Standard Deviation Calculator
Standard deviation is the typical distance of observations from their mean, measured in the original units. The squaring is what makes it algebraically useful — and the n-1 is what makes it honest for a sample.
Standard deviation
6.0581
- Variance
- 36.7
- Mean
- 16.8
- Spread relative to the mean
- 36.06%
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
The formula this calculator uses
SD = sqrt(Sum of (xi - Mean)^2 / (n - 1))
- xi
- Each observation
- mu or x-bar
- Mean of the observations
- n
- Number of observations
- Sigma
- Sum of the squared deviations from the mean
How to check the result by hand
- 1
Compute the mean
Add every value and divide by how many there are: (10 + 12 + 23 + 23 + 16) / 5 = 16.8. Everything downstream depends on this being right.
- 2
Find each deviation from the mean
Subtract the mean from every observation: -6.8, -4.8, 6.2, 6.2 and -0.8. These sum to zero by construction, which is a convenient check before continuing.
- 3
Square each deviation
46.24, 23.04, 38.44, 38.44 and 0.64. Squaring removes the signs so distances of equal size on either side contribute equally.
- 4
Sum them and choose the divisor
Total is 146.8. Divide by n-1 = 4 for a sample giving 36.70, or by n = 5 for the whole population giving 29.36. This is the variance.
- 5
Take the square root
sqrt(36.70) = 6.0581. Bringing the units back to the original scale makes the figure readable as roughly six units of typical distance from the mean of 16.8.
For worked examples, common mistakes and the limits of this formula, read the full How To Calculate Standard Deviation page.