This document discusses measures of variation and box-and-whisker plots. It defines key terms like median, quartiles, interquartile range, range, and outliers. Examples are provided to demonstrate how to calculate these measures and construct box-and-whisker plots using given data sets. Outliers are identified as any data point more than 1.5 times the interquartile range above the third quartile or below the first quartile.

1. Measures of Variation. .
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Measures of Variation
Box and Whisker

2. Median
• The middle number in a data set when the data are ordered from least
to greatest.
3, 4, 4, 5, 6, 7, 8
3, 4, 4, 5, 6, 7, 8, 9
5 + 6 = 11
11 2 = 5.5
Odd Number of Data
Even Number of Data

3. Measures of Variation
• Measures of variation are used to describe the distribution of the data.
Upper and Lower Quartiles
The upper and lower quartiles are the medians of the upper half and lower half of a set of data, respectively.
Interquartile Range
The range of the middle half of the data. It is the difference between the upper quartile and the lower quartile.
Range
The difference between the greatest and least data values
20, 22, 22, 25, 27, 30, 31, 35
median
Upper quartileLower quartile
30 + 31 = 6122 + 22 = 44
44 2 = 22 61 2 = 30.5
30.5 – 22 = 8.5
35 – 20 = 15
Lower quartile = 22
Upper quartile = 30.5
median
25 + 27 = 52
52 2 = 26

4. 4, 4, 6, 7, 8, 10, 12, 15, 19
Measures of Variation
medianLower quartile Upper quartile
Interquartile range
85 13.5
8.5
Range
15
13.5 - 5
19 - 4

5. 4, 4, 6, 7, 8, 10, 12, 19
Measures of Variation
medianLower quartile Upper quartile
Interquartile range
7.55 11
6
Range
15

6. Outlier• An outlier is a data value that is either much greater or much less than the median. If a
data value is more than 1.5 times the value of the interquartile range beyond the
quartiles, is an outlier.
2, 21, 23, 23, 24, 25, 27, 31
Lower quartile Upper quartile
Interquartile range
4 X 1.5
6
22 26
4
Add 6Subtract 6
26 + 6 = 3222 – 6 = 16
If 31 is greater than
or equal to 32.
It is an outlier.
If 2 is less than
or equal to 16.
It is an outlier
Outlier Test Outlier Test
Pass Fail

7. 4, 4, 6, 7, 8, 10, 12, 19
Outlier Test
medianLower quartile Upper quartile
Interquartile range
7.55 11
6
6 X 1.5
9
9 + 11 = 205 – 9 = -4
Outlier TestOutlier Test

8. -7, 3, 5, 6, 8, 9, 11, 25
Outlier Test
medianLower quartile Upper quartile
Interquartile range
7 10
6
6 X 1.5
9
10 + 9 = 194 – 9 = -5
Outlier TestOutlier Test
4

9. -3, 4, 6, 7, 8, 9, 17, 24
Outlier Test
medianLower quartile Upper quartile
Interquartile range
7.5 13
8
8 X 1.5
12
Outlier TestOutlier Test
5
5 – 12 = -7 13 + 12 = 25

10. -3, -2, 4, 5, 5, 12, 18, 30
Outlier Test
medianLower quartile Upper quartile
Interquartile range
5 15
14
14 X 1.5
21
Outlier TestOutlier Test
1
1 – 21 = -20 15 + 21 = 36

11. Box-and-Whisker Plots
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• A box-and-whisker plot is a diagram that is constructed using the median, quartiles,
and extreme values. A box is drawn around the quartile values, and the whiskers
extend from each quartile to the extreme values. The median is marked with a vertical
line.
.. .
Upper quartileLower quartile
median
Interquartile range
42 – 30 = 12
Upper extremeLower extreme
The outlier will be
shown with a small
symbol that is off the
Box-and-Whisker Plot

12. Box-and-Whisker Plots
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.. .
Upper quartileLower quartile
median
Interquartile range
31 - 22 = 9
8, 20, 22, 22, 25, 27, 30, 31, 35, 52
median
Upper quartileLower quartile
26
Lower extreme Upper extreme OutlierOutlier

13. Box-and-Whisker Plots
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.. .
Upper quartileLower quartile
median
Interquartile range
32 - 20 = 12
8, 19, 20, 20, 21 22, 26, 27, 30, 34, 35, 44
median
Upper quartileLower quartile
Lower extreme Upper extreme

14. Box-and-Whisker Plots
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.. .
Upper quartileLower quartile
median
Interquartile range
32 - 20 = 12
6, 16, 20, 20, 21 22, 26, 27, 30, 34, 35, 44
median
Upper quartileLower quartile
Lower extreme Upper extreme
Outlier

15. Box-and-Whisker Plots
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.. .
Upper quartileLower quartile
median
12, 13, 15, 17, 21 26, 26, 27, 27, 29, 35, 38
median
Upper quartileLower quartile
Lower extreme Upper extreme

16. Box-and-Whisker Plots
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..
median
8, 9, 16, 18, 30 32, 36, 40, 44, 46, 50, 58
median
Upper quartileLower quartile
Upper extreme
Lower quartile Upper quartile
.
Lower extreme

17. Box-and-Whisker Plots
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.. .
16, 20, 21, 23, 28 29, 35, 37, 37, 49, 55, 57
median
Upper quartileLower quartile

18. Box-and-Whisker Plots
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.. .
Median_______________
Upper quartile_________
Lower quartile_________
Upper extreme________
Lower extreme________
Outlier_______________
30
40
24
50
16
no outlier

19. Box-and-Whisker Plots
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.. .
Median_______________
Upper quartile_________
Lower quartile_________
Upper extreme________
Lower extreme________
Outlier_______________
42
50
32
60
26
8

20. Box-and-Whisker Plots
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.. .
Median_______________
Upper quartile_________
Lower quartile_________
Upper extreme________
Lower extreme________
Outlier_______________
36
44
32
52
28
10

21. Box-and-Whisker Plots
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.. .
Median_______________
Upper quartile_________
Lower quartile_________
Upper extreme________
Lower extreme________
Outlier_______________
18
26
32
8
60
13