11.2B Box-and Whisker Plots

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11.2B Box-and Whisker Plots
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Objectives

• To get a more complete picture of the data.
• Be able to figure out the first quartile, third
  quartile, and interquartile range.

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Introduction

• The purpose of calculating a mean or median is to
  obtain one number that describes some
  measurements.
• That one number alone however, may not
  adequately represent the data.

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Definitions

• A box-and-whisker plot is a graph that gives a
  more complete picture of the data. It shows five
  numbers
• The smallest value
• The first quartile
• The median,
• The third quartile and
• The greatest value

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First/Third Quartile definitions

• Symbolized by Q1, the number below which
  one-quarter of the data lie. The third quartile,
  symbolized by Q3 is the number above which
  one-quarter of the data lie.

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Example

• Find the first quartile Q1 and the third quartile
  Q3 for the prices of 15 half-gallon cartons of
  deluxe ice cream.

3.26 4.71 4.18 4.45 5.49 3.18 3.86 3.58 4.29 5.44 4.83 4.56 4.36 2.39 2.66
To find the quartiles, first arrange the data
from the smallest value to the largest value.
Then find the median.

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Example

• Find the first quartile Q1 and the third quartile
  Q3 for the prices of 15 half-gallon cartons of
  deluxe ice cream.

3.26 4.71 4.18 4.45 5.49 3.18 3.86 3.58 4.29 5.44 4.83 4.56 4.36 2.39 2.66
To find the quartiles, first arrange the data
from the smallest value to the largest value.
Then find the median.
2.39 2.66 3.18 3.26 3.58 3.86 4.18 4.29 4.36 4.45 4.56 4.71 4.83 5.44 5.49
The median is 4.29.
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Example
Now separate the data into two groups. Those
values below the median and those values above
the median.

Values less than median
Values greater than median
4.36 4.45 4.56 4.71 4.83 5.44 5.49
2.39 2.66 3.18 3.26 3.58 3.86 4.18
Q3
Q1
The first quartile Q1 is the median of the lower
half of the data Q1 3.26
The third quartile Q3 is the median of the upper
half of the data Q3 4.71
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Interquartile Range Definition

• Is the difference between the third quartile Q3
  and the first quartile Q1. Interquartile range
• Q3 Q1 4.71 3.26 1.45

Fifty percent of the data in a distribution lie
in the interquartile range.
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Box-and-Whisker Plots

• Shows the data in the interquartile range as a
  box. The box-and-whisker plot for the data on
  the cost of ice cream is shown below.

Q1
Q3
Median
5.49
2.39
3.26
4.29
4.71
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Box-and-Whisker Plots

• Note that the box-and-whisker plot labels five
  values the smallest (2.39) the first quartile
  Q1 (3.26), the median (4.29) the third quartile
  Q3, 4.71 and the largest value (5.49).

Median
Recall from the last section that the difference
between the largest and smallest values is called
the RANGE. For these data Range 5.49 2.39
3.10
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Example
30 45
54 24
48 38
43 38
46 53
62 64
40 35

• For these next two examples, use the data in the
  following table. I am putting it vertically, so
  you can read it.

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Find the first quartile and third quartile for
the data in the software training table.

• Strategy
• Arrange the data from smallest to largest. Then
  find the median
• Find Q1 as the median of the lower half of the
  data.
• Find Q3 as the median of the upper half of the
  data.
• Draw a box and whiskers plot for the data in the
  software training table.

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Example
30 45
54 24
48 38
43 38
46 53
62 64
40 35

• For these next two examples, use the data in the
  following table. I am putting it vertically, so
  you can read it.

Arrange data from least to greatest.
24 30 35 38 38 40 43 45 46 48 53 54 62 64
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Find the median
24 30 35 38 38 40 43 45 46 48 53 54 62 64
Oops, there are an even number, so you must take
the two middle numbers, add them together and
divide by 2.
Median 43 45 44 2
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Next find the median of the top lower half and
the upper half.
24 30 35 38 38 40 43
Median 38 so Q1 38
45 46 48 53 54 62 64
Median 53 so Q3 53
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Draw the Line and plot points
64
24
38
44
53
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Draw a box neatly and label 1st and 3rd
quartile and median
Median
Q1
Q3
64
24
38
44
53