ToolCalcu

Standard Deviation Calculator

Paste a list of numbers to get the standard deviation and variance. Choose sample or population, because they give different answers.

Your data

Separate values with commas, spaces, semicolons or new lines. Use a period for decimals and no thousands separators.

Result

Please correct the highlighted fields.
Sample standard deviation (s)
2.13809
Variance
4.571429
Mean
5
Count
8
Sum
40
Smallest value
2
Largest value
9

How standard deviation is calculated

Standard deviation measures how far the values typically are from their mean. Subtract the mean from each value, square the differences, average them and take the square root.

The average of the squared differences is the variance. For a population it divides by n; for a sample it divides by n − 1, which corrects the tendency of a sample to underestimate the spread. The tool uses a numerically stable running method so large numbers with small differences stay accurate.

Formulas used

Mean = Σx ÷ n
Population: σ = √( Σ(x − μ)² ÷ n )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Variance = (standard deviation)²

Worked example

The values 2, 4, 4, 4, 5, 5, 7 and 9 (n = 8, mean 5):

  1. Squared differences: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32
  2. Population: √(32 ÷ 8) = √4 = 2
  3. Sample: √(32 ÷ 7) = 2.13809

Sample standard deviation = 2.13809 (population: 2).

Assumptions and limits

  • All values are weighted equally.
  • A sample needs at least two values; a population can have one.
  • Up to 10,000 values are accepted.

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