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Lesson 3 - 5

Lesson 3 - 5. The Five-Number Summary and Boxplots. Objectives. Compute the five-number summary Draw and interpret boxplots. Vocabulary. Five-number Summary – the minimum data value, Q 1 , median, Q 3 and the maximum data value. Five-number summary.

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Lesson 3 - 5

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  1. Lesson 3 - 5 The Five-Number Summary and Boxplots

  2. Objectives • Compute the five-number summary • Draw and interpret boxplots

  3. Vocabulary • Five-number Summary – the minimum data value, Q1, median, Q3 and the maximum data value

  4. Five-number summary Min Q1 M Q3 Max smallest value largest value First, Second and Third Quartiles (Second Quartile is the Median, M) LowerFence UpperFence Boxplot ] [ * Smallest Data Value > Lower Fence Largest Data Value < Upper Fence (Min unless min is an outlier) (Max unless max is an outlier) Outlier

  5. Distribution Shape Based on Boxplots: • If the median is near the center of the box and each horizontal line is of approximately equal length, then the distribution is roughly symmetric • If the median is to the left of the center of the box or the right line is substantially longer than the left line, then the distribution is skewed right • If the median is to the right of the center of the box or the left line is substantially longer than the right line, then the distribution is skewed left

  6. Why Use a Boxplot? • A boxplot provides an alternative to a histogram, a dotplot, and a stem-and-leaf plot. Among the advantages of a boxplot over a histogram are ease of construction and convenient handling of outliers. In addition, the construction of a boxplot does not involve subjective judgements, as does a histogram. That is, two individuals will construct the same boxplot for a given set of data - which is not necessarily true of a histogram, because the number of classes and the class endpoints must be chosen. On the other hand, the boxplot lacks the details the histogram provides. • Dotplots and stemplots retain the identity of the individual observations; a boxplot does not. Many sets of data are more suitable for display as boxplots than as a stemplot. A boxplot as well as a stemplot are useful for making side-by-side comparisons.

  7. Example 1 Consumer Reports did a study of ice cream bars (sigh, only vanilla flavored) in their August 1989 issue. Twenty-seven bars having a taste-test rating of at least “fair” were listed, and calories per bar was included. Calories vary quite a bit partly because bars are not of uniform size. Just how many calories should an ice cream bar contain? Construct a boxplot for the data above.

  8. Example 1 - Answer Q1 = 182 Q2 = 221.5 Q3 = 319 Min = 111 Max = 439 Range = 328 IQR = 137 UF = 524.5 LF = -23.5 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 Calories

  9. Example 2 The weights of 20 randomly selected juniors at MSHS are recorded below: a) Construct a boxplot of the data b) Determine if there are any mild or extreme outliers.

  10. Example 2 - Answer Q1 = 130.5 Q2 = 138 Q3 = 145.5 Min = 121 Max = 213 Range = 92 IQR = 15 UF = 168 LF = 108 Extreme Outliers( > 3 IQR from Q3) * * 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 Weight

  11. Example 3 The following are the scores of 12 members of a woman’s golf team in tournament play: a) Construct a boxplot of the data. b) Are there any mild or extreme outliers? c) Find the mean and standard deviation. d) Based on the mean and median describe the distribution?

  12. Example 3 - Answer Q1 = 84.5 Q2 = 88.5 Q3 = 93 Min = 79 Max = 111 Range = 32 IQR = 18.5 UF = 120.75 LF = 56.75 Golf Scores 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120 123 126 No Outliers Mean= 90.67 St Dev = 9.85 Distribution appears to be skewed right (mean > median and long whisker)

  13. Example 4 Comparative Boxplots: The scores of 18 first year college women on the Survey of Study Habits and Attitudes (this psychological test measures motivation, study habits and attitudes toward school) are given below: The college also administered the test to 20 first-year college men. There scores are also given: Compare the two distributions by constructing boxplots. Are there any outliers in either group? Are there any noticeable differences or similarities between the two groups?

  14. Example 4 - Answer Q1 = 12698 Q2 = 138.5114.5 Q3 = 154 143 Min = 10170 Max = 200187 Range = 99 117 IQR = 2845 UF = 196210.5 LF = 59 30.5 Comparing Men and Women Study Habits and Attitudes Women * 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 Men Women’s median is greater and they have less variability (spread) in their scores; the women’s distribution is more symmetric while the men’s is skewed right. Women have an outlier; while the men do not.

  15. Summary and Homework • Summary • Boxplots are used for checking for outliers • Use comparative boxplots for two datasets • Constructing a boxplot is not subjective • Identifying a distribution from boxplots or histograms is subjective! • Homework:pg 181-183: 5-7, 15

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