# 2.2.10 - Five Number Summary

2.2.10 - Five Number SummaryA **five number summary** can be used to communicate some key descriptive statistics. It is comprised of five values, presented in the following order:

**Minimum**: Smallest value**First quartile (Q**: 25th percentile (value that separates the bottom 25% of the distribution from the top 75%)_{1})**Median**: Middle value (50th percentile)**Third quartile (Q**: 75th percentile (value that separates the bottom 75% of the distribution from the top 25%)_{3})**Maximum**: Largest value

- Five Number Summary
- Minimum, Q
_{1}, Median, Q_{3}, 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

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

A teacher wants to examine studentsâ€™ 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.

\(IQR = Q3 - Q1 = 90 - 80 = 10\)