A standard deviation calculator identifies how spread out the values in a dataset are from the mean.
How to calculate the standard deviation:
The standard deviation is a measure of dispersion — it tells you how much individual values differ from the average.
Step-by-step method:
- List all values in the dataset
- Calculate the mean (average) of the values
- Subtract the mean from each value and square the result
- Sum all the squared differences
- Divide by n (population) or n−1 (sample)
- Take the square root — that is the standard deviation
Formula:
For a dataset X = {x₁, x₂, …, xₙ} with mean μ:
Population standard deviation:
σ = √( Σ(xᵢ − μ)² / n )
Sample standard deviation:
s = √( Σ(xᵢ − x̄)² / (n − 1) )
Where n is the number of values, μ is the population mean, and x̄ is the sample mean.
Examples
Example 1: Dataset {2, 4, 4, 4, 5, 5, 7, 9}
- Mean = (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
- Squared differences: 9, 1, 1, 1, 0, 0, 4, 16 → Sum = 32
- Population variance = 32 / 8 = 4
- Population Standard Deviation σ = √4 = 2.0000
Example 2: Dataset {10, 20, 30}
- Mean = 60 / 3 = 20
- Squared differences: 100, 0, 100 → Sum = 200
- Population variance = 200 / 3 ≈ 66.67 → σ ≈ 8.1650
- Sample variance = 200 / 2 = 100 → s = 10.0000
Key features include:
- Input for up to 50 values
- Automatic calculation
- Toggle between population and sample standard deviation
- Flexible list management — add or remove values freely
- Support for decimals and negative numbers
- Dynamic recalculation on every change