Standard Deviation Calculator
This free standard deviation calculator finds the standard deviation, mean, and variance of any data set, and shows the formula and steps. Standard deviation measures how spread out your data is around the mean — a small value means the data is tightly clustered, while a large value means it is widely dispersed. Enter your numbers and the calculator returns the mean and standard deviation using either the population or sample formula.
How to Use the Calculator
- Enter your data values separated by commas or spaces.
- Choose population or sample.
- Calculate — see the mean, variance, and standard deviation with steps.
The Standard Deviation Formula
Population: σ = √( Σ(xᵢ − μ)² ÷ N )
Sample: s = √( Σ(xᵢ − x̄)² ÷ (n − 1) )
You subtract the mean from each value, square the differences, average them (dividing by N for a population or n − 1 for a sample), and take the square root.
Step-by-Step Example
For the data 4, 8, 6, 5, 3: the mean is 26 ÷ 5 = 5.2. The squared differences sum to 14.8. Dividing by 5 gives a population variance of 2.96, so the population standard deviation is √2.96 ≈ 1.72.
Population vs Sample
| Type | Divides By | Use When |
|---|---|---|
| Population | N | You have the entire group |
| Sample | n − 1 | You have a sample of a larger group |
Why Standard Deviation Matters
- Spread: shows how consistent or variable your data is.
- Statistics: underpins confidence intervals and hypothesis tests.
- Finance: measures the volatility (risk) of returns.
- Quality control: tracks process consistency.
Frequently Asked Questions
What is the standard deviation formula?
For a population, σ = √(Σ(xᵢ − μ)² ÷ N); for a sample, s = √(Σ(xᵢ − x̄)² ÷ (n − 1)). You square the deviations from the mean, average them, and take the square root.
How do you calculate standard deviation?
Find the mean, subtract it from each value and square the result, average those squares (variance), then take the square root. The calculator shows each step.
What is the difference between population and sample standard deviation?
Population divides by N and is used for the whole group, while sample divides by n − 1 to give an unbiased estimate from a sample.
How is standard deviation related to variance?
Standard deviation is the square root of the variance, expressed in the same units as the data, which makes it easier to interpret.
Is this standard deviation calculator free?
Yes — it is completely free, requires no signup, and shows the mean, variance, and standard deviation with steps.