2.6: Boxplots Objective: To find the five-number summaries of data

and create and analyze boxplots

CHS Statistics

The Five-Number Summary

The five-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum).

Example: The five-number summary for the daily wind speed is:

Max 8.67

Q3 2.93

Median 1.90

Q1 1.15

Min 0.20

Making Boxplots

A boxplot is a graphical display of the five-number summary.

Boxplots are useful when comparing groups.

Boxplots are particularly good at pointing out outliers.

Constructing Boxplots

1. Draw a single vertical axis spanning the range of the data. Draw short horizontal lines at the lower and upper quartiles and at the median. Then connect them with vertical lines to form a box.

Constructing Boxplots (cont.)

2. Draw “fences” around the main part of the data.

The upper fence is 1.5*(IQR) above the upper quartile.

The lower fence is 1.5*(IQR) below the lower quartile.

Note: the fences only help with constructing the boxplot and should not appear in the final display.

Constructing Boxplots (cont.)

3. Use the fences to grow “whiskers.”

Draw lines from the ends of the box up and down to the most extreme data values found within the fences.

If a data value falls outside one of the fences, we do not connect it with a whisker.

Constructing Boxplots (cont.)

4. Add the outliers by displaying any data values beyond the fences with special symbols.

We often use a different symbol for “far outliers” that are farther than 3 IQRs from the quartiles.

Overview of Boxplots

Extreme Outlier more than (3*IQR) above and below 3rd and 1st quartiles

Outliers more than (1.5*IQR) above and below 3rd and 1st quartiles

Upper Fence

Max value within fence

Q3Median

Q1

Min value within fence

Lower Fence

Wind Speed: Making Boxplots (cont.) Compare the histogram and boxplot for daily wind

speeds:

How does each display represent the distribution?

What Do Boxplots Tell Us?

The center of the boxplot shows us the middle half of the data between the quartiles.

The height of the box is equal to the IQR.

If the median is roughly centered between the quartiles, then the middle half of the data is roughly symmetric. Thus, if the median is not centered, the distribution is skewed.

The whiskers also show the skewness if they are not the same length.

Outliers are out of the way to keep you from judging skewness, but give them special attention.

Comparing Groups

What do these boxplots tell you?

Example: Construct a Boxplot:

The following are test scores for the written portion of the physical education final exam.

40, 73, 81, 95, 97, 32, 17, 107, 50, 51, 57, 67, 72

Example: Construct a Boxplot:

The following are test scores for the CHS final exam.

75, 70, 71, 72, 80, 73, 71, 74, 70, 72, 74, 73, 71, 70, 72

Assignment

pp.102 # 2, 7