Population standard deviation calculator
Separate values with commas, spaces, or new lines. Any mix works.
This page leads with the population result: sigma divides the sum of squared deviations by n. Use it only when your list is the whole group you care about. For sample data, the divide-by-n-minus-1 result on the homepage solver is the right choice.
Show the step-by-step calculation
Population Standard Deviation Calculator
Population standard deviation (sigma) divides the sum of squared deviations by n, the full count, and is the right spread measure when your data list is the entire group.
This calculator is a preset view of the same engine that powers the homepage, fixed to lead with the population result. Sigma is computed by finding the mean, squaring every deviation from it, averaging those squares by dividing by n, and taking the square root. Because it divides by the full count rather than n - 1, the population standard deviation is always a little smaller than the sample standard deviation for the same numbers. The sample results still appear lower in the table for reference.
Choosing between the two formulas comes down to one question: is your dataset the whole group or a piece of it? If it is the whole group, divide by n and report sigma. If it is a sample meant to generalize, divide by n - 1 and report s. Our sample versus population standard deviation guide walks through the reasoning, Bessel's correction, and real world decision examples with worked numbers. For both results side by side, use the standard deviation solver.
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Worked example
Take the complete dataset 2, 4, 4, 4, 5, 5, 7, 9, for example the scores of every member of a small team. Here is the full population standard deviation calculation.
Step 1, find the mean. The sum is 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40, and there are 8 values, so the mean is 40 / 8 = 5.
Step 2, square each deviation from the mean:
- 2 - 5 = -3, and (-3) squared is 9
- 4 - 5 = -1, and (-1) squared is 1
- 4 - 5 = -1, and (-1) squared is 1
- 4 - 5 = -1, and (-1) squared is 1
- 5 - 5 = 0, and 0 squared is 0
- 5 - 5 = 0, and 0 squared is 0
- 7 - 5 = 2, and 2 squared is 4
- 9 - 5 = 4, and 4 squared is 16
Step 3, sum the squared deviations: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
Step 4, divide by n, because this is the whole population: population variance = 32 / 8 = 4. Population standard deviation = sqrt(4) = 2. For comparison, the sample version would divide by n - 1 = 7, giving a variance of about 4.5714 and a standard deviation of about 2.1381. Paste the same eight numbers into the calculator above and the results table shows exactly these values, with the population result on top.
Frequently asked questions
When should I use the population standard deviation?
Use it when your dataset contains every member of the group you care about and you are only describing that group. Examples: the test scores of all 28 students in one class, the closing prices of a stock over a fixed 30 day record, or the ages of every employee in one small company. If your data are a subset drawn from a larger group, use the sample standard deviation instead.
What is the difference between population and sample standard deviation?
Both follow the same steps: find the mean, square every deviation from it, average the squares, and take the square root. The only difference is the divisor. Population standard deviation, written as sigma, divides the sum of squared deviations by n. Sample standard deviation, written as s, divides by n - 1, a change called Bessel's correction that removes the downward bias when estimating from a sample.
Why does the sample formula divide by n - 1?
In a sample, deviations are measured around the sample mean, which is fitted to that very sample, so the raw average of squared deviations comes out slightly too small. Dividing by n - 1 instead of n inflates the result just enough to make the sample variance an unbiased estimate of the population variance.
What symbol is used for population standard deviation?
The Greek letter sigma, in its lowercase form. Population variance is sigma squared. The sample equivalents are s and s squared. When you see sigma in a formula or report, the author divided by n; when you see s, they divided by n - 1.
Can the population standard deviation be zero?
Yes, but only in one situation: every value in the dataset is identical. Then every deviation from the mean is zero, the variance is zero, and its square root is zero. Any spread at all produces a population standard deviation greater than zero.
Does the population standard deviation work with a single value?
Yes. With one value, the mean equals that value, the only squared deviation is zero, and the population standard deviation is zero. The sample version needs at least 2 values because dividing by n - 1 with one value would mean dividing by zero.