Standard Deviation Calculator
Paste a list of numbers to get the sample or population standard deviation, variance, mean and standard error, with a table of every deviation.
Standard deviation measures how spread out numbers are around their mean: a small value means the data cluster tightly, a large one means they vary widely. Paste a list of values to get the sample or population standard deviation, the variance, mean and standard error, plus a table of every deviation so you can follow — or check — a hand calculation.
How to use this tool
- Paste or type the values, separated by commas, spaces or new lines.
- Choose sample if the data are a subset of a larger group, or population if they cover every member.
- Read the standard deviation and variance; the other type is shown as well.
- Use the deviation table to see how far each value sits from the mean.
Formula
Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Subtract the mean from each value, square the differences, add them up, divide, then take the square root. A sample divides by n − 1 (Bessel’s correction) because a sample’s own mean sits closer to its values than the true population mean does, which would otherwise understate the spread.
Example
Eight test scores: 72, 85, 90, 66, 78, 95, 88 and 81. They add up to 655, so the mean is 81.875, and the squared deviations sum to 650.875. As a sample, the variance is 650.875 ÷ 7 ≈ 92.982 and the standard deviation is ≈ 9.6427. Treated as the whole population, it is √(650.875 ÷ 8) ≈ 9.0199.
Frequently asked questions
Should I use the sample or population standard deviation?
Use population only when you have every value you care about, such as all employees of a small company. Survey answers, experiment results and production batches are samples, so use the n − 1 version.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root. Standard deviation is in the same units as the data, which makes it easier to interpret.
What does the standard error of the mean tell me?
It estimates how much the sample mean would vary from sample to sample: s ÷ √n. It shrinks as the sample grows, which is why larger samples give more precise averages.
What counts as a high standard deviation?
It depends on the scale of the data. Compare it with the mean using the coefficient of variation (SD ÷ mean × 100); for the scores above it is about 11.8%, a fairly tight spread.
Last updated: 2026-10-30