# What is the basis for the Box and Whisker Plot definition of an outlier?

The standard definition of an outlier for a Box and Whisker plot is points outside of the range $\left\{Q1-1.5IQR,Q3+1.5IQR\right\}$, where $IQR= Q3-Q1$ and $Q1$ is the first quartile and $Q3$ is the third quartile of the data.

What is the basis for this definition? With a large number of points, even a perfectly normal distribution returns outliers.

For example, suppose you start with the sequence:

xseq<-seq(1-.5^1/4000,.5^1/4000, by = -.00025)


This sequence creates a percentile ranking of 4000 points of data.

Testing normality for the qnorm of this series results in:

shapiro.test(qnorm(xseq))

Shapiro-Wilk normality test

data:  qnorm(xseq)
W = 0.99999, p-value = 1

ad.test(qnorm(xseq))

Anderson-Darling normality test

data:  qnorm(xseq)
A = 0.00044273, p-value = 1


The results are exactly as expected: the normality of a normal distribution is normal. Creating a qqnorm(qnorm(xseq)) creates (as expected) a straight line of data: If a boxplot of the same data is created, boxplot(qnorm(xseq)) produces the result: The boxplot, unlike shapiro.test, ad.test, or qqnorm identifies several points as outliers when the sample size is sufficiently large (as in this example).

• what do you mean by "basis"? this is some definition, and no one says perfectly normal distribution does not have outliers – hxd1011 Feb 2 '17 at 20:48
• @hxd1011, the definition of the distribution cannot be an outlier from itself. This definition for testing for outliers on a box and whisker plot is testing /something/ to provide the result, whatever it is testing would be the test's basis. – Tavrock Feb 2 '17 at 20:55
• I think the box and whisker outlier definition is just some heuristics... Also, why definition of the distribution cannot have an outlier from self? – hxd1011 Feb 2 '17 at 20:56
• It doesn't matter what rule you choose, you'd end up saying "with a large number of points, even a perfectly normal distribution returns outliers". [Try to come up with a way of usefully identifying outliers that cannot reject any points if you sample from a normal distribution.] – Glen_b Feb 3 '17 at 10:14
• A much repeated anecdote is that John Tukey, who came up with this rule of thumb, was asked why 1.5; and said that 1 would be too little and 2 would be too much. Given the number of times I have seen it misread as somehow a definitive, oracular criterion, I would be more than happy for it to fade away. Now we all have computers that can show all the data! – Nick Cox Feb 3 '17 at 16:51

## Boxplots

Here is a relevant section from Hoaglin, Mosteller and Tukey (2000): Understanding Robust and Exploratory Data Analysis. Wiley. Chapter 3, "Boxplots and Batch Comparison", written by John D. Emerson and Judith Strenio (from page 62):

[...] Our definition of outliers as data values that are smaller than $$F_{L}-\frac{3}{2}d_{F}$$ or larger than $$F_{U}+\frac{3}{2}d_{F}$$ is somewhat arbitrary, but experience with many data sets indicates that this definition serves well in identifying values that may require special attention.[...]

$$F_{L}$$ and $$F_{U}$$ denote the first and third quartile, whereas $$d_{F}$$ is the interquartile range (i.e. $$F_{U}-F_{L}$$).

They go on and show the application to a Gaussian population (page 63):

Consider the standard Gaussian distribution, with mean $$0$$ and variance $$1$$. We look for population values of this distribution that are analogous to the sample values used in the boxplot. For a symmetric distribution, the median equals the mean, so the population median of the standard Gaussian distribution is $$0$$. The population fourths are $$-0.6745$$ and $$0.6745$$, so the population fourth-spread is $$1.349$$, or about $$\frac{4}{3}$$. Thus $$\frac{3}{2}$$ times the fourth-spread is $$2.0235$$ (about $$2$$). The population outlier cutoffs are $$\pm 2.698$$ (about $$2\frac{2}{3}$$), and they contain $$99.3\%$$ of the distribution. [...]

So

[they] show that if the cutoffs are applied to a Gaussian distribution, then $$0.7\%$$ of the population is outside the outlier cutoffs; this figure provides a standard of comparison for judging the placement of the outlier cutoffs [...].

Further, they write

[...] Thus we can judge whether our data seem heavier-tailed than Gaussian by how many points fall beyond the outlier cutoffs. [...]

They provide a table with the expected proportion of values that fall outside the outlier cutoffs (labelled "Total % Out"): So these cutoffs where never intended to be a strict rule about what data points are outliers or not. As you noted, even a perfect Normal distribution is expected to exhibit "outliers" in a boxplot.

## Outliers

As far as I know, there is no universally accepted definition of outlier. I like the definition by Hawkins (1980):

An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism.

Ideally, you should only treat data points as outliers once you understand why they don't belong to the rest of the data. A simple rule is not sufficient. A good treatment of outliers can be found in Aggarwal (2013).

## References

Aggarwal CC (2013): Outlier Analysis. Springer.
Hawkins D (1980): Identification of Outliers. Chapman and Hall.
Hoaglin, Mosteller and Tukey (2000): Understanding Robust and Exploratory Data Analysis. Wiley.

The word 'outlier' is often assumed to mean something like 'a data value that is erroneous, misleading, mistaken or broken and should therefore be omitted from analysis', but that is not what Tukey meant by his use of outlier. The outliers are simply points that are a long way from the median of the dataset.

Your point about expecting outliers in many datasets is correct and important. And there are many good questions and answers on the topic.

Removing outliers from asymmetric data

Is it appropriate to identify and remove outliers because they cause problems?

As with all outlier detection methods, care and thought must be used to determine what values are truly outliers. I think the boxplot simply provides a good visualization of the spread of data and any true outliers will be easy to catch.

I think you should be concerned if you don't get some outliers as part of a normal distribution, otherwise perhaps you should be looking for reasons there aren't any. Clearly they should be reviewed to ensure they are not recording errors, but otherwise they are to be expected.