Standard Deviation Calculator
Paste or type a list of numbers to get the standard deviation, variance, mean, median and range, with the working shown, a margin of error for the mean and a frequency table. Handles both samples and whole populations.
Separate values with commas, spaces or new lines — pasting a spreadsheet column works.
Divides by n − 1 (Bessel's correction) to estimate the spread of a larger group.
- 1. Mean: 144 ÷ 8 = 18
- 2. Squared deviations: subtract the mean from each value, square it, and add them up: Σ(x − μ)² = 192
- 3. Variance: 192 ÷ (8 − 1) = 27.428571
- 4. Standard deviation: s = √27.428571 = 5.237229
| Confidence | Margin | Range for the mean |
|---|---|---|
| 68.3% | ±1.85164 | 16.14836 to 19.85164 |
| 90% | ±3.045678 | 14.954322 to 21.045678 |
| 95% | ±3.629148 | 14.370852 to 21.629148 |
| 99% | ±4.769509 | 13.230491 to 22.769509 |
| 99.9% | ±6.092872 | 11.907128 to 24.092872 |
Uses the normal distribution (z-scores). For small samples a t-distribution gives slightly wider ranges.
| Value | Count | Share |
|---|---|---|
| 23 | 3 | 37.5% |
| 16 | 2 | 25% |
| 10 | 1 | 12.5% |
| 12 | 1 | 12.5% |
| 21 | 1 | 12.5% |
How to use this calculator
- Enter your numbers separated by commas, spaces or new lines. You can paste a column straight from a spreadsheet.
- Say whether they are a sample or the whole population.This decides whether the variance divides by n − 1 or by n.
- Read the results as you type. Anything that is not a number is skipped and listed so you can fix it.
The formulas
Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Standard error = SD ÷ √n
Margin of error = z × standard error
The calculator finds the mean first and then sums the squared distances from it. This two-pass method stays accurate even when the values are large and close together, where the shortcut formula can lose most of its precision.
Worked example
Eight test results: 10, 12, 23, 23, 16, 23, 21, 16.
- Mean: 144 ÷ 8 = 18
- Squared deviations: 64, 36, 25, 25, 4, 25, 9, 4 — total 192
- Population variance: 192 ÷ 8 = 24, so σ = √24 ≈ 4.899
- Sample variance: 192 ÷ 7 ≈ 27.43, so s ≈ 5.237
Frequently asked questions
Choose population when your numbers are every member of the group you care about — the marks of every student in one class, for example. Choose sample when they are a subset you are using to learn about a larger group, such as a survey of 200 customers.
If you are unsure, sample is the safer choice and the usual default in science and statistics courses.
A sample's own mean sits closer to its values than the true population mean does, so squared deviations from it come out slightly too small. Dividing by n − 1 instead of n corrects that bias. It is known as Bessel's correction.
The difference matters for small samples and fades away as n grows.
Roughly how far a typical value sits from the average. A small standard deviation means the values cluster tightly around the mean; a large one means they are spread out.
For data shaped like a bell curve, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.
Variance is the average squared distance from the mean; standard deviation is its square root. Standard deviation is usually easier to interpret because it is in the same units as your data — centimetres rather than square centimetres.
It estimates how far the true mean of the whole population might be from the mean of your sample. At 95% confidence, the true mean is expected to fall within the margin on either side of your sample mean in 95 out of 100 samples.
It is the standard error multiplied by a z-score: 1.96 for 95%, 2.576 for 99%.