# Five-Number Summary Calculator

Free five-number summary calculator. Calculate the minimum, Q1, median, Q3, and maximum from a normal distribution for box plot construction.

## What this calculates

Calculate the five-number summary (minimum, Q1, median, Q3, maximum) from a normal distribution. These five values completely describe the shape and spread of the data for a box plot.

## Inputs

- **Mean (μ)** — The mean of the distribution.
- **Standard Deviation (σ)** — min 0.0001 — The standard deviation of the distribution.
- **Sample Size (n)** — min 5 — The number of observations (used for estimating min/max).

## Outputs

- **Minimum (estimated)** — Estimated minimum based on sample size.
- **Q1 (25th Percentile)** — First quartile.
- **Median (Q2)** — The median (equals the mean for a normal distribution).
- **Q3 (75th Percentile)** — Third quartile.
- **Maximum (estimated)** — Estimated maximum based on sample size.
- **Range** — Maximum - Minimum.
- **IQR** — Q3 - Q1.

## Details

The five-number summary provides a compact description of a dataset.

The Five Numbers

- Minimum - the smallest value

- Q1 - 25th percentile (lower quartile)

- Median - 50th percentile (middle value)

- Q3 - 75th percentile (upper quartile)

- Maximum - the largest value

For a Normal Distribution

- Q1 ≈ μ - 0.6745σ

- Median = μ

- Q3 ≈ μ + 0.6745σ

- Min/Max estimated from sample size

The five-number summary is the basis for box-and-whisker plots.

## Frequently Asked Questions

**Q: Why is the five-number summary useful?**

A: It provides a quick snapshot of the data distribution: center (median), spread (IQR and range), and symmetry (comparing distances from median to Q1 vs. Q3). It is robust to outliers (unlike mean and standard deviation) and is the basis for box plots.

**Q: How are min and max estimated?**

A: For a normal distribution, the expected extreme values depend on sample size. As n increases, the expected min and max move further from the mean. The approximation uses √(2 × ln(n)) standard deviations from the mean. For n = 100, this is about 3.03 standard deviations.

**Q: What is the difference between range and IQR?**

A: Range = max - min, covering the full spread. IQR = Q3 - Q1, covering the middle 50%. Range is affected by outliers; IQR is robust to outliers. For comparing variability, IQR is generally preferred because extreme values disproportionately affect the range.

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Source: https://vastcalc.com/calculators/statistics/five-number-summary
Category: Statistics
Last updated: 2026-04-21