Five-number Summary

TL;DR

• Condenses a distribution into five key statistics: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
• Computed after ordering the data; useful for quick inspection of spread and center.
• Helps identify potential outliers by comparing values to these extremes and quartiles.

Definition

A five-number summary is a method of summarizing a dataset by providing a concise description of its key features. It consists of the minimum value, the maximum value, the median, the first quartile, and the third quartile.

Explanation

To compute a five-number summary:

• Order the observations from least to greatest.
• Report:
  • The minimum (lowest) value.
  • The first quartile (Q1), which divides the lower half of the dataset into two parts.
  • The median, the middle value of the ordered dataset.
  • The third quartile (Q3), which divides the upper half of the dataset into two parts.
  • The maximum (highest) value.

These five numbers provide a rough sketch of the data distribution and can be used to identify potential outliers.

Examples

Generic example (100 numbers)

For a dataset of 100 numbers, after ordering:

• Minimum: 1
• Maximum: 100
• Median: 50
• First quartile (Q1): 25
• Third quartile (Q3): 75

Using these five numbers, one can quickly summarize the data and identify potential outliers (for example, a value much lower than 1 or much higher than 100).

Class test scores (20 students)

Raw scores: 80, 85, 90, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15

Ordered: 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 80, 85, 85, 90, 90, 95

Five-number summary:

• Minimum: 15
• Maximum: 95
• Median: 60
• First quartile (Q1): 40
• Third quartile (Q3): 85

Using these five numbers, a score of 100 would be identified as an outlier because it is much higher than the maximum value of 95.

Use cases

• Quickly summarizing the central tendency and spread of a dataset.
• Identifying potential outliers.
• Supporting simple visualizations of distribution (e.g., as an input to a boxplot).

Notes or pitfalls

• The five-number summary provides a rough sketch and may not capture all distribution details.
• The source describes identifying outliers by comparing values to the reported minimum and maximum (e.g., values much lower than 1 or much higher than 100 in the generic example, or a score of 100 compared with a maximum of 95 in the test-score example).

Related terms

• Median
• Quartile
• Outlier