Standard Deviation Calculator
Calculate the standard deviation, variance, and mean of a list of numbers.
Try an example
2.0656
About this calculator
What Is Standard Deviation?
Standard deviation, usually written σ (sigma) for a population or s for a sample, measures how spread out a set of numbers is around its mean. A small standard deviation means the values cluster tightly near the average; a large one means they range widely. Enter a list of numbers separated by commas and this calculator returns the standard deviation along with the variance, mean, count and sum — the basic toolkit of descriptive statistics.
Population vs. Sample Standard Deviation
Use the population version when your data includes every member of the group you care about. It is the square root of the variance: σ = √[ Σ(xᵢ − μ)² ÷ N ], where μ is the mean and N is the number of values. Use the sample version when your numbers are only a subset drawn from a larger group. It divides by n − 1 instead of n: s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ]. That adjustment (Bessel's correction) compensates for the fact that a sample tends to understate the true spread.
Worked Example
For the values 4, 8, 6, 5, 3, 8: the mean is 5.67, and the sample standard deviation is 2.07 (population standard deviation: 1.89). The sample variance is 4.27.
How to Calculate It Step by Step
- Find the mean of the data.
- Subtract the mean from each value to get its deviation.
- Square each deviation, so negatives do not cancel positives.
- Add the squares and divide by N (population) or n − 1 (sample) — this is the variance.
- Take the square root to return to the original units.
How to Read the Result
For data that is roughly bell-shaped, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three — the 68–95–99.7 rule. A value more than two or three standard deviations from the mean is unusual, which is the idea behind z-scores.
Where It Is Used
- Science and engineering, to describe measurement error and consistency.
- Finance, where standard deviation of returns is a basic measure of risk.
- Education, to describe how widely test scores are spread.
- Quality control, to catch processes drifting out of tolerance.
For the other basics of a data set, use the Mean, Median, Mode Calculator.
Worked example
For 4, 8, 6, 5, 3, 8: the mean is 5.6667, and the sample standard deviation is about 2.07 — a measure of how spread out the values are around that mean.
Frequently asked questions
What's the difference between sample and population standard deviation?
Population standard deviation treats your numbers as the entire group you care about; sample standard deviation treats them as a sample used to estimate a larger population, and divides by one less (n − 1) to correct for that. Use sample unless you're certain you have every value in the group.
What does a small vs. large standard deviation mean?
A small standard deviation means values cluster tightly around the mean; a large one means they're spread out widely — two data sets can have the same mean but very different standard deviations.
How is variance related to standard deviation?
Variance is the standard deviation squared — standard deviation is just the square root of variance, converted back to the same units as your original data, which is why it's usually the more intuitive number to read.
