Standard deviation measures how spread out a set of numbers is. This works it out from a list you paste in — and shows both the population and sample versions, because they are genuinely different numbers, not two ways of writing the same one.
How it works
Population and sample standard deviation
σ = √(Σ(x−μ)² / n) · s = √(Σ(x−x̄)² / (n−1))
The only difference is the divisor: n for a population, n−1 for a sample.
Why the divisor changes, and why it matters
Dividing by n treats your list as the entire group you care about — a population. Dividing by n−1 (“Bessel’s correction”) treats your list as a sample drawn from a larger group you are trying to describe, and the smaller divisor deliberately produces a slightly larger figure: a sample’s own spread systematically underestimates the true population spread, and n−1 corrects for that bias. The gap between the two is largest at small sample sizes and shrinks as the list grows.
How to use this calculator
- Paste or type your numbers, separated by commas, spaces or new lines.
- Read both standard deviations — use population if your list is everything you care about, sample if it is a sample representing a larger group.
Frequently asked questions
Which one should I use — population or sample?
Population, if your list is the complete set you are interested in (every student in one specific class, say). Sample, if your list is a subset standing in for a larger group you cannot measure entirely (a survey sample representing a whole population).
Why is variance just standard deviation squared?
Variance is defined first, in squared units — which makes it awkward to interpret directly (a variance of “16 square kilograms” means little intuitively). Standard deviation, its square root, comes back into the original units, which is why it is the more commonly quoted figure.
Why can’t sample standard deviation be computed from a single number?
Because n−1 becomes zero when n is 1, and dividing by zero is undefined. A single data point has no meaningful spread to estimate from a sample perspective, even though the population version (which is always zero for one number) is still defined.
Does a bigger standard deviation always mean “worse” data?
Not necessarily — it depends entirely on context. A larger spread might mean more variability in a manufacturing process (usually undesirable) or more diversity in a survey response (often perfectly fine, or even the point of the survey).
What’s a quick way to sanity-check the result?
For roughly bell-shaped data, about two-thirds of values typically fall within one standard deviation of the mean. If that rough rule looks wildly off for your data, it is worth checking the list was entered correctly.
