(10) Box-Plot With Fences

7/30/2019 (10) Box-Plot With Fences

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Applied Statistics and Computing Lab

BOX-PLOT WITH FENCES


Indian School of Business


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Learning goals

Why go beyond a basic box-plot?

What are fences?

How is box-plot with fences constructed?

How does one interpret such a plot? What are the gains and limitations?

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Box-plot with fences

Can we modify the basic box-plot so that it

helps in detecting unusual observations?

Box-plot with fences can be useful

What are fences? Let us take a look at a figure!

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Box-plot with fences (contd.)

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Source: http://en.wikipedia.org/wiki/Boxplot


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Basis for fences

From the previous figure, we see that for a

normally distributed data, 99.3% of the datalies in the interval

Also, only 3 out of a million or 0.003%

observations are expected to be present

outside the interval

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))(5.1),(5.1( 133131 QQQQQQ +

))(3),(3( 133131 QQQQQQ +


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Box-plot with fences

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Outlier

Suspected outlier

Visuals from Aczel A., Sounderpandian J. Complete business statistics


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Box-plot with fences Box-plot with fences are useful in identifying unusual observations

What are unusual observations?

Box-plot serves only as a diagnostic. It is not a test of significance.

Caution: Even for a random sample from a normal distribution, about 7

out of thousand sample points can lie outside the inner fence and 3 out of

a million can lie outside the outer fence. Thus when dealing with large

data sets, one has to be careful about declaration of outliers on the basis

of a Box-plot. Sometimes, simulation-based methods are used for this

purpose. For more information one may see Robert Dawson (2011)

Sometimes only the inner fence is used (as is the default in R)

The default for Box-plot command in R produces Box-plot with inner fence

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Comparison of data

8Visuals from Aczel A., Sounderpandian J. Complete business statistics


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Box-plot of final exam scores

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Box-plots of all the scores

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Box-plots of three minors

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Box-plots indicating means

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Interpretation of the Box-plot

In the Box-plot corresponding to the scores in the second semester exam, wehave 3 unusual observations among 50. Under normal situation, we expect tohave about 7 in a thousand observations. Thus one needs to probe into theseunusual observations.

The distribution of scores of second semester exam appears to be symmetric,but may have slightly longer tails in view of the unusual observations, situatedsymmetrically below and above the fences.

From the box-plots corresponding to the three minors, it appears that

The distribution of scores in First minor is skewed to the right,

The distributions of scores in Second and Third minors are symmetric and aresomewhat similar, and

The median scores of the three minors seem to be close (we shall examinethis further when we deal with the notched box-plots)

There is an unusual observation in the Box-plot of scores of First semesterexam, with a value of about 18. We know that the GPA is out of 10. Thus this isan outlier!

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Gain from a Box-plot with fence

As we saw,

We can identify unusual observations

We can examine the tail behaviour

We can compare two or more variables or datasets more easily

However we cannot get modal information from these plots!

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R-codesPlot R-code

Boxplot (of single variable) boxplot(variable name)

Boxplot (of all the variables in a dataset) boxplot(name of data as input in R)

Boxplot (of k distinct variables from a

dataset)

boxplot(dataname$variable 1 name,

dataname$variable 2 name,,

dataname$variable k name)

Boxplot with means (can be drawn for

one or many variables at the same time)

boxplot(variable specification)

points(y=colMeans(variables

specification),x=1:(total number of

variables in a box-plot))

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Thank you

Documents

(10) Box-Plot With Fences