Section 7.7

Integration: Statistics

Box-and-Whisker Plots

What you should learn

To display and interpret data on box-and-whisker plots

NCTM Curriculm Standards 2, 5 - 10

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Box-and-Whisker Plot

Extreme Values

Whiskers

Introduction: In 1996, the site of the Summer Games of the XXVI Olympiad was Atlanta, Georgia. The table below shows the number of gold medals won by the top 10 medal-winning teams.

 1996 SUMMER OLYMPICS Top 10 Medal Winners Team Number of Medals Won Gold Total USA 44 101 Russia 26 63 Germany 20 65 China 16 50 France 15 37 Italy 13 35 Australia 9 41 South Korea 7 27 Cuba 9 25 Ukraine 9 23

We can describe this data using the mean, median, and mode. We can also use the quartiles and interquartile range, to obtian a graphic representation of the data. A type of diagram, or graph, that shows quartiles and extreme values of data is called a box-and-whisker plot.

Suppose we wanted to make a box-and-whisker plot of the numbers of gold medals won by each of the nations in the table. First, arrrange the data in numberical order. Next, compute the median and quartiles. Also, identify the extreme values.

7, 9, 9, 9, 13, 15, 16, 20, 26, 44

The median for this set of data is the average of the eighth and ninth values.

median = (13+ 15) / 2 = 14

Recall that the lower quartile (Q1) is the median of the lower half of the distribution of values. The upper quartile (Q3) is the median of the upper half of the data.

7, 9, 9, 9, 13

Where Q1 = 9

and

15, 16, 20, 26, 44

Where Q3 = 20

The extreme values are the least value (LV), 7, and the greatest value (GV), 44.

Now we have the information we need to draw a box-and-whisker plot.

Step 1: Draw a number line. Assign a scale to the number line that includes the extreme values. Plot dots to represent the extreme values (LV and GV), the upper and lower quartile points (Q3 and Q1), and the median (Q2).

Step 2: Draw a box to designate the data falling between the upper and lower quartiles. Draw a vertical line through the point representing the median. Draw a segment from the lower quartile to the least value and one from the upper quartile to the greatest value. These segments are the whiskers of the plot.

Even though the whiskers are different lengths, each whisker contains at least one fourth of the data while the box contains one half of the data. Compound inequalities can be used to describe the data in each fourth. Assume that the replacement set for x is the set of data.

1st fourth {x|x < 9}

2nd fourth {x|9 < x < 14}

3rd fourth {x|14 < x < 20}

4th fourth {x|x > 20}

Step 3: Before finishing the box-and-whisker plot, check for outliers. In Lesson 5-7, you learned that an outlier is any element of the set of data that is at least 1.5 interquartile ranges above the upper quartile or below the lower quartile. Recall that the interquartile range (IQR) is the difference between the upper and lower quartiles, or in this case, 20 - 9 or 11.

x Q3 + 1.5 (IQR)

x 20 + 1.5 (11)

x 20 + 16.5

x 36.5

or

x Q1 - 1.5 (IQR)

x9 - 1.5 (11)

x9 - 16.5

x-7.5

Step 4: If x is an outlier in this set of data, then the outliers an be described as {x|x-7.5 or x36.5}. In this case, there are no data less than -7.5. However, 44 is greater than 36.5, so it is an outlier. We now need to revise the box-and-whisker plot. Outliers are plotted as isolated points, and the right whisker is shortened to stop at 26.

Exercise 1: Refer to the application at the beginning of the lesson. Use the box-and-whisker plot for the gold medal to answer each question.

a. What percent of the teams won between 14 and 20 medals?

b. What does the box-and-whisker plot tell us about the upper half of the data compared to the lower half?

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 9 - 15 odd, 16 - 20

Alternative Homework: Enriched: 8 - 14 even, 15 - 20

Extra Practice: Students book page 773 Lesson 7-7

Extra Practice Worksheet: Click Here.