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Standard Deviation Calculator

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

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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