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Standard Deviation Calculator

The standard deviation, variance and mean of a list of numbers — sample or population — with each step laid out so you can check it.

Enter or paste a list of numbers to see their standard deviation.

Separate them with commas, spaces or new lines — or paste a column from a spreadsheet. Leave out thousands separators: 1200, not 1,200.

Your data is a

A sample is part of a larger group — 30 people surveyed out of a town. Most data is a sample. Divides by n − 1.

Exact calculation. Given your inputs, this is the answer — no estimation involved.

How this calculator works

Standard deviation measures how spread out numbers are around their average. A small one means most values sit close to the mean; a large one means they are scattered.

It takes four steps. Find the mean. Measure each value’s distance from it. Square those distances and average them, which gives the variance. Then take the square root, which brings the answer back to the original units.

The one choice is sample or population. If your numbers are the whole group you care about — every score in one class — divide by n, the number of values. If they are a sample from a larger group — 30 people surveyed in a town — divide by n − 1. Each value is measured against the sample’s own mean rather than the true one, which makes a sample understate the spread slightly, and the smaller divisor corrects for it. Most real data is a sample, so that is the default.

The arithmetic is done the careful way: the mean first, then each distance from it. A common shortcut formula subtracts two large, nearly equal numbers, and for values such as 1000000.1, 1000000.2 and 1000000.3 it can lose every digit of the answer.

The formula

Exactly what happens to your numbers, step by step.

  1. Mean

    x̄ = Σx ÷ n

  2. Sum of squares

    SS = Σ(x − x̄)²

  3. Sample variance

    s² = SS ÷ (n − 1)

    Bessel’s correction: dividing by n − 1 removes a sample’s bias.

  4. Population variance

    σ² = SS ÷ n

  5. Standard deviation

    s = √s²    σ = √σ²

A worked example

Five quiz scores out of 10 — 6, 8, 7, 10 and 9 — treated as a sample.

What you enter

Numbers
6, 8, 7, 10, 9
Data is a
Sample

Step by step

Mean
40 ÷ 5 = 8
Distances from the mean
−2, 0, −1, 2, 1
Squared
4, 0, 1, 4, 1
Sum of squares
10
Variance
10 ÷ (5 − 1) = 2.5

Sample standard deviation = √2.5 = 1.581139

The scores sit about 1.6 points either side of the average of 8. Treated as the whole population instead, the divisor is 5 and the standard deviation is 1.414214.

Assumptions

Every result here rests on these. Change your inputs and the result changes with them.

  • Values can be separated by commas, spaces, tabs, semicolons or new lines, so thousands separators cannot be used: 1200, not 1,200.
  • Up to 10,000 values. Negative numbers and decimals are fine.
  • Results are shown to up to six decimal places.

What this cannot tell you

  • It describes the spread of the numbers you enter. It does not test whether they follow a bell curve, or flag outliers.
  • For data in a frequency table, enter each value as many times as it occurs.

Questions people ask

Should I use the sample or population standard deviation?

Use population only when your data is every member of the group you are describing — all 24 students in a class, and you care about that class alone. If the data stands in for something larger, such as a survey, a batch of test measurements or a season of results, use sample. When in doubt, sample is the safer choice.

What is the difference between variance and standard deviation?

Variance is the average squared distance from the mean, and standard deviation is its square root. Variance is in squared units — square dollars, square inches — which are hard to picture, so the standard deviation, in the original units, is the one usually reported.

Can a standard deviation be negative?

No. It is the square root of an average of squares, so the smallest it can be is 0, and it is 0 only when every value is the same.

How do I paste data from a spreadsheet?

Copy the column and paste it into the box; each cell lands on its own line. Remove commas used as thousands separators first, or 1,200 will be read as two values, 1 and 200.

Related calculators

Sources and review

Sources

Methodology version 1.0.0 · Last reviewed

Cite this calculator

For coursework, a resource list or your own site. How every calculator is built and checked is set out on the methodology page.

Citation

Chalk Calculator. (2026, October 1). Standard Deviation Calculator (Methodology version 1.0.0). https://chalkcalculator.com/standard-deviation-calculator/

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