# 2.2.10 - Five Number Summary

A five number summary can be used to communicate some key descriptive statistics. It is comprised of five values, presented in the following order:

1. Minimum: Smallest value
2. First quartile (Q1): 25th percentile (value that separates the bottom 25% of the distribution from the top 75%)
3. Median: Middle value (50th percentile)
4. Third quartile (Q3): 75th percentile (value that separates the bottom 75% of the distribution from the top 25%)
5. Maximum: Largest value
Five Number Summary
Minimum, Q1, Median, Q3, Maximum

Five number summaries are used to describe some of the key features of a distribution. Using the values in a five number summary we can also compute the range and interquartile range.

Range
The difference between the maximum and minimum values.
Range
$$Range = Maximum - Minimum$$
Note:
The range is heavily influenced by outliers. For this reason, the interquartile range is often preferred because it is resistant to outliers.
Interquartile range (IQR)
The difference between the first and third quartiles.
Interquartile Range
$$IQR = Q_3 - Q_1$$

## Example: Hours Spent Studying Section

A professor asks a sample of students how many hours they spent studying for the final. The five number summary for their responses is (5, 7, 9, 11, 13).

Range

The maximum is 13 and the minimum is 5.

$$Range = 13 - 5 = 8$$

Interquartile Range

The third quartile is 11 and the first quartile is 7.

$$IQR = Q_3 - Q_1 = 11 - 7 = 4$$

## Example: Test Scores Section

A teacher wants to examine students’ test scores. The five number summary for their scores is (74, 80, 89, 90, 98).

Range

The highest score is 98. The lowest score is 74.

$$Range = 98 - 74 = 24$$

Interquartile Range

The third quartile is 90 and the first quartile is 80.

$$IQR = Q3 - Q1 = 90 - 80 = 10$$