Free standard deviation calculator
Calculate population or sample standard deviation and variance from a dataset — enter your numbers to see all descriptive stats, updated live, as you type.
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Sample std dev divides by N−1 (Bessel's correction); population divides by N. Source: NIST/SEMATECH e-Handbook of Statistical Methods.
Results are estimates. Consult a professional.
How the standard deviation calculator works
Standard deviation measures how spread out values are around the mean. A low standard deviation means values cluster tightly near the mean; a high value means they are widely scattered. The formula differs slightly depending on whether the data is a complete population or a sample drawn from a larger population.
Worked example: {2, 4, 4, 4, 5, 5, 7, 9}
Data set: {2, 4, 4, 4, 5, 5, 7, 9}. Calculate the mean, then the sum of squared deviations, and finally both the population and sample standard deviations.
The empirical rule (68–95–99.7 rule)
For data that follows a normal (bell-curve) distribution, the empirical rule tells you what percentage of values fall within 1, 2, or 3 standard deviations of the mean.
| Range | Percentage of data | Approximate coverage |
|---|---|---|
| μ ± 1σ | 68.27% | About 2 in 3 values |
| μ ± 2σ | 95.45% | About 19 in 20 values |
| μ ± 3σ | 99.73% | Almost all values |
| Beyond μ ± 3σ | 0.27% | Rare outliers |
Source: empirical rule for normal distributions (Gauss, 1809).
Tips for standard deviation
Choosing between population and sample standard deviation — and interpreting the result correctly — are the two most important skills.
- Population vs sample — use σ (divide by N) when you have every value in the group; use s (divide by n−1) when your data is a random sample and you want to estimate the population spread.
- Check the units — standard deviation is in the same units as the original data; variance is in squared units. Use std dev for interpretation, variance for further calculations.
- Outliers inflate std dev — a single extreme value can dramatically increase standard deviation. Consider the median absolute deviation (MAD) for robustness against outliers.
- Low n makes s unreliable — with fewer than ~30 data points, sample standard deviation is a rough estimate; collect more data when precision matters.
- Compare with the mean — the coefficient of variation (CV = σ/μ × 100%) lets you compare spread across data sets with different units or scales.
Accuracy and limitations
Standard deviation is calculated using a two-pass algorithm (mean first, then deviations) to minimise floating-point cancellation errors. Results are accurate to approximately 14 significant digits for typical data. Standard deviation does not fully describe non-normal distributions — asymmetric or multi-modal data requires additional measures (skewness, kurtosis, interquartile range). The empirical rule applies only to approximately normal distributions; do not apply it to skewed data.
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