Why divide by n − 1?

Because the sample mean is calculated from the same data, deviations from it are slightly too small. Dividing by n − 1 (Bessel's correction) corrects that bias.

What does a standard deviation of 0 mean?

All values are identical — there is no spread at all.

Can standard deviation be negative?

No. It is the square root of an average of squared numbers, so it is always zero or positive.

How is variance related to standard deviation?

Variance is the standard deviation squared. For 2, 4, 4, 4, 5, 5, 7, 9 the population variance is 4 and σ is 2.

Which does Excel use?

STDEV.S (or STDEV) gives the sample SD; STDEV.P gives the population SD.

Math · Free tool

Standard Deviation Calculator

Sample and population standard deviation, variance and mean for any list of numbers.

Standard deviation measures how spread out numbers are around their mean. A small value means the data cluster tightly; a large one means they vary widely. Two classes can have the same average mark of 70 while one has every student between 65 and 75 and the other ranges from 30 to 100 — the standard deviation is what tells them apart.

There are two versions, and choosing the right one matters. Use the population standard deviation (σ, divide by n) when your data include every member of the group you care about. Use the sample standard deviation (s, divide by n − 1) when your data are a sample used to estimate a larger population — the n − 1, known as Bessel's correction, removes the bias of estimating spread from a sample.

Paste your numbers separated by commas, spaces or new lines. The calculator shows both versions, the variance, the mean and the sum of squared deviations, so you can check homework or a report step by step.

By DailyCalc Editorial Team · Updated 2026-10-03 · 6-min read · 3 sources cited

● Std dev calculator

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Standard deviation formulas

σ = √( Σ(x − μ)² ÷ n ); s = √( Σ(x − x̄)² ÷ (n − 1) )

First find the mean. Subtract it from every value, square each difference so negatives do not cancel positives, and add them up — the sum of squares. Divide by n for the population variance or by n − 1 for the sample variance. The standard deviation is the square root of the variance, which brings the answer back to the original units.

x: Each value in the data set.
μ or x̄: Mean of the values (population or sample).
n: Number of values.
σ / s: Population / sample standard deviation.

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

The classic textbook data set:

Mean = (2+4+4+4+5+5+7+9) ÷ 8 = 40 ÷ 8 = 5.
Deviations squared: 9, 1, 1, 1, 0, 0, 4, 16; sum of squares = 32.
Population variance = 32 ÷ 8 = 4, so σ = √4 = 2.
Sample variance = 32 ÷ 7 ≈ 4.5714, so s ≈ 2.1381.
The sample SD is larger because dividing by n − 1 corrects for estimating the mean from the same data.

Result: Mean 5 · population σ = 2 · sample s ≈ 2.138 · sum of squares 32.

How to use this calculator

Paste, calculate, choose the right version.

Step 1: Enter your data

Separate numbers with commas, spaces or line breaks. You can paste a column from a spreadsheet.

Step 2: Calculate

The result shows s (sample) as the headline, with σ (population), mean, variance and count.

Step 3: Pick σ or s

Whole group measured? Use σ. A sample standing in for a bigger group? Use s.

Where standard deviation is used

Common real-world questions:

Exam results

Quiz scores 72, 85, 91, 68 and 77 have a mean of 78.6 and a sample SD of about 9.40 — most scores sit within roughly 9 marks of the mean.

Quality control

Manufacturers track the SD of part sizes; a rising SD signals a process drifting out of control.

Investment risk

The SD of yearly returns is a standard measure of volatility.

Science and lab work

Report the mean ± SD of repeated measurements to show precision.

Comparing consistency

Two delivery services with the same average time can differ hugely in reliability; the SD shows which is more predictable.

Pro tips

Read the number correctly:

For roughly bell-shaped data, about 68% of values fall within one SD of the mean and about 95% within two.
Spreadsheet STDEV.S matches s and STDEV.P matches σ.
Compare spread across different scales with the coefficient of variation: SD ÷ mean.
A single extreme value can inflate the SD a lot; check for outliers first.

Common mistakes

Errors that give the wrong spread:

Using σ for a sample

Dividing by n on sample data underestimates the population spread, especially for small samples.

Forgetting to square

Adding raw deviations always gives zero, because positive and negative differences cancel.

Reporting variance as SD

Variance is in squared units; take the square root to get the standard deviation.

Ignoring skewed data

For strongly skewed data, the median and interquartile range can describe spread better than the SD.

Sources & references

The formula, thresholds and guidance on this page were checked against these primary sources. Written and reviewed by DailyCalc Editorial Team; last updated 2026-10-03. Spotted an error? Tell us.

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