Standard Deviation Calculator

Calculate standard deviation, variance, and mean for a sample or population data set.

Numbers (comma or newline separated)

Data Type

About the Std Deviation Calculator

Standard deviation measures how spread out a set of numbers is from its average — a small standard deviation means values cluster tightly around the mean, while a large one means they're scattered widely. This calculator computes standard deviation, variance, and mean, with support for both sample and population data.

How It Works

The calculator finds the mean of the data set, then computes each value's squared deviation from that mean, and averages those squared deviations to get variance — dividing by n for a full population, or by n-1 for a sample (Bessel's correction, which produces a better estimate of population variance from a subset of data). Standard deviation is the square root of variance.

Variance = Σ(x − mean)² / (n or n−1); Standard Deviation = √Variance.

Worked example

Scenario: The sample 12, 15, 11, 18, 14, 10.

Mean: 80 ÷ 6 = 13.333.
Squared deviations sum to 43.333.
Sample variance: 43.333 ÷ (6 − 1) = 8.667.
Standard deviation: √8.667 = 2.944.

Result: A sample standard deviation of about 2.944.

Sample vs. population standard deviation

Population standard deviation describes an entire population and divides the variance calculation by N. Sample standard deviation estimates population spread from a sample and uses N − 1 (Bessel’s correction). Choose the setting that matches what your data represents; using the wrong one changes the result.

Tips and common mistakes

Use 'sample' when your data is a subset of a larger group; use 'population' when you have every value.
The population SD for the same data would be √(43.333 ÷ 6) = 2.687.
Roughly 68% of normally distributed values fall within one standard deviation of the mean.

Assumptions & Limitations

'Sample' mode divides by n−1 (Bessel's correction), appropriate when your data is a sample drawn from a larger population.
'Population' mode divides by n, appropriate only when your data represents the entire population being studied, not a subset.

Requires at least two numbers to compute a meaningful standard deviation.

Frequently Asked Questions

Should I use sample or population standard deviation?

Use 'sample' whenever your data is a subset drawn from a larger group you're trying to draw conclusions about, which is the overwhelming majority of real-world statistics; use 'population' only when your data set truly represents every member of the group you care about.

How is variance different from standard deviation?

Variance is the average of squared deviations from the mean, while standard deviation is the square root of variance — standard deviation is more commonly reported because it is in the same units as the original data, unlike variance.

Why divide by n − 1 for a sample?

It corrects the tendency of a sample to underestimate the spread of the whole population (Bessel's correction).

What does a large standard deviation mean?

That values are widely spread around the mean; a small one means they cluster closely.

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