Standard deviation measures how spread out a data set is — the typical distance of each value from the mean. A small standard deviation means the numbers cluster tightly around the average; a large one means they are widely scattered. This calculator takes any list of numbers and returns the sample standard deviation, the population standard deviation, the variance for each, the mean, range, standard error of the mean and the coefficient of variation — and, crucially, it shows the full working so you can check or learn each step rather than trusting a black box.
Paste your data as numbers separated by commas, spaces or new lines. Everything is computed instantly in your browser as you type, and nothing you enter is uploaded.
How it works
The calculator follows the standard textbook procedure:
- Mean x̄ = sum of values ÷ count (n).
- For each value, compute the deviation
x − x̄and then square it to get(x − x̄)². - Sum the squared deviations — this total is often written Σ(x − x̄)².
- Variance = that sum ÷ n for a population, or ÷ (n − 1) for a sample.
- Standard deviation = √variance.
The population version uses the formula σ = √( Σ(x − x̄)² ⁄ n ), and the sample version uses s = √( Σ(x − x̄)² ⁄ (n − 1) ). The only difference is the divisor. Dividing a sample by n − 1 rather than n — known as Bessel’s correction — compensates for the fact that a sample’s own mean sits closer to its data than the true population mean would, which otherwise makes the spread look slightly too small. The tool also reports the standard error of the mean (s ⁄ √n) and the coefficient of variation (standard deviation as a percent of the mean), two derived measures that frequently accompany a standard-deviation report. Sample mode requires at least two values because it divides by n − 1; population mode needs at least one.
Worked example
For the data set 10, 12, 23, 23, 16, 23, 21, 16 (n = 8):
- Mean x̄ = 144 ÷ 8 = 18
- Squared deviations: 64, 36, 25, 25, 4, 25, 9, 4
- Sum of squared deviations Σ(x − x̄)² = 192
- Population variance = 192 ÷ 8 = 24 → population SD = √24 ≈ 4.899
- Sample variance = 192 ÷ 7 ≈ 27.4286 → sample SD ≈ 5.2372
| Measure | Sample (n − 1) | Population (n) |
|---|---|---|
| Variance | 27.4286 | 24.0000 |
| Standard deviation | 5.2372 | 4.8990 |
| Mean | 18 | 18 |
Notice the sample figures are slightly larger — that is Bessel’s correction at work. The step table inside the tool reproduces exactly these per-value deviations and squares, with the running sums shown in the footer, so the maths above is fully auditable.
Formula note: population σ = √( Σ(x − x̄)² ⁄ n ); sample s = √( Σ(x − x̄)² ⁄ (n − 1) ); standard error of the mean = s ⁄ √n; coefficient of variation = (s ⁄ x̄) × 100%.
Every figure is calculated locally in your browser, so the numbers you paste are never sent anywhere.