How this calculator works
Standard deviation measures how spread out a set of numbers is around its mean. A small standard deviation means the values cluster tightly near the average; a large one means they're spread widely. This calculator returns both the sample and population versions, since they use slightly different formulas and it matters which one applies to your data.
Use the population formula only when your list is the entire group you care about — every test score in a class, every measurement in a finished batch. Use the sample formula, which this calculator treats as the primary result, whenever your numbers are a subset drawn from a larger population you're trying to estimate, which is the far more common situation in practice.
How do you calculate standard deviation?
Mean: μ = (Σx) ÷ nPopulation variance: σ² = Σ(x − μ)² ÷ nSample variance: s² = Σ(x − μ)² ÷ (n − 1)Standard deviation = √varianceDividing by n − 1 instead of n (Bessel's correction) compensates for the fact that a sample's mean is estimated from the same data, which would otherwise make the sample systematically underestimate the true population variance.
Worked example: 2, 4, 4, 4, 5, 5, 7, 9
- Mean: (2+4+4+4+5+5+7+9) ÷ 8 = 40 ÷ 8 = 5.
- Squared deviations from the mean: 9, 1, 1, 1, 0, 0, 4, 16 — these sum to 32.
- Population variance: 32 ÷ 8 = 4. Population standard deviation: √4 = 2.
- Sample variance: 32 ÷ 7 ≈ 4.571. Sample standard deviation: √4.571 ≈ 2.138.
Frequently asked questions
Should I use sample or population standard deviation?
Use population SD only if your numbers are the complete group you're studying. Use sample SD whenever your data is a subset used to estimate a larger population — for example, a survey of 200 customers standing in for all customers. When in doubt, sample SD is the safer default and is shown as the primary result here.
Why are the two formulas different?
The sample formula divides by n − 1 instead of n. This makes the result slightly larger, correcting for the fact that a sample's own mean is calculated from the same limited data and tends to sit closer to the sample's values than the true population mean would.
What does a standard deviation of 0 mean?
Every value in the list is identical. There's no spread at all, so both variance and standard deviation are zero.
Can I compute a sample standard deviation from a single number?
No — the sample formula divides by n − 1, which is zero when there's only one value, making the calculation undefined. The population standard deviation of a single value is always zero.
How do I interpret a large standard deviation?
It means the values are spread widely around the mean rather than clustered close to it. This page's own example data (2, 4, 4, 4, 5, 5, 7, 9) has a population standard deviation of just 2 against a mean of 5, a fairly tight spread; the same standard deviation of 2 would count as enormous for data hovering near zero, but tiny for data averaging in the millions, since 'large' only makes sense relative to the scale of what's being measured.