Measures of Variation. .

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Measures of Variation

Box and Whisker

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

Measures of Variation• Measures of variation are used to describe the distribution of the data.

Upper and Lower QuartilesThe upper and lower quartiles are the medians of the upper half and lower half of a set of data, respectively.

Interquartile RangeThe range of the middle half of the data. It is the difference between the upper quartile and the lower quartile.

RangeThe difference between the greatest and least data values

20, 22, 22, 25, 27, 30, 31, 35

median

Upper quartileLower quartile30 + 31 = 6122 + 22 = 44

44 2 = 22 61 2 = 30.5

30.5 – 22 = 8.5

35 – 20 = 15

Lower quartile = 22Upper quartile = 30.5

median25 + 27 = 5252 2 = 26

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

4, 4, 6, 7, 8, 10, 12, 19

Measures of Variation

medianLower quartile Upper quartile

Interquartile range

7.55 11

6

Range

15

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 = 16If 31 is greater thanor 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

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

-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

-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

-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

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 range42 – 30 = 12

Upper extremeLower extreme

The outlier will be shown with a smallsymbol that is off theBox-and-Whisker Plot

Box-and-Whisker Plots

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.. .

Upper quartileLower quartile

median

Interquartile range31 - 22 = 9

8, 20, 22, 22, 25, 27, 30, 31, 35, 52

medianUpper quartileLower quartile

26

Lower extreme Upper extreme OutlierOutlier

Box-and-Whisker Plots

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.. .

Upper quartileLower quartile

median

Interquartile range32 - 20 = 12

8, 19, 20, 20, 21 22, 26, 27, 30, 34, 35, 44

medianUpper quartileLower quartile

Lower extreme Upper extreme

Box-and-Whisker Plots

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.. .

Upper quartileLower quartile

median

Interquartile range32 - 20 = 12

6, 16, 20, 20, 21 22, 26, 27, 30, 34, 35, 44

medianUpper quartileLower quartile

Lower extreme Upper extreme

Outlier

Box-and-Whisker Plots

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.. .

Upper quartileLower quartile

median

12, 13, 15, 17, 21 26, 26, 27, 27, 29, 35, 38

medianUpper quartileLower quartile

Lower extreme Upper extreme

Box-and-Whisker Plots

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..

median

8, 9, 16, 18, 30 32, 36, 40, 44, 46, 50, 58

medianUpper quartileLower quartile

Upper extreme

Lower quartile Upper quartile

.

Lower extreme

Box-and-Whisker Plots

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.. .

16, 20, 21, 23, 28 29, 35, 37, 37, 49, 55, 57

medianUpper quartileLower quartile

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

Box-and-Whisker Plots

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.. .

Median_______________

Upper quartile_________

Lower quartile_________

Upper extreme________

Lower extreme________

Outlier_______________

42

50

32

60

26

8

Box-and-Whisker Plots

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.. .

Median_______________

Upper quartile_________

Lower quartile_________

Upper extreme________

Lower extreme________

Outlier_______________

36

44

32

52

28

10

Box-and-Whisker Plots

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.. .

Median_______________

Upper quartile_________

Lower quartile_________

Upper extreme________

Lower extreme________

Outlier_______________

18

26

32

8

60

13