The Five-Number Summary

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 gt Lower Fence
Largest Data Value lt 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.

342 377 319 353 295 234 294 286
377 182 310 439 111 201 182 197
209 147 190 151 131 151
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 40
0 425 450 475 500
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.

121 126 130 132 143 137 141 144 148 205
125 128 131 133 135 139 141 147 153 213
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( gt 3 IQR from Q3)


100 110 120 130 140 150 160 170 180 190 200 210 22
0 230 240 250 260
Weight
11
Example 3

• The following are the scores of 12 members of a
  womans 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
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 12
0 123 126
No Outliers Mean 90.67 St Dev
9.85 Distribution appears to be skewed right
(mean gt 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?

154 109 137 115 152 140 154 178 101
103 126 126 137 165 165 129 200 148
108 140 114 91 180 115 126 92 169 146
109 132 75 88 113 151 70 115 187 104
14
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 19
0 200 210 220
Men
Womens median is greater and they have less
variability (spread) in their scores the womens
distribution is more symmetric while the mens 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!
• Homeworkpg 181-183 5-7, 15