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Standard Deviation Calculator

Paste a list of numbers to find the sample and population standard deviation, variance, mean, sum, and count, with every step shown. Everything runs in the browser.

  • Sample & population
  • Variance & mean
  • Step-by-step work
  • Stays on your device

Population σ = √(Σ(x − μ)² / N). Sample s = √(Σ(x − x̄)² / (n − 1)).

Step-by-step work

  1. Count: n = 8
  2. Sum: Σx = 40
  3. Mean: x̄ = 40 / 8 = 5
  4. Squared deviations (x − x̄)²:
    xx − x̄(x − x̄)²
    2-39
    4-11
    4-11
    4-11
    500
    500
    724
    9416
  5. Sum of squares: Σ(x − x̄)² = 32
  6. Population variance: σ² = 32 / 8 = 4
  7. Population std dev: σ = √4 = 2
  8. Sample variance: s² = 32 / 7 = 4.5714285714
  9. Sample std dev: s = √4.5714285714 = 2.1380899353

Numbers stay on this device. Nothing is uploaded, and there is no account.

Standard deviation is the square root of the average squared distance from the mean. For a population of N values, σ = √(Σ(x − μ)² / N). For a sample of n values, s = √(Σ(x − x̄)² / (n − 1)). The calculator shows both so you can pick the one that matches your data.

Example: for 2, 4, 4, 4, 5, 5, 7, 9 the count is 8, the sum is 40, and the mean is 5. The squared deviations add up to 32, so the population variance is 32 / 8 = 4 and σ = 2. The sample variance is 32 / 7 ≈ 4.5714 and s ≈ 2.1381.

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How to use the standard deviation calculator

  1. Type or paste your numbers, separated by commas, spaces, or new lines.
  2. Read the sample and population standard deviation in the result panel.
  3. Check the count, sum, mean, and both variances.
  4. Follow the step-by-step work to see each squared deviation and how the totals were found.

Frequently asked questions

What is standard deviation?

Standard deviation measures how spread out a set of numbers is around its mean. A small standard deviation means the values sit close to the average; a large one means they are spread far apart. It is the square root of the variance.

How do I calculate standard deviation by hand?

Find the mean, subtract the mean from each value, square each difference, and add the squares. Divide that sum by N for a population (or by n − 1 for a sample) to get the variance, then take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the sum of squares is 32, and the population standard deviation is √(32 / 8) = 2.

What is the difference between sample and population standard deviation?

Use population standard deviation (σ) when your list includes every member of the group you care about, and divide by N. Use sample standard deviation (s) when your list is a sample drawn from a larger group, and divide by n − 1. The sample version is slightly larger to correct for estimating the mean from the same data.

Why does sample standard deviation divide by n − 1?

Dividing by n − 1 (Bessel's correction) removes the bias that comes from using the sample mean instead of the true population mean. Without it, a sample would tend to underestimate the real spread.

What is variance and how is it related to standard deviation?

Variance is the average of the squared deviations from the mean. Standard deviation is the square root of variance, which puts the spread back into the same units as the original data. This calculator shows both for population and sample.

How do I enter numbers into the standard deviation calculator?

Type or paste your numbers separated by commas, spaces, semicolons, or new lines. Negative numbers and decimals are fine, so you can paste a column straight from a spreadsheet. Any entry that is not a number is flagged.

Can standard deviation be negative or zero?

Standard deviation is never negative because it is the square root of a sum of squares. It is zero only when every value in the list is the same. A single number has a population standard deviation of 0 and no sample standard deviation, since n − 1 would be 0.

Is my data saved or uploaded?

No. The standard deviation calculator runs entirely in your browser. Your numbers are not sent to a server or stored, and there is no account.

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