MultiConvers

Standard Deviation Calculator

Measure how spread out a list is around its mean. You pick sample (divide by n − 1) or population (divide by n); the page does not infer which list you meant.

How to use

  1. Choose Sample when the numbers are a subset of a larger group, or Population when the list is every value you care about.
  2. Paste the list. Spaces, commas or new lines are fine.
  3. Read the standard deviation as the headline, then variance, mean and count in the details.
  4. A single number has no sample SD. Switch to population if you truly have the whole group — that result is 0 when every value is identical.

How it works

Two lists can share a mean of 10 and still feel nothing alike. 9, 10, 11 is tight. 2, 10, 18 is wide. Standard deviation is the usual number for that difference: the square root of the average squared distance from the mean. Variance is that average of squares; taking the square root puts the result back on the same scale as the data, so an SD of 1 on 1, 2, 3 is “about one unit of typical scatter,” not one squared unit.

This calculator asks you to name the denominator. Sample standard deviation divides the sum of squared deviations by n − 1 (Bessel’s correction). The sample mean was fitted to the same numbers, so dividing by n would systematically understate spread if you are estimating a larger population. Population standard deviation divides by n because every member is already in the list — there is nothing left to estimate. For 1, 2, 3 the sample SD is 1 and the population SD is √(2/3) ≈ 0.8165. The page will not silently swap those two.

One observation cannot support a sample SD: n − 1 would be zero. The sample method returns no result. The population method on a single number is 0, which is honest (there is no scatter in a list of one) but rarely what a homework sheet meant if the sheet said “sample.” Identical values in a longer list also give population SD 0; sample SD is 0 as well once n ≥ 2 and every entry matches.

This is not the standard error of the mean. Standard error divides s by √n and answers a different question — how much the mean itself would jitter across repeated samples. Confidence-interval and sample-size pages use that idea. This page stops at the spread of the list you pasted.

If you want mean, median, mode and range without choosing a method, use the four-summary calculator. If you want both sample and population spread plus min, max and sum from one paste, use the statistics calculator. This page exists for the moment when the only decision that matters is n − 1 versus n.

Formula

s = √[ Σ(x − x̄)² / (n − 1) ]   (sample)
σ = √[ Σ(x − μ)² / n ]   (population)

Worked examples

  • Sample SD of 1, 2, 3

    1, 2, 3 with sample method → 1

  • Population SD of 1, 2, 3

    1, 2, 3 with population method → √(2/3) ≈ 0.8165

Useful notes

  • This is not the standard error of the mean (that divides s by √n).
  • The average calculator reports only the mean of a list.
  • Sample SD needs at least two numbers; population SD of a single value is 0.

FAQ

Should I use sample or population?

Use sample when the list is a subset of a larger group you care about. Use population when the list is every value in the group.

Why divide by n − 1?

A sample mean is fitted to the same data, so dividing by n would understate spread. n − 1 corrects that.

What if I enter only one number?

Population SD is 0. Sample SD is undefined — there is no n − 1 spread.

Does the page also show variance?

Yes. Variance is the square of the chosen standard deviation and appears in the details with the mean and the count.

Is a larger SD always “worse”?

Only if you wanted the values bunched together. A wide process, a diverse class or a volatile price series is supposed to have a large SD. Compare SD to the size of the mean, not to a universal target.

Related tools

See all Statistics tools.