Formula
mean x̄ = Σx ÷ npopulation: σ = √(Σ(x − x̄)² ÷ n)sample: s = √(Σ(x − x̄)² ÷ (n − 1))
- Variance is the standard deviation squared
- n is the number of values
The standard deviation is the typical distance of a value from the mean.
Worked example
Seven values, as a sample
- Numbers
- 4 8 6 5 3 7 9
- Treat the numbers as
- A sample (divide by n − 1)
Sample standard deviation (s)
2.1602469
Sample variance (s²): 4.66666667
Population standard deviation (σ): 2
Mean: 6
How it works
Standard deviation measures how spread out numbers are around their mean. Find the mean, square each value's distance from it, add the squares, divide, and take the square root. A small standard deviation means the values cluster near the mean; a large one means they are scattered.
The only choice is the divisor. If your numbers are the entire group you care about, divide by n (population). If they are a sample drawn from a larger group — the usual case — divide by n − 1, which corrects the tendency of a sample to understate the spread (Bessel's correction). For the data 4, 8, 6, 5, 3, 7, 9 the mean is 6 and the sample standard deviation is about 2.16. For up to 20 values the table shows each deviation so you can follow along.
The standard error shown is s ÷ √n, which estimates how far the sample mean may be from the true mean. For plain averages, the median and the mode use the Average Calculator; for weighted data try the Weighted Average Calculator, and to see probabilities for repeated trials try the Probability Calculator.
Assumptions and limits
- Results use the usual two-pass formula for numerical stability.
- Up to 5,000 values; plain numbers only, without thousands separators.