Find how spread out a set of numbers is. Enter your data set and this calculator works out the mean, variance and standard deviation step by step.
Standard Deviation Calculator
Compute sample standard deviation (s) or population standard deviation (σ), variance, mean, standard error of the mean (SEM), and 95% confidence interval from any dataset.
Enter your dataset
Separate numbers with commas, spaces, or line breaks.
Your result will appear here
Enter your values above, then press Calculate Standard Deviation — or press Enter in any field.TinyExe · Std Dev Result
Dispersion, variance, mean & confidence interval
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Name
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Mean (x̄) & Count
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Variance
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Std. Error (SEM)
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How this was calculated
What to do next
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Sample mode uses Bessel's correction (dividing by n − 1) to produce an unbiased estimator of population variance.
Sample vs. Population Standard Deviation
Choosing the right denominator depends on whether your data is a sample or the entire population.
| Statistic | Symbol | Divisor & Use Case |
|---|---|---|
| Sample Standard Deviation | s | Divides by (n − 1). Used in experiments, surveys, and subsets. |
| Population Standard Deviation | σ | Divides by n. Used when every member of the group is measured. |
| Variance | s² or σ² | Average squared deviation from the mean before taking √. |
| Standard Error (SEM) | s ÷ √n | Measures how precisely the sample mean estimates the true mean. |
The formula
The two standard deviation formulas in statistics.
Sample Standard Deviation (s)
Subtract the sample mean x̄ from each value, square each difference, sum them up, divide by n − 1, and take the square root.
Population Standard Deviation (σ)
Divides the sum of squared deviations by the full population size N.
Worked examples
Worked dispersion calculations.
Example 1 · Sample (n=8)
10, 12, 23, 23, 16, 23, 21, 16
Mean x̄ = 18.0, Σ(x − x̄)² = 192.
| Calculation | Steps |
|---|---|
| s² = 192 ÷ 7 = 27.4286 | Divide by n − 1 = 7. |
| s = √27.4286 ≈ 5.2372 | Sample standard deviation. |
s = 5.2372
Example 2 · Population (N=8)
Same dataset as full population
Mean μ = 18.0, Σ(x − μ)² = 192.
| Calculation | Steps |
|---|---|
| σ² = 192 ÷ 8 = 24.0 | Divide by N = 8. |
| σ = √24 ≈ 4.8990 | Population standard deviation. |
σ = 4.8990
Example 3 · Zero Spread
5, 5, 5, 5, 5
Identical observations.
| Calculation | Steps |
|---|---|
| Σ(x − 5)² = 0 | Every deviation from the mean is 0. |
| s = 0, σ = 0 | Zero dispersion. |
SD = 0
How to use this calculator
Choose Sample (s) or Population (σ)
Use Sample (s) for almost all scientific, survey, and classroom datasets.
Paste or type your numbers
Separate values with commas or spaces.
Press Calculate Standard Deviation
Get the standard deviation, variance, mean, SEM, and 95% confidence interval.
Frequently asked questions
Why do we divide by n − 1 for sample standard deviation?
Dividing by n − 1 (Bessel's correction) corrects for the fact that deviations around a sample mean are slightly smaller on average than deviations around the true unknown population mean.
What does a high standard deviation mean?
A larger standard deviation means values are spread out widely from the mean; a standard deviation near 0 means data points cluster tightly around the mean.