Standard Deviation Calculator
Measuring How Far Data Wanders From Its Center
Two data sets can share the exact same mean and look nothing alike — one tightly clustered, the other scattered wide. Standard deviation is the standard way to quantify that difference: it converts the average squared distance from the mean back into the original units, giving you a single number for "typical spread." A small standard deviation means values cluster near the mean; a large one means they don't.
The Formula
Sample: s = √( Σ(x − mean)² ÷ (n − 1) )
Both versions sum the squared deviation of every value from the mean. The population formula divides by N (the full data set); the sample formula divides by n−1 instead, a correction (Bessel's correction) that keeps the sample statistic from systematically underestimating the true population spread.
Which Version to Use
Use the population formula only when your data set genuinely is the entire population you care about — every student in a class, every part produced in a finished batch. Use the sample formula whenever your data is a subset drawn from a larger group you're trying to draw conclusions about, which describes most real-world statistics work: a survey sample, a set of test measurements, an experiment's results.
The Empirical Rule
For data that follows a roughly normal (bell-curve) distribution, standard deviation has a well-known interpretation:
| Range | Approx. share of data |
|---|---|
| Mean ± 1 standard deviation | 68.27% |
| Mean ± 2 standard deviations | 95.45% |
| Mean ± 3 standard deviations | 99.73% |
Where This Matters
- Manufacturing quality control — monitoring how tightly a production process holds to a target dimension or weight.
- Finance — standard deviation of returns is the standard measure of volatility for a stock or portfolio.
- Standardized testing — converting raw scores to z-scores relies directly on this value.
- Research — reporting spread alongside a mean gives readers a sense of how representative that mean actually is.
How to Use This Calculator
- Choose Sample Standard Deviation or Population Standard Deviation from the dropdown.
- Enter your numbers into the values field, separated by commas.
- Select Calculate to see the standard deviation and the mean it was measured from.
Related Calculations
Standard deviation is the square root of the Variance Calculator's result. To convert an individual data point into standard-deviation units, use the Z-Score Calculator.