Standard Deviation & Variance Calculator
Paste a list of numbers to get mean, sample and population variance, sample and population standard deviation, and range — instantly.
Mean
5
8 values
Sample std dev
2.1381
Sample variance
4.5714
Population std dev
2
Population variance
4
Range
7
Min / Max
2 / 9
Use sample statistics when your numbers are a sample drawn from a larger population (the usual case in surveys and experiments) — it divides by n − 1, which corrects for the fact that a sample tends to slightly underestimate the true population spread. Use population statistics only when your numbers are the entire population you care about, not a sample of it.
About this tool
How to use it
- Paste or type your numbers into the box, separated by commas, spaces, or new lines — any mix works.
- Press Calculate Statistics, or just keep typing — the results update live.
- Read the mean at the top, then check both the sample and population versions of variance and standard deviation below it.
- Use the note to decide which of sample or population applies to your data.
Why it matters
Standard deviation and variance measure how spread out a set of numbers is, but there are two different formulas — sample and population — depending on whether your numbers are the entire group you care about or just a sample drawn from a larger one. Using the wrong one is a common statistics mistake, and computing either by hand for more than a handful of numbers is slow and error-prone.
When to use it
Useful for checking statistics homework, for a quick spread-of-data check on survey results or measurements, or for verifying a calculation you've already done by hand or in a spreadsheet.
Good to know
Sample statistics divide the sum of squared differences by n − 1 (Bessel's correction), which corrects for the fact that a sample's own mean is slightly closer to its data points than the true population mean would be — this is the version to use almost any time your numbers are a sample rather than a complete population. Population statistics divide by n instead, and are only correct when your numbers genuinely are the entire population you're interested in, not a sample of it. With only one number, standard deviation and variance are shown as sample-inapplicable ('n/a'), since a spread measure needs at least two values to mean anything for the sample formula.
Common questions
What's the difference between sample and population standard deviation?
Sample standard deviation divides by n − 1 instead of n, which corrects for a real statistical bias: a sample's own mean sits slightly closer to its own data points than the true population mean would, so dividing by n − 1 gives a less biased estimate of the population's actual spread. Use sample statistics almost any time your numbers are a subset of a larger group you're trying to draw conclusions about.
When should I use population statistics instead?
Only when your numbers are the complete set you care about — for example, the ages of every employee at a small company, if you want that company's own age spread rather than to generalize to a wider population.
Why is variance harder to interpret than standard deviation?
Variance is in squared units of your original data (if your numbers are in dollars, variance is in dollars-squared), which makes it hard to interpret directly. Standard deviation is the square root of variance, bringing it back into the same units as your original data, which is why it's usually the more intuitively useful number.
Why does the calculator show 'n/a' for sample statistics with only one number?
Sample standard deviation and variance divide by n − 1, which is zero when there's only one value — dividing by zero is undefined, so the calculator reports it as not applicable rather than showing an error or an infinite result.