1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.5. Quantitative Techniques


Introduction 
An outlier is an observation that appears to deviate markedly from
other observations in the sample.
Identification of potential outliers is important for the following reasons.


Labeling, Accomodation, Identification 
Iglewicz and Hoaglin
distinguish the three following issues with regards to outliers.


Normality Assumption 
Identifying an observation as an outlier depends on the underlying
distribution of the data. In this section, we limit the discussion
to univariate data sets that are assumed to follow an approximately
normal distribution. If the normality assumption for the data being
tested is not valid, then a determination that there is an outlier
may in fact be due to the nonnormality of the data rather than the
prescence of an outlier.
For this reason, it is recommended that you generate a normal probability plot of the data before applying an outlier test. Although you can also perform formal tests for normality, the prescence of one or more outliers may cause the tests to reject normality when it is in fact a reasonable assumption for applying the outlier test. In addition to checking the normality assumption, the lower and upper tails of the normal probability plot can be a useful graphical technique for identifying potential outliers. In particular, the plot can help determine whether we need to check for a single outlier or whether we need to check for multiple outliers. The box plot and the histogram can also be useful graphical tools in checking the normality assumption and in identifying potential outliers. 

Single Versus Multiple Outliers 
Some outlier tests are designed to detect the prescence of a
single outlier while other tests are designed to detect the
prescence of multiple outliers. It is not appropriate to apply
a test for a single outlier sequentially in order to detect
multiple outliers.
In addition, some tests that detect multiple outliers may require that you specify the number of suspected outliers exactly. 

Masking and Swamping 
Masking can occur when we specify too few outliers in the test. For
example, if we are testing for a single outlier when there are in
fact two (or more) outliers, these additional outliers may influence
the value of the test statistic enough so that no points are declared
as outliers.
On the other hand, swamping can occur when we specify too many outliers in the test. For example, if we are testing for two or more outliers when there is in fact only a single outlier, both points may be declared outliers (many tests will declare either all or none of the tested points as outliers). Due to the possibility of masking and swamping, it is useful to complement formal outlier tests with graphical methods. Graphics can often help identify cases where masking or swamping may be an issue. Swamping and masking are also the reason that many tests require that the exact number of outliers being tested must be specified. Also, masking is one reason that trying to apply a single outlier test sequentially can fail. For example, if there are multiple outliers, masking may cause the outlier test for the first outlier to return a conclusion of no outliers (and so the testing for any additional outliers is not performed). 

ZScores and Modified ZScores 
The Zscore of an observation is defined as
Although it is common practice to use Zscores to identify possible outliers, this can be misleading (particularly for small sample sizes) due to the fact that the maximum Zscore is at most \((n1)/\sqrt{n}\) Iglewicz and Hoaglin recommend using the modified Zscore
with MAD denoting the median absolute deviation and \(\tilde{x}\) denoting the median. These authors recommend that modified Zscores with an absolute value of greater than 3.5 be labeled as potential outliers. 

Formal Outlier Tests 
A number of formal outlier tests have proposed in the
literature. These can be grouped by the following characteristics:


Lognormal Distribution  The tests discussed here are specifically based on the assumption that the data follow an approximately normal disribution. If your data follow an approximately lognormal distribution, you can transform the data to normality by taking the logarithms of the data and then applying the outlier tests discussed here.  
Further Information 
Iglewicz and Hoaglin
provide an extensive discussion of the outlier tests given above
(as well as some not given above) and also give
a good tutorial on the subject of outliers.
Barnett and Lewis
provide a book length treatment of the subject.
In addition to discussing additional tests for data that follow an approximately normal distribution, these sources also discuss the case where the data are not normally distributed. 