Standard Deviation Calculator Formula
Understand the math behind the standard deviation calculator. Each variable explained with a worked example.
Formulas Used
Sample Standard Deviation
samp_sd = sqrt(ss / 4)Population Standard Deviation
pop_sd = sqrt(ss / 5)Mean
data_mean = mnVariables
| Variable | Description | Default |
|---|---|---|
v1 | Value 1 | 4 |
v2 | Value 2 | 8 |
v3 | Value 3 | 6 |
v4 | Value 4 | 5 |
v5 | Value 5 | 7 |
mn | Derived value= (v1 + v2 + v3 + v4 + v5) / 5 | calculated |
ss | Derived value= pow(v1 - mn, 2) + pow(v2 - mn, 2) + pow(v3 - mn, 2) + pow(v4 - mn, 2) + pow(v5 - mn, 2) | calculated |
How It Works
How to Calculate Standard Deviation
Formula
Population SD (sigma) = sqrt( Sum of (xi - mean)^2 / N )
Sample SD (s) = sqrt( Sum of (xi - mean)^2 / (n - 1) )
Standard deviation is the square root of the variance and is expressed in the same units as the original data. About 68% of data in a normal distribution falls within one standard deviation of the mean.
Worked Example
Find the standard deviation of 4, 8, 6, 5, 7.
v1 = 4v2 = 8v3 = 6v4 = 5v5 = 7
- 01Mean = 30 / 5 = 6
- 02Squared deviations: 4, 4, 0, 1, 1
- 03Sum of squares = 10
- 04Sample SD = sqrt(10 / 4) = sqrt(2.5) ≈ 1.5811
- 05Population SD = sqrt(10 / 5) = sqrt(2) ≈ 1.4142
Ready to run the numbers?
Open Standard Deviation Calculator