Standard Deviation Calculator

Calculate population or sample standard deviation, variance, mean, count and sum from a list of numbers.

Separate values with commas, spaces, or new lines.

Your statistics will appear here.

About the Standard Deviation Calculator

Two data sets can share the same mean yet have very different levels of spread. This calculator measures that dispersion with population or sample standard deviation and also exposes the variance behind it.

The correct formula depends on whether your data represent the entire population of interest or a sample used to estimate a larger population. That choice changes the denominator.

How to Use the Standard Deviation Calculator

  1. Enter the numerical observations that belong to the same data set.
  2. Choose population or sample standard deviation deliberately.
  3. Calculate the mean, variance and standard deviation.
  4. Interpret the standard deviation in the original measurement units and alongside the shape of the data.

How the Calculation Works

For a population, variance = Σ(x − μ)² ÷ N and population standard deviation σ is the square root of that variance.

For a sample, variance commonly uses Σ(x − x̄)² ÷ (n − 1), and sample standard deviation s is the square root. Subtracting the mean centers each observation before the squared deviations are averaged.

Worked example

Standard Deviation Calculator Worked Example

Population values 2, 4, 4, 4, 5, 5, 7, 9

The mean is 5. The squared deviations sum to 32. Population variance = 32 ÷ 8 = 4, so population standard deviation = √4 = 2.

Result guide

Understanding Your Standard Deviation Calculator Result

Standard deviation uses the same units as the data, which makes it easier to interpret than variance. Its meaning still depends on context; there is no universal value that counts as “high.”

Before you rely on the answer

Assumptions and Limitations

  • Population or sample treatment is selected correctly.
  • Observations are numerical and belong to the intended data set.
  • Standard deviation alone does not describe skewness, clusters or outliers.

Practical tip

Using This Result in Practice

Plot or inspect the raw values as well as calculating standard deviation. Two data sets can share the same mean and standard deviation while having very different distributions.

Related calculations

Useful Calculations to Continue With

If you need the squared-dispersion measure itself, use the Variance Calculator.

To compare variability across data sets with different scales, use the Coefficient of Variation Calculator.

Authoritative references

Useful Official Sources

For standards, definitions or real-world guidance related to this calculation, these primary sources may be useful:

Common questions

Standard Deviation Calculator FAQ

What is the difference between sample and population standard deviation?

Population standard deviation divides by N. Sample standard deviation commonly divides by n−1 to estimate population variability from sample data.

Can standard deviation be negative?

No. It is the square root of a non-negative variance, so standard deviation cannot be negative.

Does a low standard deviation mean the data are good?

Not automatically. It only means the values are relatively close to their mean; whether that is desirable depends on the context.