Box Plots

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Box Plots

• A Modern View of the Data

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The Five-number Summary

• One common way of summarizing information is by
  giving what is called the five-number summary
• The minimum value Min
• The first quartile Q1
• The median M
• The third quartile Q3
• The maximum value Max

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From Numbers to Pictures
A Box Plot is a way to visualize the data using
the five numbers and other numbers derived from
them. This type of plot was invented by John
Tukey, who also devised the Stem-and-Leaf
plot. We now present a step-by-step procedure
illustrating the construction of a box plot.
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Some Simple Calculations

• From the given 5 numbers we can compute a couple
  of other numbers
• The IQR Q3 Q1
• The Upper Fence Q3 1.5 IQR
• The Lower Fence Q1 1.5 IQR
• Sometimes, two additional numbers are
  calculated
• Extreme Upper Fence Q3 3 IQR
• Extreme Lower Fence Q1 3 IQR

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Sample Data for the Box Plot
To illustrate the construction of a box plot, we
shall use the data provided by the authors for
Problem 12 in Chapter 5. The data shows the
number of campsites in the various parks of
Vermont.

• Count 46
• Min 0
• Q1 28

• Median 43.5
• Q3 78
• Max 275

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1. Do Required Calculations

• For our sample data
• IQR Q3 Q1 78 28 50
• Upper Fence Q3 1.5 IQR 78 (1.5)(50)
  78 75 153
• Lower Fence Q1 75 28 75 -27, which we
  can replace by 0 since you cannot have a negative
  number of campsite.
• We shall not use the other two numbers.

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2. Set up the Axis

• It is possible to use box plots to compare
  different distributions as well as simply
  describe a single distribution.
• If more than one distribution is being shown,
  then be sure that the axis is long enough to
  encompass all of the distributions being plotted.
• Note that the axis may be horizontal or vertical.
  This presentation will use the horizontal
  orientation.
• Label the axis suitably I.e. so that the max
  and min values will be able to be shown on the
  plot.
• Our sample data goes from 0 to 275, so an axis
  running from 0 to 300 or something similar would
  be appropriate.

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Sample Axis

• The sample data contains values from 0 to 275,
  therefore we construct an axis from 0 to 300 to
  accommodate all the values.

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3. Draw the Rectangle

• Next one draws a rectangle of suitable height
  with its ends at Q1 and Q3, and a line going
  through the rectangle at the value corresponding
  to the median.
• Remember in our case Q128, M43.5, and Q378.

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4. Draw the Whiskers

• Whiskers are drawn out from the rectangle going
  up and down.
• The whisker going down extends to the smallest
  value which is within the lower fence. This
  whisker is terminated with a small vertical line.
• Note that the fence is not drawn unless it
  happens to be one of the values.
• Similarly the whisker going up is extended to the
  largest value which is still at or under the
  upper fence. Also terminated with a small line.
• Again the fence itself is not drawn unless it is
  one of the values.

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Whiskers Drawn

• In our case there are values going all the way to
  the whiskers. This is not always the case.
• The fences are not drawn on the final plot!

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5. Include Any Outliers

• After the whiskers are drawn, then any outliers
  are indicated on the plot by means of circles.
• This is your box plot complete with whiskers!