Standard Deviation Calculator
Standard deviation measures how spread out a set of numbers is around its mean: a small value means the data clusters tightly, a large value means it scatters widely. This calculator reports both flavours that statistics courses and spreadsheets distinguish — the sample standard deviation, used when your numbers are a sample of a larger group, and the population standard deviation, used when they are the entire group. Enter up to eight numbers to see the mean and both measures of spread side by side.
Calculate
Default result: 2.1381
Standard Deviation Calculator · Result
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Sample standard deviation (s)
2 × 4 × 4 × 4 × 5 × 5 × 7 × 9
- Population standard deviation (σ)
- 2.0000
- Mean
- 5.0000
- How many values
- 8
Reviewed by the calculators.dev team · Last updated 2026-06-23
How to calculate
Enter at least two numbers; leave the remaining boxes blank. The calculator first finds the mean, then measures how far each value sits from that mean, squares those distances so positives and negatives do not cancel, and adds them up. Dividing that total by the number of values and taking the square root gives the population standard deviation; dividing by one less than the number of values gives the sample standard deviation. Both appear together so you can pick the one your situation calls for, and they update as you type.
Population: σ = √( Σ(x − mean)² ÷ n ). Sample: s = √( Σ(x − mean)² ÷ (n − 1) ). The only difference is the divisor — n for a whole population, n − 1 for a sample (Bessel's correction), which makes the sample value slightly larger to account for estimating the mean from the same data. Variance is the figure inside the square root; standard deviation puts it back into the original units.
Example calculation
For the eight numbers 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. Squaring each deviation from the mean and adding gives 32. Dividing by n (8) gives a population variance of 4, so the population standard deviation σ is √4 = 2. Dividing instead by n − 1 (7) gives a sample variance of about 4.571, so the sample standard deviation s is about 2.138 — slightly larger, because the sample version corrects for estimating the mean from the data itself.
- sampleStdDev
- 2.1381
- populationStdDev
- 2
- mean
- 5
- count
- 8
Assumptions
- Use the sample version (n − 1) when your numbers are a subset of a larger group; use the population version (n) only when you have every member of the group.
- At least two values are required, because the sample formula divides by n − 1 and a single value has no spread to measure.
- Standard deviation is in the same units as the data, while variance is in squared units — the two describe the same spread differently.
Common mistakes
- Using the population formula on a sample, which understates the true spread of the wider group.
- Forgetting to square the deviations, so positive and negative distances cancel and the spread looks like zero.
- Comparing standard deviations across data sets with very different means without also looking at the relative spread.
Frequently asked questions
What is the difference between sample and population standard deviation?
The population version divides the squared deviations by n; the sample version divides by n − 1. Use the sample version when your data is a sample of a larger group, which makes the result slightly larger.
When should I use n − 1 instead of n?
Use n − 1 (the sample standard deviation) whenever your numbers are a sample meant to estimate a larger population. Use n only when you have measured the entire population.
What does a standard deviation actually tell me?
It is the typical distance of a value from the mean. A small standard deviation means values cluster near the average; a large one means they are widely scattered.
Is variance the same as standard deviation?
Variance is the average of the squared deviations; standard deviation is its square root. Standard deviation is in the original units, which is why it is easier to interpret.