Standard Deviation Calculator
Paste a list of numbers to get the sample and population standard deviation, the variance and the mean. Separate numbers with commas, spaces or new lines.
| Count | 5 |
|---|---|
| Sum | 45 |
| Minimum | 4 |
| Maximum | 15 |
| Range | 11 |
| Variance (population) | 12.8 |
| Variance (sample) | 16 |
The formulas
mean = sum / n population variance = sum of (x - mean)^2 / n sample variance = sum of (x - mean)^2 / (n - 1) standard deviation = square root of the variance
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The squared differences are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32. The population variance is 32 / 8 = 4, so the population standard deviation is 2. The sample variance is 32 / 7 = 4.5714, so the sample standard deviation is 2.1381.
For the mean, median and mode of the same list, use the average calculator.
Frequently asked questions
What is standard deviation?
A measure of how spread out numbers are around their mean. A small value means the numbers cluster near the mean; a large value means they are widely scattered.
What is the difference between sample and population standard deviation?
Use the population version when your numbers are the whole group you care about, and divide by n. Use the sample version when they are only a sample of a larger group, and divide by n - 1, which corrects for a sample underestimating the spread.
How do I calculate it by hand?
Find the mean, subtract it from each number and square the result, add the squares, divide by n (population) or n - 1 (sample), and take the square root.
What is variance?
The standard deviation before the square root: the average of the squared differences from the mean. Standard deviation is easier to read because it is in the same units as your data.
Why does it need at least two values for the sample version?
Dividing by n - 1 with one value means dividing by zero. One value has no spread to estimate, so the sample result is not defined.