Five-Number Summary Calculator

What is a Five-Number Summary?

The five-number summary is a fundamental descriptive statistic that provides a concise snapshot of a dataset's distribution. It divides a set of observations into four equal segments, giving key insights into central tendency, spread, and overall range without assuming a normal distribution.

The Five Values in the Summary

A standard five-number summary consists of five values arranged in ascending order:

• Minimum (Min): The smallest data point in the dataset.
• First Quartile ($Q_1$): The 25th percentile value, marking the boundary below which 25% of the data falls.
• Median ($Q_2$): The 50th percentile value or middle number, dividing the dataset into two equal halves.
• Third Quartile ($Q_3$): The 75th percentile value, marking the boundary below which 75% of the data falls.
• Maximum (Max): The largest data point in the dataset.

Interquartile Range and Outlier Detection

Beyond the five numbers, the summary allows for important secondary statistical metrics:

• Range: Calculated as $\text{Range} = \text{Max} - \text{Min}$.
• Interquartile Range (IQR): Calculated as $\text{IQR} = Q_3 - Q_1$, representing the spread of the middle 50% of data.
• Lower Fence: $Q_1 - 1.5 \times \text{IQR}$. Data points below this threshold are potential low outliers.
• Upper Fence: $Q_3 + 1.5 \times \text{IQR}$. Data points above this threshold are potential high outliers.

How the Five-Number Summary Powers Box Plots

The five-number summary is the direct foundation for constructing a box plot (also called a box-and-whisker plot). In a box plot:

• The box spans from $Q_1$ to $Q_3$, containing the middle 50% of values.
• A vertical line inside the box marks the median ($Q_2$).
• The whiskers extend outward from $Q_1$ and $Q_3$ to the Minimum and Maximum non-outlier values.