1. Analyzing One-Variable Data
Lesson 1.8
Summarizing Quantitative Data:
Boxplots and Outliers
Statistics and Probability with Applications, 3rd Edition
Starnes & Tabor
Bedford Freeman Worth Publishers

2. Boxplots and Outliers Learning Targets
After this lesson, you should be able to:
Use the 1.5 x IQR rule to identify outliners.
Make and interpret boxplots of quantitative data.
Compare distributions of quantitative data with boxplots.

3. Boxplots and Outliers
Barry Bonds set the major league record by hitting 73 home runs in a single season in The dotplot below shows the number of home runs that Bonds hit in each of his 21 complete seasons:
Bonds’s 73 home run season stands out (in red) from the rest of the distribution. Should this value be classified as an outlier?

4. Boxplots and Outliers
Besides serving as a measure of variability, the interquartile range (IQR) is used as a ruler for identifying outliers.
The 1.5 x IQR Rule
Call an observation an outlier if it falls more than 1.5 × IQR above the third quartile or below the first quartile. That is,
Low Outliers < Q1 −1.5 × IQR
High Outliers > Q × IQR

5. Boxplots and Outliers
It is important to identify outliers in a distribution for several reasons:
They might be inaccurate data values.
They can indicate a remarkable occurrence.
They can heavily influence the values of some summary statistics, like the mean, range, and standard deviation.

6. To Determine Outliers
Find Quartile 1 & Quartile 3
Determine Interquartile Range :
IQR = Q Q1
Multiply 1.5 x IQR
Set up “fences” Q1-(1.5 IQR) and Q3+(1.5 IQR)
Observations “outside” the fences are outliers.
Why 1.5? According to John Tukey, 1 IQR seemed like too little and 2 IQRs seemed like too much...

7. IDENTIFYING OUTLIERS
USE THE 1.5 IQR RULE TO DECIDE IF THERE ARE ANY OUTLIERS IN THE FOLLOWING DATA SET:

8. Boxplots and Outliers
You can use a dotplot, stemplot, or histogram to display the distribution of a quantitative variable. Another graphical option for quantitative data is a boxplot (sometimes called a box-and-whisker plot). A boxplot summarizes a distribution by displaying the location of 5 important values within the distribution, known as its five-number summary.
Five-Number Summary, Boxplot
The five-number summary of a distribution of quantitative data consists of the minimum, the first quartile Q1, the median, the third quartile Q3, and the maximum.
A boxplot is a visual representation of the five-number summary.

9. TO FIND THE 5-NUMBER SUMMARY ON THE CALCULATOR: ENTER DATA INTO A LIST .
USING THE CALCULATOR
TO FIND THE 5-NUMBER SUMMARY ON THE CALCULATOR:
ENTER DATA INTO A LIST
USING THE STAT MENU SCROLL TO STAT
AND RUN 1-VARS STATS ON LIST
Find the 5-number summary for the data list:

10. Exploratory data analysis tool.
Box-and-Whisker Plot
Exploratory data analysis tool.
Highlights important features of a data set.
Requires (five-number summary):
Minimum entry
First quartile Q1
Median Q2
Third quartile Q3
Maximum entry
Whisker
Maximum entry
Minimum entry
Box
Median, Q2 Q3 Q1
© 2012 Pearson Education, Inc. All rights reserved.
10 of 149

11. Drawing a Box-and-Whisker Plot
Find the five-number summary of the data set.
Construct a horizontal scale that spans the range of the data.
Plot the five numbers above the horizontal scale.
Draw a box above the horizontal scale from Q1 to Q3 and draw a vertical line in the box at Median.
Draw whiskers from the box to the minimum and maximum entries if there are no outliers.
Whisker
Maximum entry
Minimum entry
Box
Median, Q2 Q3 Q1
© 2012 Pearson Education, Inc. All rights reserved.
11 of 149

12. Example: Drawing a Box-and-Whisker Plot
Draw a box-and-whisker plot that represents 15 test scores with the following summary statistics: Min = 5 Q1 = 10 Q2 = 15 Q3 = 18 Max = 37
Solution: 5 10 15 18 37
About half the scores are between 10 and 18. By looking at the length of the right whisker, you can conclude 37 is a possible outlier.
Larson/Farber 4th ed. 12

13. Boxplots and Outliers
The top dotplot in the figure below shows Barry Bonds’s home run data.
The first quartile, the median, and the third quartile are marked with lines.
The process of testing for outliers with the 1.5 × IQR rule is shown in red.
Because there are no outliers, we draw the whiskers to the maximum and minimum data values, as shown in the finished boxplot at the bottom of the figure.

14. ALWAYS use modified boxplots in this class!!!
display outliers
fences mark off mild & extreme outliers
whiskers extend to largest (smallest) data value inside the fence
ALWAYS use modified boxplots in this class!!!

15. Construct a modified boxplot. Describe the distribution.
A report from the U.S. Department of Justice gave the following percent increase in federal prison populations in 20 northeastern & mid-western states in 1999.
Construct a modified boxplot. Describe the distribution.

16. Why use boxplots? ease of construction convenient handling of outliers
Used with medium or large size data sets (n > 10)
useful for comparative displays

17. Boxplots and Outliers
Boxplots provide a quick summary of the center and variability of a distribution and can also quickly be constructed on the calculator. Boxplots do not display each individual value in a distribution. And boxplots don’t show gaps, clusters, or peaks.

18. Boxplots and Outliers
Boxplots are especially effective for comparing the distribution of a quantitative variable in two or more groups.

19. Which is best at reducing stress?
LESSON APP 1.8
Which is best at reducing stress?
If you are a dog lover, having your dog with you may reduce your stress level. Does having a friend with you reduce stress? To examine the effect of pets and friends in stressful situations, researchers recruited 45 women who said they were dog lovers. Fifteen women were assigned at random to each of three groups: to do a stressful task (1) alone, (2) with a good friend present, or (3) with their dogs present. The stressful task was to count backward by 13s or 17s. The woman’s average heart rate during the task was one measure of the effect of stress. The following table shows the data.
Identify any outliers in the three groups. Show your work.
Make parallel boxplots to compare the heart rates of the women in the three groups.
Based on the data, does it appear that the presence of a pet or friend reduces heart rate during a stressful task? Justify your answer.
Alone
62.6
70.9
73.3
75.5
77.8
80.4
84.5
84.7
84.9
87.2
87.4
87.8
90.0
91.8
99.0
Friend
76.9
80.3
81.6
83.4
87.0
88.0
89.8
91.4
92.5
97.0
98.2
99.7
100.9
101.1
102.2
Pet
58.7
64.2
65.4
68.9
69.2
69.5
70.2
70.1
72.3
76.0
79.7
85.0
86.4
97.5

20. Boxplots and Outliers Learning Targets
After this lesson, you should be able to:
Use the 1.5 × IQR rule to identify outliners.
Make and interpret boxplots of quantitative data.
Compare distributions of quantitative data with boxplots.