What your result means
A small standard deviation means your values sit close to the mean; a large one means they are spread out. Use the sample version (dividing by n − 1) when your numbers are a subset of a larger group, which is the usual case for homework, surveys and experiments, and the population version (dividing by n) only when you have every value there is.
How to use the Standard Deviation Calculator
- Paste or type your numbers, separated by commas, spaces or new lines.
- Choose sample or population.
- Read the standard deviation, variance, mean, count and sum.
- Open "Show the working" to see every step, ready to copy into your notes.
Formula and worked example
mean (x̄) = Σx ÷ npopulation SD (σ) = √( Σ(x − x̄)² ÷ n )sample SD (s) = √( Σ(x − x̄)² ÷ (n − 1) ) ← Bessel’s correctionvariance = SD²
Worked example: 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5 and the squared deviations add up to 32. As a population, variance = 32 ÷ 8 = 4 and σ = 2. As a sample, variance = 32 ÷ 7 = 4.5714 and s = 2.1381.
Sample vs population standard deviation
| Data | Mean | Population σ | Sample s |
|---|---|---|---|
| 1, 2, 3, 4, 5 | 3 | 1.4142 | 1.5811 |
| 10, 10, 10, 10 | 10 | 0 | 0 |
| 2, 4, 4, 4, 5, 5, 7, 9 | 5 | 2 | 2.1381 |
| 1, 100 | 50.5 | 49.5 | 70.0036 |
Computed by the calculator above.
Assumptions and limits
- Use "sample" when your numbers are a subset of a bigger group (most homework and survey data); use "population" when you have every value.
- The calculator uses a two-pass method (mean first, then squared deviations), which avoids the rounding errors of the shortcut formula on large numbers.
Frequently asked questions
What is the difference between sample and population standard deviation?
The population formula divides the sum of squared deviations by n, the number of values. The sample formula divides by n − 1, which corrects the tendency of a sample to underestimate the spread of the whole group. With large data sets the two get very close.
How do you calculate standard deviation by hand?
Find the mean, subtract it from each value and square the results, add the squares, divide by n (population) or n − 1 (sample) to get the variance, then take the square root. The working panel shows each of these steps for your data.
What is variance?
Variance is the average of the squared deviations from the mean, which is the standard deviation squared. It is in squared units (for example dollars squared), which is why the standard deviation is usually easier to interpret.
Can standard deviation be zero?
Yes, when every value is the same, because nothing differs from the mean. Standard deviation can never be negative.
How many numbers can I enter?
Up to 5,000 values, separated by commas, spaces, semicolons or new lines. Each value can be negative or a decimal.
Related calculators
More free calculators
About this calculator
Written and maintained by Ahsan Khan. See how every number is calculated, and the test cases checked on every update, on the methodology page.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.6 Measures of Scale, National Institute of Standards and Technology (NIST) (accessed Oct 11, 2026)
- Introductory Statistics 2e, 2.7 Measures of the Spread of the Data, OpenStax, Rice University (Barbara Illowsky and Susan Dean) (accessed Oct 11, 2026)
Found an error? Email a correction and include the link from “Copy link” so we can reproduce it.