Statistics Calculator

How This Calculator Works

This calculator finds the mean, median, mode, range, variance, and standard deviation of any list of numbers. Paste or type your numbers into the box below — separated by commas, spaces, or new lines, in any combination — and every common descriptive statistic is calculated at once.

Mean is the arithmetic average. Median is the middle value when the numbers are sorted (or the average of the two middle values, for an even-sized list). Mode is whichever value (or values) appear most often — if every number appears exactly once, there’s no mode.

Standard deviation measures how spread out the numbers are from the mean — a small standard deviation means the numbers are clustered close together; a large one means they’re spread widely. This calculator reports it two ways: population standard deviation (when your numbers are the entire group you care about) and sample standard deviation (when your numbers are a sample used to estimate a larger population) — the sample version is slightly larger, correcting for the fact that a sample tends to underestimate the true spread of the full population.

The Formula

$$\vC{\text{Mean}} = \frac{\sum \vA{\text{Numbers}}}{\vB{\text{Count}}}$$ $$\vD{\text{Population Variance}} = \frac{\sum (\vA{\text{Number}} - \vC{\text{Mean}})^2}{\vB{\text{Count}}}$$ $$\vD{\text{Sample Variance}} = \frac{\sum (\vA{\text{Number}} - \vC{\text{Mean}})^2}{\vB{\text{Count}} - 1}$$ $$\text{Standard Deviation} = \sqrt{\vD{\text{Variance}}}$$

Population or sample, matching whichever variance is used.

The only difference between the population and sample versions is dividing by the count itself versus the count minus one (known as Bessel’s correction) — everything else is identical.

Worked Example

The numbers 2, 4, 4, 4, 5, 5, 7, 9:

1. Count: $\vB{8}$. Sum: $\vA{40}$. Mean: $\vA{40} \div \vB{8} = \vC{5}$.
2. Sorted: 2, 4, 4, 4, 5, 5, 7, 9 — median is the average of the two middle values (4 and 5): $(4 + 5) \div 2 = 4.5$.
3. Mode: 4 (appears three times, more than any other value).
4. Range: $9 - 2 = 7$.
5. Population standard deviation: 2.0. Sample standard deviation: ≈2.14.