6-9 Data Distributions

Objective

• Create and interpret box-and-whisker plots.

Another way to describe a data set is how the data values are spread out from the center.

Quartiles divide a data set into four equal parts.

Each quartile contains one-fourth of the values in the set.

First quartile: median of the lower half of the data set

Second quartile: median of the whole data set

Third quartile: median of the upper half of the data set

Reading Math

The first quartile is sometimes called the lower quartile, and the third quartile is sometimes called the upper quartile.

Interquartile range (IQR) is the difference between the third and first quartiles. It represents the range of the middle half of the data.

A box-and-whisker plot can be used to show how the values in a data set are distributed.

You need five values to make a box and whisker plot

1. minimum (or lowest value)

2. first quartile

3. Median

4. third quartile

5. maximum (or greatest value).

Median

First quartile Third quartile

Minimum Maximum

Example 1: Application

The number of runs scored by a softball team in 19 games is given. Use the data to make a box-and-whisker plot.

3, 8, 10, 12, 4, 9, 13, 20, 12, 15, 10, 5, 11, 5, 10, 6, 7, 6, 11

Step 1 Order the data from least to greatest.

3, 4, 5, 5, 6, 6, 7, 8, 9, 10, 10, 10, 11, 11, 12, 12, 13, 15, 20

Step 2 Identify the five needed values.

3, 4, 5, 5, 6, 6, 7, 8, 9, 10, 10, 10, 11, 11, 12, 12, 13, 15, 20

Q1

6

Q3

12

Q2

10

Minimum

3

Maximum

20

Example 1 Continued

0 8 16 24

Median

First quartile Third quartile

Minimum Maximum

Step 3 Draw a number line and plot a point above each of the five needed values.

Draw a box through the first and third quartiles and a vertical line through the median.

Draw lines from the box to the minimum and maximum.

Use the data to make a box-and-whisker plot.

13, 14, 18, 13, 12, 17, 15, 12, 13, 19, 11, 14, 14, 18, 22, 23

You Try! Example 2

Step 1 Order the data from least to greatest.

11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23

Step 2 Identify the five needed values.

11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23

Q1

13

Q3

18

Q2

14

Minimum

11

Maximum

23

You Try! Example 2 Continued

8 16 24

Median

First quartile Third quartile

• ••• •

Minimum Maximum

Step 3 Draw a number line and plot a point above each of the five needed values.

The box-and-whisker plots show the number of mugs sold per student in two different grades.

Example 3: Reading and Interpreting Box-and-Whisker Plots

A. About how much greater was the median number of mugs sold by the 8th grade than the median number of mugs sold by the 7th grade?

about 5B. Which data set has a greater maximum? Explain.

8th grade; point for maximum is farther to the right for the 8th grade than for the 7th grade

50% of all the numbers are between Q1 and Q3

This is called the Inter-Quartile Range (IQR)

17 - 9 = 8

3 7 9 12 14 15 17 18 40

IQR = 8

min maxmedian Q1Q2

3 7 9 12 14 15 17 18 40

IQR = 8

To determine if a number is an outlier, multiply the IQR by 1.5

8 • 1.5 = 12

An outlier is any number that is 12 less than Q1 or 12 more than Q3

min maxmedian Q1Q2

Example 4Given a 5 number summary, provide two outliers. One outlier should be greater than Q3 and one outlier should be less than Q1.

Step 1 Identify Q1 and Q3.

Q1

3, 6, 10, 12, 20

Q3

Step 2 Calculate the IQR. 12 – 6 =6

Step 3 Calculate the outlier factor. 6(1.5) =9

Any number that is 9 less than Q1 or 9 more than Q3

Examples: -3 or 21

Classwork/Homework

6-9 Worksheet