Standard Deviation Calculator
Paste a list of numbers and get both population and sample standard deviation — clearly labelled — plus variance and mean, with the working shown.
Result
How It Works
Standard deviation measures how spread out numbers are around their mean. The recipe: find the mean, subtract it from each value, square each difference, average those squares — that average is the variance — and finally take the square root. The square root brings the answer back into the original units — test marks in, test marks out.
There are two versions, and they differ by one step. Population standard deviation divides the squared differences by n, the count of values — use it when your list is the entire group. Sample standard deviation divides by n−1 instead, which corrects for the fact that a sample usually underestimates the spread of the full group — use it when your list is a sample. The sample value is always slightly larger.
Worked example: for 2, 4, 4, 4, 5, 5, 7, 9, the mean is 40 / 8 = 5. The squared differences from 5 are 9, 1, 1, 1, 0, 0, 4, 16, totalling 32. Population variance is 32 / 8 = 4, so population SD is √4 = 2. Sample variance is 32 / 7 ≈ 4.5714, so sample SD is ≈ 2.1381.
How to Use This Calculator
Paste or type your numbers into the box — the results update live as you type.
- Enter your numbers separated by commas, spaces or new lines. Decimals and negative numbers are fine.
- Read both standard deviations in the table — population (÷n) and sample (÷n−1) — plus both variances and the mean.
- Pick the version that fits — whole group, use population; sample, use sample.
Sample standard deviation needs at least two values; with a single value it is shown as unavailable. The Reset button restores the example list so you can start over.
Tips and Practical Advice
Standard deviation only means something when read next to the mean.
- Read SD next to the mean. An SD of 2 around a mean of 5 means the values are fairly tight; the same SD around a mean of 100 means they are very tight indeed.
- Use sample SD for survey data. If you measured 30 customers out of thousands, your list is a sample — the n−1 version is the honest one.
- Remember the rough rule. In bell-shaped data, about two-thirds of values fall within one SD of the mean — a quick sanity check on whether a result looks plausible.
- Watch for outliers. One extreme value inflates the SD a lot, because differences are squared — check whether an outlier is real or a typo before trusting the spread.
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population SD divides the squared differences by n; sample SD divides by n−1. Use population when your list is the whole group, sample when it is a subset of a larger group.
What is variance?
Variance is the average of the squared differences from the mean. Standard deviation is the square root of the variance, which returns the answer to the original units.
Can standard deviation be zero?
Yes — if every value is identical, there is no spread at all, so the standard deviation is 0.
Why divide by n−1 for a sample?
A sample's values cluster slightly tighter around their own mean than the full population's would, so dividing by n−1 corrects that underestimate. It is known as Bessel's correction.
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