The Five-Number Summary and Boxplots 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, Q1, median, Q3 and the maximum data value

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

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

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.

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. 342 377 319 353 295 234 294 286 182 310 439 111 201 197 209 147 190 151 131

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

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. 121 126 130 132 143 137 141 144 148 205 125 128 131 133 135 139 147 153 213

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

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? 89 90 87 95 86 81 111 108 83 88 91 79

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)

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? 154 109 137 115 152 140 178 101 103 126 165 129 200 148 108 140 114 91 180 115 126 92 169 146 109 132 75 88 113 151 70 187 104

Example 4 - Answer Q1 = 126 98 Q2 = 138.5 114.5 Q3 = 154 143 Min = 101 70 Max = 200 187 Range = 99 117 IQR = 28 45 UF = 196 210.5 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.

Summary and Homework Summary Homework: pg 181-183: 5-7, 15 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