Standard deviation and variance

Separate numbers with commas, spaces or new lines.

How it works: find the mean, subtract it from every value and square each difference, average those squares to get the variance, then take the square root for the standard deviation. Population divides by n; sample divides by n − 1.

This free Standard Deviation Calculator measures how spread out a set of numbers is around its average. Paste any list — test scores, daily sales, rainfall readings, experiment results — and it returns the mean, the population standard deviation and variance, and the sample standard deviation and variance, all in one result. A small standard deviation means the values huddle close to the mean, while a large one means they are widely scattered, which is often the single most useful thing to know about a data set after the average itself.

The calculator reports both flavours because the right one depends on your data. Use the population figures when your list is the complete set you care about, such as every student in one class. Use the sample figures when your list is a sample drawn from something bigger, such as thirty voters surveyed out of a whole city; the sample version divides by n minus one to correct for the fact that a sample usually understates the true spread. The method note under the tool shows the steps, so students can follow the working. Everything is computed locally in your browser and your numbers are never uploaded.

How to use

Type or paste your list of numbers into the box.
Separate values with commas, spaces or new lines.
Click Calculate to see the mean, variances and standard deviations.
Use the population result if your list is the whole group.
Use the sample result if your list represents a larger group.

Frequently asked questions

What does a standard deviation of zero mean?

It means every value in the list is identical, so there is no spread at all. If all scores were exactly seventy, the mean is seventy and the standard deviation is zero, because nothing deviates from the average.

Why are there two different answers?

Population and sample formulas differ in one step: the sample version divides the squared differences by n minus one instead of n. This small correction, called Bessel's correction, makes sample estimates more accurate for the larger group they came from.

What is variance, then?

Variance is the step before the square root: the average of the squared differences from the mean. Because it is measured in squared units it is harder to interpret directly, which is why the standard deviation, in the original units, is the number most people quote.

What counts as a large standard deviation?

There is no universal cut-off; it depends on the scale of your data. A standard deviation of two is large for exam marks out of twenty but tiny for annual salaries. Compare it with the mean and the range of your values to judge.