Standard deviation measures how far values typically sit from their average. Two classes can both average 75% on a test, but if one class’s scores range from 40 to 100 and the other’s from 70 to 80, the first has a much larger standard deviation. This calculator works out both the sample and population versions and shows every intermediate number.
How to use the calculator
- Paste or type your numbers into the data box. Commas, spaces, tabs and line breaks all work as separators.
- Choose Sample if your data is a subset of a bigger group (the usual case in surveys and experiments) or Population if it is the entire group.
- Read the standard deviation on the tape. The deviations table below lists each value’s distance from the mean and its square.
Standard deviation formulas
Population standard deviation
Sample standard deviation
Here μ or x̄ is the mean, Σ means “add up over every value”, and N or n is the number of values.
Step-by-step example
Take the data set 10, 12, 23, 23, 16, 23, 21, 16.
- Mean: the values add to 144, and 144 ÷ 8 = 18.
- Deviations: subtract 18 from each value: −8, −6, 5, 5, −2, 5, 3, −2.
- Squares: 64, 36, 25, 25, 4, 25, 9, 4.
- Sum of squares: 192.
- Variance: population 192 ÷ 8 = 24; sample 192 ÷ 7 ≈ 27.43.
- Standard deviation: population √24 ≈ 4.899; sample √27.43 ≈ 5.237.
Reading the result
For data that is roughly bell-shaped, the 68–95–99.7 rule gives a quick sense of scale:
| Range | Share of values (approx.) |
|---|---|
| Mean ± 1 SD | 68% |
| Mean ± 2 SD | 95% |
| Mean ± 3 SD | 99.7% |
So if delivery times average 30 minutes with a standard deviation of 5, about two-thirds of deliveries arrive between 25 and 35 minutes. A value more than 2 or 3 standard deviations from the mean is unusual — the z-score calculator measures exactly how unusual.
Related measures on the tape
- Variance is the standard deviation squared. It is useful in formulas but harder to interpret because its units are squared.
- Coefficient of variation (SD ÷ mean × 100%) compares spread across data sets with different scales.
- Standard error of the mean (SD ÷ √n) estimates how much the sample mean itself would vary from sample to sample; it shrinks as you collect more data.
Frequently asked questions
What is the difference between sample and population standard deviation?
Use population standard deviation (σ, divide by N) when your data includes every member of the group you care about. Use sample standard deviation (s, divide by n − 1) when the data is a sample used to estimate a larger population. The n − 1, called Bessel's correction, compensates for a sample's tendency to underestimate spread.
What does a high standard deviation mean?
It means the values are spread far from the mean. A low standard deviation means they cluster tightly around it. The number is in the same units as the data, so a standard deviation of 5 minutes is directly comparable to an average of 30 minutes.
How is standard deviation related to variance?
Variance is the average squared distance from the mean; standard deviation is its square root. Taking the square root returns the measure to the original units of the data.
How many numbers do I need?
At least two. With a single value there is no spread to measure, and the sample formula would divide by zero.