Standard Deviation Calculator
Calculate population and sample standard deviation, variance, mean and range for any list of numbers.
What standard deviation measures
Standard deviation describes how spread out values are around their mean. A small standard deviation means values cluster near the average; a large one means they are widely scattered.
Step by step
- Find the mean: (4 + 8 + 6 + 5 + 3 + 7 + 9) ÷ 7 = 6.
- Subtract the mean from each value and square the result: 4, 4, 0, 1, 9, 1, 9.
- Add the squares: 28. This is the sum of squares.
- Divide by N for a population (28 ÷ 7 = 4) or by N − 1 for a sample (28 ÷ 6 = 4.67). This is the variance.
- Take the square root: population σ = 2; sample s ≈ 2.16.
Population or sample?
Use the population formula when your list includes every member of the group you care about, such as the scores of every student in one class. Use the sample formula when the list is a sample used to estimate a larger group, as in surveys and experiments. Dividing by N − 1 (Bessel's correction) avoids underestimating the spread.
The 68–95–99.7 rule
For data that follows a normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. If IQ scores have a mean of 100 and a standard deviation of 15, about 95% of people score between 70 and 130.
Frequently asked questions
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean; standard deviation is its square root, in the same units as the data.
When do I use N − 1?
When your data is a sample from a larger population. Use N when you have the whole population.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is zero or positive.
Last reviewed October 2026. Results are estimates; check important decisions with a qualified professional.