# 2.2.10 - Five Number Summary

Five Number Summary
Minimum, Q1, Median, Q3, Maximum

Q1 is the first quartile, this is the 25th percentile
Q3 is the third quartile, this is the 75th percentile

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.

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