Standard Deviation Calculator
Find the population or sample standard deviation of a list of numbers, with the variance, standard error, margin of error and every step.
Change the values and press Calculate to work out your own figures.
Enter numbers and say whether they are a whole population or a sample.
| Confidence | z | Margin of error | Interval |
|---|---|---|---|
| 90% | 1.644854 | ± 7.15457469767 | 15.0954253023 to 29.4045746977 |
| 95% | 1.959964 | ± 8.52519970936 | 13.7248002906 to 30.7751997094 |
| 99% | 2.575829 | ± 11.2040101972 | 11.0459898028 to 33.4540101972 |
Show the working
- Mean = 178 ÷ 8 = 22.25
- Sum of squared deviations = Σ(x − mean)² = 1,059.5
- σ² = 1,059.5 ÷ (N) = 132.4375
- σ = √132.4375 = 11.5081492865
How to use it
Enter numbers and say whether they are a whole population or a sample.
Paste or type the numbers separated by commas, spaces or new lines. Results are worked out exactly and shown to 12 significant figures; Show the working lists each step.
Key facts
σ = √(Σ(x − μ)² ÷ N); s = √(Σ(x − x̄)² ÷ (n − 1)) SE = s ÷ √n; margin of error = z × SE
What standard deviation tells you
It is the typical distance of the values from the mean, in the same units as the data. For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean and 95% within two.
Questions
What is the difference between sample and population standard deviation?
The sample version divides by n − 1 instead of N, which makes it slightly larger to correct for estimating the mean from the same data.
Can the standard deviation be negative?
No; it is 0 only when every value is the same.
Formulas
σ = √(Σ(x − μ)² ÷ N); s = √(Σ(x − x̄)² ÷ (n − 1)) SE = s ÷ √n; margin of error = z × SE
Sources
Limitations
- Up to 5,000 values. Values are separated by commas, spaces or new lines, so write 1250 rather than 1,250.