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# Quartiles

## Introducing data for box-and-whisker plots

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Quartiles

### Credit: Alexander Stein Source: https://pixabay.com/en/calculator-paperclip-pen-office-178127/ License: CC BY-NC 3.0 [Figure1]

William is making a sales report for the week. The number of items sold were as follows:

Monday - 42

Tuesday - 32

Wednesday - 80

Thursday - 75

Friday - 90

Saturday - 65

Sunday - 22

What measure of data does William need to build a box-and-whisker plot for his report?

In this concept, you will learn how to find measures of data to build a box-and-whisker plot.

### Quartiles

Once you have collected data, you can graph the information with a box-and whisker plot. Before building a box-and-whisker plot, you will need to find the median, quartile, and extremes of the data.

The median is the middle number in a set of data that is ordered from least to greatest. If there is an odd number of values, the middle number is the median. If there is an even number of values, the average of the two middle values is the median.

Here is a data set from a survey of the number of hours worked by teenagers with part-time jobs.

16, 10, 8, 8, 11, 11, 12, 15, 10, 20, 6, 16, 8

First, order the data from least to greatest including any repeated numbers.

6, 8, 8, 8, 10, 10, 11, 11, 12, 15, 16, 16, 20

Then, identify the median. There are 13 values in the data set. Remember that the median is the middle number in a data set with an odd number of values. The median is 11.

6, 8, 8, 8, 10, 10, 11, 11, 12, 15, 16, 16, 20

Use the median to divide the data set into two parts. The first half are the values from 6 to 10 and the second half are the values starting at the second 11 up to 20.

6, 8, 8, 8, 10, 10, 11, 11, 12, 15, 16, 16, 20

A quartile divides the data set into 4 parts. The median of the first half of the data is called the lower quartile. The median of the second half of the data is called the upper quartile.

The lower quartile is the average between 8 and 8. The lower quartile is 8.

The upper quartile is the average between 15 and 16. The upper quartile is 15.5.

6, 8, 8 | 8, 10, 10, 11, 11, 12, 15 | 16, 16, 20

The extremes are the lowest value in a data set (the lower extreme) and the highest value in a data set (the upper extreme).

In the data set, 6 is the lower extreme and 20 is the upper extreme.

This information will be used to make a box-and-whisker plot.

### Examples

#### Example 1

Earlier, you were given a problem about William's sales report.

William needs to identify measure of data to build a box-and-whisker plot for his sales report.

Monday - 42

Tuesday - 32

Wednesday - 80

Thursday - 75

Friday - 90

Saturday - 65

Sunday - 22

Find the median, lower quartile, upper quartile, lower extreme, and upper extreme of the data set.

First, order the data values from least to greatest.

22, 32, 42, 65, 75, 80, 90

Then, identify the median. The middle value is 65.

Next, identify the lower and upper quartile. The median of the first data set is 32. The median of the second data set is 80.

After that, identify the lower and upper extremes. The lowest value is 22 and the highest value is 90.

The measures of data are as follows.

median: 65

lower quartile: 32

upper quartile: 80

lower extreme: 22

upper extreme:90

#### Example 2

What is the median of this data set?

4, 5, 12, 11, 9, 8, 7, 4, 3

First, order the values from least to greatest.

3, 4, 4, 5, 7, 8, 9, 11, 12

Then, identify the median. There are nine values in this data set. The median is the middle number.

3, 4, 4, 5, 7, 8, 9, 11, 12

The median is 7.

#### Example 3

What is the median of this data set?

4, 4, 5, 6, 7, 8, 11, 13, 16

The values are already listed in order from least to greatest.

Identify the middle value.

4, 4, 5, 6, 7, 8, 11, 13, 16

The median is 7.

#### Example 4

What is the lower and upper quartile of this data set?

4, 4, 5, 6, 7, 8, 11, 13, 16

First, identify the median of the first half of the data set.

4, 4 | 5, 6, 7, 8, 11, 13, 16

The lower quartile is between 4 and 5.

Then, identify the median of the second half of the data set.

4, 4 | 5, 6, 7, 8, 11 | 13, 16

The upper quartile is between 11 and 13.

The lower quartile is 4.5 and the upper quartile is 12.

#### Example 5

What are the extremes of this data set?

4, 4, 5, 6, 7, 8, 11, 13, 16

The lowest value is 4 and the highest value is 16.

The lower extreme is 4 and the upper extreme is 16.

### Review

Use each data set to answer the questions following it.

3, 5, 6, 8, 11, 13, 15, 17, 19

1. How many values are there in this data set?
2. What is the median of the data?
3. What is the range?
4. What is the upper quartile?
5. What is the lower quartile?
6. What are the extremes?

100, 112, 115, 122, 123, 126, 130, 131

1. How many values are there in this data set?
2. What is the median of the data?
3. What is the range?
4. What is the upper quartile?
5. What is the lower quartile?
6. What are the extremes?

113, 120, 131, 142, 150, 155, 157, 161, 167

1. How many values are there in this data set?
2. What is the median of the data?
3. What is the range?
4. What is the upper quartile?
5. What is the lower quartile?
6. What are the extremes?

To see the Review answers, open this PDF file and look for section 6.16.

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Extremes The extremes are the maximum and minimum values in a data set.
first quartile The first quartile, also known as $Q_1$, is the median of the lower half of the data.
Lower quartile The lower quartile, also known as $Q_1$, is the median of the lower half of the data.
Median The median of a data set is the middle value of an organized data set.
Quartile A quartile is each of four equal groups that a data set can be divided into.
second quartile The second quartile, also known as $Q_2$, is the median of the data.
third quartile The third quartile, also known as $Q_3$, is the median of the upper half of the data.
Upper Quartile The upper quartile, also known as $Q_3$, is the median of the upper half of the data.