Standard Deviation Calculator
Mean, variance, and standard deviation of a list of numbers.
Input sheet
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Standard deviation measures how spread out a set of numbers is around their average. A small value means the data clusters tightly; a large one means it scatters widely — essential for grades, quality control, and finance.
How it works
Variance is the average squared distance from the mean (dividing by n−1 for a sample); standard deviation is its square root.
There are two versions. Sample standard deviation divides the squared distances by n−1 and is used when your data is a sample of a larger group; population standard deviation divides by n and is used when you have every member. This tool reports both, headlining the sample figure.
The n−1 divisor (Bessel's correction) compensates for the fact that a sample tends to underestimate the true spread of the population it was drawn from.
Standard deviation = square root of [ sum of (value − mean)² ÷ (n − 1 for a sample, or n for a population) ]
Worked examples
The list 2, 4, 4, 4, 5, 5, 7, 9. → Sample ≈ 2.138, population = 2
Mean is 5; squared deviations sum to 32. Population: √(32/8) = 2. Sample: √(32/7) ≈ 2.138.
The list 10, 12, 14 (sample). → ≈ 2.0
Mean is 12; squared deviations 4 + 0 + 4 = 8; √(8 ÷ 2) = √4 = 2.
Tips & gotchas
- Use the sample value when your numbers are a subset of a bigger group; use the population value only when you have the entire group.
- Standard deviation shares the units of the data, unlike variance, which is in squared units — that is why it is more interpretable.
- For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and 95% within two.
FAQ
What is the difference between sample and population standard deviation?
Sample divides by n−1 and estimates spread from a subset; population divides by n and applies when you have every data point. Sample values are slightly larger.
How does standard deviation relate to variance?
Variance is the average squared distance from the mean; standard deviation is its square root, bringing the result back into the original units.
Why divide by n−1 for a sample?
It corrects for the bias that arises because the sample mean sits closer to the sample's own points than the true population mean would, which otherwise understates the spread.
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