worldofcalcsStandard Deviation Calculator 📉
Standard deviation
4.899
Mean
18
Sum of squared deviations
192
Variance
24
Count
8
What is standard deviation?
Standard deviation measures how spread out numbers are around their mean. A small value means the numbers are close to the mean; a large one means they are widely spread. This calculator gives the standard deviation, variance and mean, for a whole population or a sample.
Standard deviation formula
σ = √(Σ(x − mean)² ÷ n) · s = √(Σ(x − mean)² ÷ (n − 1))
Find the mean, subtract it from each value, square the differences and add them up. Divide by n for a whole population (σ), or by n − 1 for a sample (s), to get the variance; the standard deviation is its square root. School textbooks usually use the population formula.
Good to know
- Use the population formula when you have every value; use the sample formula when your data is a sample from a bigger group.
- In a normal distribution, about 68% of values lie within one standard deviation of the mean and about 95% within two.
- Standard deviation is in the same units as the data; variance is in squared units.
- In investing, standard deviation of returns is a common measure of risk.
Worked example
- Data: 10, 12, 23, 23, 16, 23, 21, 16
- Mean = 144 ÷ 8 = 18
- Sum of squared deviations = 192
- Population variance = 192 ÷ 8 = 24, so σ = √24 = 4.899
- Sample variance = 192 ÷ 7 = 27.43, so s = 5.237
Step-by-step working
Deviations from the mean (18) for the example data.
| Value (x) | x − mean | (x − mean)² |
|---|---|---|
| 10 | -8 | 64 |
| 12 | -6 | 36 |
| 23 | 5 | 25 |
| 23 | 5 | 25 |
| 16 | -2 | 4 |
| 23 | 5 | 25 |
| 21 | 3 | 9 |
| 16 | -2 | 4 |
The squared deviations add up to 192.