Foundations · Spread
Standard Deviation Calculator
How far, typically, your numbers sit from their average. Paste your data to get the standard deviation and variance — both the sample (÷ n−1) and population (÷ n) versions — with every deviation laid out so you can see exactly where the number comes from, and a plain-English guide to which one you actually want.
Most real data are a sample from a larger group — that is the usual choice, and the default. Pick “population” only when your numbers are the entire group (every employee, all six dice faces).
Result
In plain English
The standard deviation answers one question: how spread out are the numbers? It is, near enough, the typical distance of a value from the average. A small standard deviation means the data huddle close to the mean; a large one means they are scattered. It is measured in the same units as the data, which is why it is usually quoted instead of its squared cousin, the variance.
- standard deviation (s or σ)
- The typical gap between a value and the mean. Computed as the square root of the variance, so it is back in the original units (pounds, seconds, marks).
- variance (s² or σ²)
- The average of the squared deviations from the mean. Squaring keeps everything positive and is mathematically convenient, but it leaves the answer in squared units — hence taking the root to get the standard deviation.
- sample vs population (n−1 vs n)
- If your data are a sample, dividing by n−1 (Bessel's correction) corrects a built-in tendency to understate the spread; if your data are the whole population, divide by n. The two differ most when n is small.
- coefficient of variation (CV)
- The standard deviation as a percentage of the mean — a unit-free way to compare spread between datasets of different size or scale.
- standard error (SEM)
- Not the spread of the data, but the spread of the sample mean: s ÷ √n. It shrinks as you collect more data; the standard deviation does not.
Frequently asked
Should I use the sample or population standard deviation?
Use the sample standard deviation (divide by n−1) when your data are a sample drawn from some larger group you want to describe — which is almost always the case. Use the population version (divide by n) only when your numbers are the entire group, with no one left out. The sample version is slightly larger; the gap matters most for small datasets and vanishes as n grows.
Why divide by n−1 instead of n?
Because the deviations are measured from the sample mean, not the true population mean, they are on average a touch too small — the sample mean is, by construction, the point closest to its own data. Dividing by n−1 rather than n (Bessel's correction) compensates, giving an unbiased estimate of the population variance. With large n the correction is negligible; with n = 2 it is the difference between dividing by 1 and by 2.
What is the difference between standard deviation and standard error?
The standard deviation describes how spread out the individual values are, and does not shrink as you gather more data. The standard error of the mean (s ÷ √n) describes how precisely you have pinned down the average, and does shrink with more data. Reporting the standard error when you mean the standard deviation makes data look far tidier than it is — a common and misleading slip.
What does a standard deviation of zero mean?
That every value is identical — there is no spread at all, so each point sits exactly on the mean. Any variation whatsoever gives a positive standard deviation, and there is no upper limit: the more scattered the data, the larger it grows. A standard deviation can never be negative, because it is the square root of an average of squared distances, and squared quantities are never below zero.