Standard Deviation Calculator

Measure how far values spread from the mean, with the sample or population denominator shown clearly.

Drag & drop or upload an image or PDF

Enter data, such as population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9

Enter numbers separated by commas, spaces, or line breaks. State whether the list is a sample or the full population when that distinction matters.

Try an example

Choose the denominator before doing the arithmetic

Standard deviation describes spread around the mean in the same units as the original data. Values packed close to the mean produce a small result. Values spread across a wider range produce a larger one.

The population formula applies when the list contains every value in the group of interest. The sample formula applies when the list is being used to estimate variability in a larger population. That choice changes the denominator, so it belongs at the start of the calculation.

How to use the standard deviation calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Population and sample standard deviation

Both formulas square distances from the mean and then take a square root. The sample formula uses n minus 1 because the population mean was estimated from the sample.

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

A reliable way to work through it

Identify sample or population

Use the population formula for the whole group and the sample formula when the data estimate a larger group.

Build the squared deviations

Find the mean, subtract it from each value, square each deviation, and add those squares.

Divide, then take the square root

Divide by N or n minus 1 to get variance, then take the principal square root to return to the original units.

Worked example

Find the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9

μ = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5

Find the population mean.

\sum (x_{i} - μ)^{2} = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32

Square every deviation from 5 and add the results.

σ^{2} = \frac{32}{8} = 4

Divide by N because the list is being treated as the full population.

σ = 4 = 2

Take the square root of the population variance.

The population variance is 4 and the population standard deviation is 2. If the same values were a sample, the standard deviation would be √(32/7), about 2.138.

Common mistakes to check

Using the wrong denominator

Population standard deviation divides by N. Sample standard deviation divides by n minus 1.

Averaging signed deviations

Deviations from the mean sum to zero, so they must be squared before they are averaged.

Stopping at variance

Variance uses squared units. Standard deviation is the square root of variance and returns to the data's units.

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Questions students ask

Should I use sample or population standard deviation?

Use population standard deviation when your data includes the entire group you want to describe. Use sample standard deviation when the data is a subset used to estimate a larger group.

Why does sample standard deviation use n minus 1?

The sample mean is estimated from the same data. Dividing the squared-deviation sum by n minus 1 corrects the downward bias in the sample variance estimator.

Can standard deviation be negative?

No. Squared deviations are nonnegative, and the principal square root of their average is also nonnegative.

Is standard deviation the same as standard error?

No. Standard deviation describes variation among observations. Standard error describes variation in an estimate across repeated samples.