Standard Deviation Calculator

Find the mean, variance and standard deviation of a data set, for a sample or a whole population.

Try an example:

What standard deviation measures

Standard deviation measures how spread out numbers are around their mean. A small value means the data sit close to the average; a large one means they're widely scattered. It's in the same units as the data, which makes it easier to interpret than the variance.

How to use the standard deviation calculator

Type or paste your problem into the box above — for example Sample standard deviation of 4, 8, 6, 5, 3, 7. Use ^ for powers, / for fractions and sqrt() for square roots, or describe it in words.
Press Enter. The solver works through the problem with every step shown; switch Steps off if you only want the final answer.
Ask a follow-up in the same chat — "why did you do that?" or "now check it" — and it answers with the earlier working in context.

Sample or population?

Use the population formula when your data include every member of the group you care about, and the sample formula when the data are a sample used to estimate a larger population. The sample version divides by $n - 1$ instead of $n$ — Bessel's correction — which makes the estimate slightly larger.

σ = \frac{\sum ( x _{i} - μ ) ^{2}}{N} s = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

Step by step

Find the mean: add the values and divide by how many there are.
Subtract the mean from each value and square the result.
Add the squared deviations and divide by $n$ for a population or $n - 1$ for a sample to get the variance, then take the square root.

Worked example: sample standard deviation of 4, 8, 6, 5, 3, 7

Find the mean $x ˉ = \frac{4 + 8 + 6 + 5 + 3 + 7}{6} = \frac{33}{6} = 5.5$
Square each deviationThe deviations are $- 1.5, 2.5, 0.5, - 0.5, - 2.5, 1.5$ ; their squares are $2.25, 6.25, 0.25, 0.25, 6.25, 2.25$ .
Add them up $\sum (x_{i} - x ˉ)^{2} = 17.5$
Divide by $n - 1$ $s^{2} = \frac{17.5}{5} = 3.5$
Take the square root $s = 3.5 \approx 1.87$

Common mistakes to avoid

Using the wrong divisor

Dividing by $n$ when the data are a sample underestimates the spread. Check which version the question wants.

Forgetting to square

Deviations from the mean always add up to zero. Squaring them stops the positives and negatives cancelling out.

Stopping at the variance

The variance is in squared units. Take the square root to get the standard deviation.

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Questions students ask

What's the difference between variance and standard deviation?

Variance is the average squared deviation from the mean; standard deviation is its square root. Standard deviation is in the original units, so it's usually the one you report.

Should I use sample or population standard deviation?

Population if you have data for every member of the group; sample if your data are a subset used to draw conclusions about a bigger group. Most real-world data are samples.

How do I enter my data?

Type the values separated by commas and say whether you want the sample or population version. If you don't say, the solver states which one it used.

Can it calculate other statistics?

Yes — mean, median, mode, range, quartiles, z-scores and more. Ask in the same chat and it reuses the data you already entered.