Standard Deviation Calculator
Calculate standard deviation and variance, with the formula, a worked example, and the population-versus-sample distinction.
Measuring spread
Standard deviation measures how spread out a set of numbers is around their average — a small standard deviation means the values cluster near the mean, a large one means they are scattered widely. It is the most important measure of spread in statistics, and this calculates it for your data.
How it's calculated
Standard deviation is built from the variance, which averages the squared distances of each value from the mean. The standard deviation is the square root of the variance, which brings it back to the original units:
1. Find the mean of the data2. For each value, find its distance from the mean, and square it3. Average those squared distances -> this is the variance4. Standard deviation = square root of the varianceWorked example
Take the small set 2, 4, 4, 6 (mean 4):
Population versus sample
There is one subtlety that matters: whether your data is the whole population or a sample drawn from a larger one changes the calculation slightly.
| Situation | Divide the squared distances by |
|---|---|
| Whole population | the number of values, n |
| Sample of a larger group | n minus 1 |
Dividing by n minus 1 for a sample corrects a bias that would otherwise make the spread look smaller than the true population's. Using the wrong one is a common mistake. Standard deviation matters because it gives meaning to individual values — combined with the mean, it tells you whether a value is typical or unusual, and it underlies the bell curve, z-scores, and much of statistics. Where the mean tells you the centre, the standard deviation tells you how much to trust that the centre represents the data.
Standard Deviation Calculator FAQ
What does standard deviation measure?
How spread out data is around its mean — small means values cluster near the average, large means they are widely scattered.
How is it calculated?
Average the squared distances of each value from the mean to get the variance, then take its square root to return to the original units.
What is the difference between population and sample standard deviation?
For a whole population you divide by the number of values; for a sample of a larger group you divide by one less, which corrects a bias that otherwise understates the spread.
Why is standard deviation so important?
It gives meaning to individual values against the mean, and it underlies the bell curve, z-scores and much of statistics.
Is anything uploaded?
No. The calculation happens in your browser.