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Standard Deviation Calculator

Find the standard deviation and variance of any list of numbers.

Result

1.87
1.71
3.50
2.92
5.50
6

How it is calculated

The tool reads numbers separated by commas, spaces or line breaks and ignores anything that is not a number. It finds the mean x̄ = Σx / n and the sum of squared deviations S = Σ(x − x̄)². Population variance is σ² = S / n and sample variance is s² = S / (n − 1); each standard deviation is the square root of its variance. Use the population figures when your list is the whole group, and the sample figures when it is only a sample of a larger one. With a single value the sample variance is shown as 0. Results are rounded to two decimals.

Worked example

Data: 4, 8, 6, 5, 3, 7. Count n = 6, mean = 33 / 6 = 5.5. Deviations from the mean: −1.5, 2.5, 0.5, −0.5, −2.5, 1.5. Squared: 2.25, 6.25, 0.25, 0.25, 6.25, 2.25, sum S = 17.5. Sample variance = 17.5 / 5 = 3.50 and sample SD = 1.87. Population variance = 17.5 / 6 = 2.92 and population SD = 1.71.

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Common mistakes

  • Using the population figure for a sample. It is smaller and understates the spread, especially with few values.
  • Pasting numbers with thousands separators such as 1,200, which are read as two separate numbers (1 and 200).
  • Including labels or units; text items are skipped, so the count n may be lower than you expect. Check the count shown.
  • Reading the variance as if it were in the original unit. Variance is in squared units; only the SD is in the original unit.
  • Expecting a meaningful sample SD from one value, which is shown as 0.

About this calculator

This calculator finds the sample standard deviation (using Bessel's correction, dividing by n−1) and the population standard deviation (dividing by n) of a list of numbers, along with the variance and arithmetic mean. Standard deviation measures how spread out the values are around the mean — a low value means the data clusters tightly, a high value means it's scattered.

Statisticians, researchers, teachers, and students use it constantly: in quality control to check manufacturing consistency, in finance to measure volatility, and in science to report the uncertainty of measurements. Use the sample formula when your numbers are a subset drawn from a larger population; use the population formula when your list is the entire population itself.

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