Standard Deviation Calculator
Paste a dataset to get the mean, variance, and both sample and population standard deviation.
Standard deviation is the square root of the average squared distance from the mean: s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ]. For the data 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared deviations sum to 32, giving a sample SD of √(32 ÷ 7) = 2.138.
How standard deviation measures spread
Standard deviation answers one question: on average, how far do the values sit from their mean? You find each value’s distance from the mean, square those distances so positives and negatives don’t cancel, average the squares to get the variance, then take the square root to return to the original units. A small result means the data cluster tightly; a large result means they’re scattered.
sample SD divides by n − 1; population σ divides by n
Worked example
Eight values: 2, 4, 4, 4, 5, 5, 7, 9.
- 1 Find the mean. (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 40 ÷ 8 = 5.
- 2 Sum the squared deviations. 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
- 3 Divide, then take the root. Sample: 32 ÷ 7 = 4.571, √4.571 = 2.138. Population: 32 ÷ 8 = 4, √4 = 2.
Sample vs. population
Which divisor to use, and what the result describes.
| Version | Divisor | Use when |
|---|---|---|
| Sample (s) | n − 1 | Your data is a sample from a larger group |
| Population (σ) | n | Your data is the entire group |
Reading and using the result
It shares the units of your data. Variance is in squared units (e.g. cm²); taking the square root puts standard deviation back in the original units (cm), which is why it’s the figure most often reported.
Pair it with the mean. A standard deviation only means something next to its mean — an SD of 5 is tiny on a scale averaging 1,000 and huge on one averaging 10. The coefficient of variation (SD ÷ mean) makes that comparison explicit. Once you have the SD, you can standardize any single value with a z-score or build a confidence interval around the mean.
Don’t confuse SD with standard error. Standard deviation describes the spread of individual values; the standard error (SD ÷ √n) describes the spread of the sample mean and shrinks as your sample grows.
Outliers pull it up fast. Because deviations are squared, one extreme value can inflate the SD dramatically. Scan your data for typos and genuine outliers before trusting the number.