Standard Deviation Calculator (Sample & Population)
This standard deviation calculator computes sample and population standard deviation, variance, mean, sum, and count from a list of numbers, and shows each deviation so you can check the steps.
How to use
- Paste numbers separated by commas, spaces, or new lines. Blank gaps are ignored.
- Read the count, sum, and mean. Compare sample standard deviation (divide by n − 1) with population standard deviation (divide by n).
- Use the deviation table to check each value minus the mean and the squared deviation.
- Sample standard deviation needs at least two numbers. A single number still has a population standard deviation of 0.
Sample and population formulas
Mean = sum of the values ÷ n. Subtract the mean from each value and square those differences. Population variance averages the squares by dividing by n, and population standard deviation is the square root. Sample variance divides by n − 1, and sample standard deviation is that square root. Variance is the standard deviation squared, so it stays in squared units.
Example. For 2, 4, 4, 4, 5, 5, 7, 9 the sum is 40 and the mean is 5. The squared deviations add to 32. Population variance is 32 ÷ 8 = 4 and the population standard deviation is 2. Sample variance is 32 ÷ 7 ≈ 4.5714 and the sample standard deviation is about 2.1381.
Frequently asked questions
What is standard deviation?
A measure of how spread out values are around the mean. A small SD means values cluster close to the average.
Sample or population: which should I use?
Use population SD (divide by n) when you have every member of the group. Use sample SD (divide by n - 1) when your data is a sample of a larger group.
How is it calculated?
Find the mean, subtract it from each value, square the differences, average them (n or n - 1), and take the square root.
What is variance?
Variance is the standard deviation squared. It is in squared units, which is why SD is easier to interpret.
What does the 68-95-99.7 rule mean?
For roughly normal data, about 68% of values fall within 1 SD of the mean, 95% within 2, and 99.7% within 3.
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