Standard Deviation Calculator

Find the standard deviation and variance of a list of numbers, with every step shown.

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Standard deviation (σ)2
Variance4
Mean5
Count (n)8
Worked steps
  1. Mean = 5 — sum of all values ÷ n = 40 ÷ 8 = 5.
  2. Subtract the mean from each value and square it:
    (2 − 5)² = 9(4 − 5)² = 1(4 − 5)² = 1(4 − 5)² = 1(5 − 5)² = 0(5 − 5)² = 0(7 − 5)² = 4(9 − 5)² = 16
  3. Sum of squared deviations Σ(x − x̄)² = 32.
  4. Variance = Σ ÷ n = 32 ÷ 8 = 4.
  5. Standard deviation = √variance = √4 = 2.

Population standard deviation (σ) divides the sum of squared deviations by n; sample standard deviation (s) divides by n − 1 (Bessel's correction) to better estimate the spread of a wider population from a sample. Everything is computed in your browser — nothing is uploaded.

How to calculate standard deviation

Paste your numbers, then choose Population (÷ n) or Sample (÷ n − 1). The standard deviation, variance, mean and count appear immediately, with the full arithmetic — mean, each squared deviation, the sum, the division and the square root — shown underneath.

Using the worked steps

The step-by-step breakdown lets you check homework or follow the method by hand. Each value's deviation from the mean is squared and listed, so you can see exactly where the variance and standard deviation come from.

Questions

How is standard deviation calculated?
Find the mean, subtract it from each value and square the result, add those squared deviations together, divide by n (population) or n − 1 (sample) to get the variance, then take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squared deviations sum to 32, the population variance is 4 and the population standard deviation is 2.
Should I use the population or sample formula?
Use the population formula (÷ n) when your numbers represent the entire group you care about. Use the sample formula (÷ n − 1) when they are a sample meant to estimate a larger population — it gives a slightly larger, less biased estimate of the spread.
Why does sample standard deviation need at least two numbers?
The sample formula divides by n − 1, which is zero when you have a single value, so the result is undefined. Enter two or more numbers, or switch to the population option.