Standard Deviation Calculator
Compute sample and population standard deviation and variance from a pasted list of numbers, with the mean and formulas shown.
Result
- Count (n)8
- Mean5
- Sum of squared deviations32
- Sample variance (s²)4.571429
- Sample SD (s)2.13809
- Population variance (σ²)4
- Population SD (σ)2
Step-by-step
- Mean = 5. Sum of squared deviations Σ(x−mean)² = 32.
- Population variance = 32 ÷ 8 = 4; population SD = √ = 2.
- Sample variance = 32 ÷ 7 = 4.571429; sample SD = √ = 2.13809.
How to use this calculator
- Paste your numbers separated by commas, spaces, or new lines.
- Read the sample standard deviation (the usual choice) front and center.
- Use the population standard deviation only if your data is the whole population.
- Check the variance and sum of squared deviations in the breakdown for the working.
About this calculator
Standard deviation measures how spread out a set of numbers is around their mean: a small standard deviation means the values cluster tightly around the average, a large one means they are widely scattered. It is the square root of the variance, which is the average of the squared distances from the mean. There are two versions. The population standard deviation divides by N and is used when your data covers the entire population. The sample standard deviation divides by n − 1 (Bessel's correction) and is the right choice when your data is a sample drawn from a larger population — the most common situation. This calculator reports both, along with the variance, mean, and sum of squared deviations, from a list you paste in any common format.
How it works — the formula
Mean μ = Σx ÷ n
Population: σ² = Σ(x−μ)² ÷ N, σ = √σ²
Sample: s² = Σ(x−x̄)² ÷ (n−1), s = √s²Both square the distance of each value from the mean, average those squares, and take the root. The only difference is dividing by N (population) versus n−1 (sample).
Worked examples
- Inputs:
- data=2,4,4,4,5,5,7,9
- Output:
- mean 5, pop SD 2, sample SD 2.138
- Inputs:
- data=10,12,14,16,18
- Output:
- mean 14, pop SD 2.828, sample SD 3.162
- Inputs:
- data=5,5,5,5
- Output:
- SD 0 (no spread)
Limitations
- Sample SD requires at least two values.
- Treats the input as raw, unweighted data.
- Very large datasets are summarized in full precision but rounded for display.
Reports the spread of the entered data; inferential use requires a representative sample.