Standard Deviation Calculator
Paste a data set and get the sample and population standard deviation, the variance, the mean and the count — with the formula and steps shown.
Quick answer: Standard deviation measures spread. Sample s = √(Σ(x−x̄)² ÷ (n−1)); population σ = √(Σ(x−x̄)² ÷ n). For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, population σ = 2 and sample s ≈ 2.138.
Standard deviation summarises how far a set of values typically sits from their mean. A small value means the data cluster tightly; a large one means they're spread out. It's the square root of the variance and shares the units of the original data, which is why it's the most-used measure of spread in coursework and research.
The formulas
Mean x̄ = Σx ÷ n
Sample s = √( Σ(x − x̄)² ÷ (n − 1) )
Population σ = √( Σ(x − x̄)² ÷ n )
Sample s = √( Σ(x − x̄)² ÷ (n − 1) )
Population σ = √( Σ(x − x̄)² ÷ n )
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9: mean = 40 ÷ 8 = 5. The squared deviations sum to 32. Population σ = √(32 ÷ 8) = √4 = 2. Sample s = √(32 ÷ 7) = √4.571 ≈ 2.138.
Frequently asked questions
Should I use sample or population standard deviation?
Use sample (n − 1) when your numbers are a sample drawn from a larger group. Use population (n) only when you have every member of the group.
What's the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root, so it's back in the original units.
Why divide by n − 1 for a sample?
Dividing by n − 1 (Bessel's correction) removes bias, because a sample tends to underestimate the true spread.
Can standard deviation be negative?
No. It's a square root of a sum of squares, so it's always zero or positive. Zero means every value is identical.
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