Standard deviation calculator

Standard deviation measures how far a set of numbers is spread from the mean. For 2, 4, 4, 4, 5, 5, 7 and 9 the mean is 5, the population standard deviation is 2 and the sample standard deviation is about 2.138.

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Result

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How it is calculated

  1. Mean x̄ = Σx ÷ n.
  2. Median: sort the numbers and take the middle one. When n is even, take the mean of the two middle numbers.
  3. Mode: the number that appears most often. There may be more than one, and there is none when each number appears once. Range = maximum − minimum.
  4. Population variance σ² = Σ(x − x̄)² ÷ n and sample variance s² = Σ(x − x̄)² ÷ (n − 1). The standard deviation is the square root of the variance. Use the population formula when the numbers cover the whole group, and the sample formula when they are part of it (the same as STDEV.P and STDEV.S in Excel).
  5. Results are shown to 10 significant digits.
  6. All calculations are done in your browser. The details you enter are not sent to any server.

Worked examples

Eight values: 2, 4, 4, 4, 5, 5, 7, 9

Details
2, 4, 4, 4, 5, 5, 7, 9
Result
Mean x̄ 5, Median 4.5, Sample standard deviation s 2.138089935, Population standard deviation σ 2

1 to 10

Details
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Result
Mean x̄ 5.5, Median 5.5, Sample standard deviation s 3.027650354, Population standard deviation σ 2.872281323

Three marks: 85, 90, 78

Details
85, 90, 78
Result
Mean x̄ 84.33333333, Median 85, Sample standard deviation s 6.027713773, Population standard deviation σ 4.921607687

Two modes: 1, 2, 2, 3, 3, 4

Details
1, 2, 2, 3, 3, 4
Result
Mean x̄ 2.5, Median 2.5, Sample standard deviation s 1.048808848, Population standard deviation σ 0.9574271078