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Standard Deviation Calculator

Find the sample or population standard deviation of a list, with variance, mean and standard error.

By Nitin Raturi · Last reviewed

Your data

How to use the Standard Deviation Calculator

Measure how spread out a list of numbers is, as a sample or a whole population.

  1. Paste or type your values.
  2. Choose sample or population.
  3. Read the standard deviation, with the variance, mean and standard error beneath.

Good to know

  • Most survey and experiment data are samples.
  • Your list stays in this browser; it is kept after the # in the link, which is not sent.

Your list stays in this browser.

Values

The numbers whose spread you want to measure.

How to fill it in

  • Separate numbers with commas, spaces or new lines.
  • Paste up to 1,000 values, such as a column from a spreadsheet.

Example

For 10000001, 10000003, 10000002 the sample SD is exactly 1.

1 to 1,000 values, separated by commas, spaces or new lines.

Sample or population

Sample or population

Whether your numbers are the whole group or a sample drawn from a bigger one.

How to fill it in

  • Choose sample (divide by n − 1) when the data stand for a larger group.
  • Choose population (divide by N) when you have every value.

Example

For 2, 4, 4, 4, 5, 5, 7, 9: population σ = 2, sample s = 2.13809.

Sample standard deviation

2.13809

s = 2.13809 for 8 values with mean 5.

Variance (s²)
4.571429
Mean
5
Count
8
Sum of squares
32
Standard error of the mean
0.755929
Coefficient of variation
42.761799%
Range
2 to 9

s = √(Σ(x − mean)² ÷ (n − 1)). Rounded to six decimals for display.

ChangelogHow we test this calculatorEmbed this calculator

How this calculator works

  1. Step 1Meanx̄ = Σx ÷ nFind the centre of the data.
  2. Step 2Squared distancesΣ(x − x̄)²Square how far each value is from the mean.
  3. Step 3Average and root√(… ÷ (n − 1))Divide by n − 1 for a sample, or N for a population.

Standard deviation measures how far values typically sit from their mean. A small value means the data cluster tightly; a large one means they spread out.

The calculator first finds the mean, then the squared distances, so large values that differ only slightly still give exact answers.

Worked example: 2, 4, 4, 4, 5, 5, 7, 9

Standard deviation of eight values
Population
σ = √(Σ(x − μ)² ÷ N)
Sample
s = √(Σ(x − x̄)² ÷ (n − 1))
  • nCount 8
  • x̄Mean 5
  1. Squared distances9 + 1 + 1 + 1 + 0 + 0 + 4 + 1632
  2. Population√(32 ÷ 8)2
  3. Sample√(32 ÷ 7)2.13809

Dividing by n − 1 makes the sample figure slightly larger. It corrects for the sample mean sitting closer to its own data than the true mean would.

Sample or population

  • Sample (n − 1)2.13809The data stand for a larger group.
  • Population (N)2The data are the whole group.

Read the result

Michelson's 100 speed-of-light readings have a mean of 299.8524 and s = 0.079011, in millions of m/s.

These match the values NIST certifies for that dataset. For roughly bell-shaped data, about two-thirds of values lie within one standard deviation of the mean.

Assumptions and limits

  • Exact on hard dataNIST's NumAcc1 gives s = 1 exactly.
  • Two or moreA sample needs at least two values.
  • Up to 1,000 valuesEach within ±10 lakh crore.
  • DisplayRounded to six decimals.

Standard deviation measures spread. Scores, intervals and tests build on it.

To place one value against the mean, use the z-score calculator. The confidence interval calculator gives a margin of error, and the average calculator the mean, median and mode.

Common mistakes

  • Mistake: Using N for a survey sample.

    Instead: Divide by n − 1 when the data are a sample.

  • Mistake: Reporting the variance as the spread.

    Instead: Variance is in squared units; take the square root.

  • Mistake: Mixing units in one list.

    Instead: Convert every value to the same unit first.

Checking the method

Every golden case is re-worked with exact fractions, and two NIST certified datasets are reproduced.

An independent script uses exact fractions and the textbook formula. The NIST StRD values for NumAcc1 and Michelso are matched to 13 significant digits.

Read how we test our calculators, or browse more maths calculators.

Frequently asked questions

What is the difference between sample and population standard deviation?

A sample divides by n − 1; a population by N. For 2, 4, 4, 4, 5, 5, 7, 9: s = 2.13809, σ = 2.

How do I calculate standard deviation?

Find the mean, square each distance from it, add them, divide by n − 1 (or N), and take the square root.

What is variance?

The square of the standard deviation. For the population example above it is 4.

How accurate is this calculator?

It reproduces NIST certified values: NumAcc1 gives s = 1 exactly, and Michelson's data give 0.079011.

What is the standard error of the mean?

The sample standard deviation divided by √n. It estimates how much the sample mean would vary between samples.

Sources and methodology

The formulas are standard time-value-of-money results. We check every figure against golden values worked out independently, by closed form and by month-by-month simulation; see the methodology.

Written and maintained by Nitin Raturi, a software engineer (about the author). Last reviewed 3 Nov 2026; next review due by . Found an error? Tell us.

Changelog

  • : Initial release.