginally less efficient, it is often sensible to keep outliers and use these methods rather than removing outliers.

I fully agree with Kenneth on the point that detection of outliers does not mean to discard them. Under the Paretian thinking, outliers tend to have an extremely high impact, too high to ignore them. For example. an 8 degree of earthquake is considered to be an outlier, but it has an devastated effect. It is in line with so called the head/tail breaks thinking:

Presentation Scaling Law and Tobler's Law for Characterizing Asymmetry in Geography

Dear Wulf,

when I am performing a statistical analysis, I always calculate the minimum, the maximum, the mean and the median. With these very simple parameters, I can discover the most outliers. This is a very simple, but pragmatical way to make the statistical analysis ready.

Best

Tanja

Calculate mean and standard deviation (SD).

The data falling above

mean + 3 x SD

and below

mean - 3 x SD

can be identified, usually, as outliers

Grubbs Test is one good option.

Grubbs Test, as mentioned by Debopam Ghosh, is no doubt one good option.

However, the method,as I already mentioned, is based on the assumption that the outliers deviate from normal behavior i.e. the outliers do not fall within the natural limit within which the non-outliers fall.

The nature of outlier is to to be deviated from normality. Hence the 3 sigma limits of normal distribution can be used to detect outliers.

Thanks to Muneeb Faiq and Debopam Ghosh for their recommendations of my answer.