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Standard deviation and variance with every step of the formula shown.
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Standard Deviation Calculator
Population or sample standard deviation with steps
Separate values with commas, spaces or new lines.
Population standard deviation (σ)
4.898979
- Count (n)
- 8
- Sum
- 144
- Mean (μ)
- 18
- Variance (σ²)
- 24
- Sample SD (for comparison)
- 5.237229
- Standard error of the mean
- 1.732051
Margin of error uses the normal (z) distribution with the standard error SD / √n. For small samples a t-distribution gives slightly wider intervals.
How it was calculated
- σ = √( Σ(x - μ)² / n )
- Mean = 144 / 8 = 18
- Σ(x - mean)² = (10 - 18)² + (12 - 18)² + (23 - 18)² + (23 - 18)² + (16 - 18)² + (23 - 18)² + (21 - 18)² + (16 - 18)²
- = 192
- Variance = 192 / 8 = 24
- σ = √24 = 4.898979
| Value | Frequency | Relative | Deviation (x - mean) | (x - mean)² | Frequency × (x - mean)² |
|---|---|---|---|---|---|
| 10 | 1 | 12.5% | -8 | 64 | 64 |
| 12 | 1 | 12.5% | -6 | 36 | 36 |
| 16 | 2 | 25% | -2 | 4 | 8 |
| 21 | 1 | 12.5% | 3 | 9 | 9 |
| 23 | 3 | 37.5% | 5 | 25 | 75 |
| Confidence | Margin of error | Range |
|---|---|---|
| 68.3% | ± 1.732051 | 16.267949 to 19.732051 |
| 90% | ± 2.848971 | 15.151029 to 20.848971 |
| 95% | ± 3.394757 | 14.605243 to 21.394757 |
| 99% | ± 4.461467 | 13.538533 to 22.461467 |
| 99.9% | ± 5.69936 | 12.30064 to 23.69936 |
| 99.99% | ± 6.738703 | 11.261297 to 24.738703 |
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How the standard deviation calculator works
Standard deviation measures how spread out numbers are from their mean. Find the mean, subtract it from each value and square the result, add those squares up, divide, and take the square root.
For a population (every member of the group) divide by n. For a sample (a subset used to estimate a bigger group) divide by n − 1, which is called Bessel's correction and makes the estimate unbiased. The sample result is always a little larger.
In a normal distribution about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. The margin of error table uses the standard error (SD ÷ √n) to show how precisely the mean is known.
Using and checking your result
Published by JustYourCalculator. Check the units and assumptions above, and compare a known example before relying on the output. Calculations use browser arithmetic and may round displayed values.
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