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