. . Box and Whisker Measures of Variation Measures of Variation 8 12 16 20 24 28 32 36 40 44 48 52 56 60

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

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

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

Measures of Variation 4, 4, 6, 7, 8, 10, 12, 19 Lower quartile median Upper quartile 5 7.5 11 Interquartile range 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 Upper quartile Lower quartile 22 26 Interquartile range 4 Outlier Test Outlier Test 22 – 6 = 16 26 + 6 = 32 If 2 is less than or equal to 16. It is an outlier If 31 is greater than or equal to 32. It is an outlier. 4 X 1.5 6 Subtract 6 Add 6 Pass Fail

9 4, 4, 6, 7, 8, 10, 12, 19 5 11 6 Outlier Test 7.5 Lower quartile 4, 4, 6, 7, 8, 10, 12, 19 Lower quartile median Upper quartile 5 7.5 11 Interquartile range 6 5 – 9 = -4 9 + 11 = 20 Outlier Test Outlier Test 6 X 1.5 9

9 -7, 3, 5, 6, 8, 9, 11, 25 4 10 6 Outlier Test 7 Lower quartile -7, 3, 5, 6, 8, 9, 11, 25 Lower quartile median Upper quartile 4 7 10 Interquartile range 6 4 – 9 = -5 10 + 9 = 19 Outlier Test Outlier Test 6 X 1.5 9

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

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

Box-and-Whisker Plots 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. The outlier will be shown with a small symbol that is off the Box-and-Whisker Plot . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 median Lower extreme Upper extreme Lower quartile Upper quartile Interquartile range 42 – 30 = 12

Box-and-Whisker Plots 8, 20, 22, 22, 25, 27, 30, 31, 35, 52 Lower quartile Upper quartile median . . 26 . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 Outlier Lower extreme median Upper extreme Outlier Lower quartile Upper quartile Interquartile range 31 - 22 = 9

Box-and-Whisker Plots 8, 19, 20, 20, 21 22, 26, 27, 30, 34, 35, 44 Lower quartile Upper quartile median . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 median Upper extreme Lower extreme Lower quartile Upper quartile Interquartile range 32 - 20 = 12

Box-and-Whisker Plots 6, 16, 20, 20, 21 22, 26, 27, 30, 34, 35, 44 Lower quartile Upper quartile median . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 median Lower extreme Upper extreme Lower quartile Upper quartile Outlier Interquartile range 32 - 20 = 12

Box-and-Whisker Plots 12, 13, 15, 17, 21 26, 26, 27, 27, 29, 35, 38 Lower quartile Upper quartile median . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 median Lower extreme Upper extreme Lower quartile Upper quartile

Box-and-Whisker Plots 8, 9, 16, 18, 30 32, 36, 40, 44, 46, 50, 58 Lower quartile Upper quartile median . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 median Lower quartile Upper quartile Lower extreme Upper extreme

Box-and-Whisker Plots 16, 20, 21, 23, 28 29, 35, 37, 37, 49, 55, 57 Lower quartile Upper quartile median . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60

Box-and-Whisker Plots . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 Median_______________ Upper quartile_________ Lower quartile_________ Upper extreme________ Lower extreme________ Outlier_______________ 30 40 24 50 16 no outlier

Box-and-Whisker Plots . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 Median_______________ Upper quartile_________ Lower quartile_________ Upper extreme________ Lower extreme________ Outlier_______________ 42 50 32 60 26 8

Box-and-Whisker Plots . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 Median_______________ Upper quartile_________ Lower quartile_________ Upper extreme________ Lower extreme________ Outlier_______________ 36 44 32 52 28 10

Box-and-Whisker Plots . . . 8 12 16 20 24 28 32 36 40 44 48 52 56 60 Median_______________ Upper quartile_________ Lower quartile_________ Upper extreme________ Lower extreme________ Outlier_______________ 18 26 13 32 8 60