# Standard Deviation Calculator

> Standard deviation calculator: compute sample and population SD, variance, mean, and margin of error from any data set. Free, runs in your browser, no signup.

- **Canonical URL:** https://boringtoolskit.com/standard-deviation-calculator/
- **Category:** Everyday Calculators
- **Evidence level:** authored-methodology
- **Reviewed:** August 2026
- **Last verified:** 2026-10-12
- **Privacy:** Inputs and results stay in this browser unless the page clearly labels an external request.

## How it works

Standard deviation measures the spread of values around the mean. For a sample: s = √(Σ(xᵢ − x̄)² ÷ (n − 1)). For a population: σ = √(Σ(xᵢ − μ)² ÷ n). The standard error of the mean is s/√n, and the 95% margin of error is 1.96 × that.

## Worked example

For the values 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The sample standard deviation is about 2.138, the population standard deviation is 2.000, and the standard error of the mean is about 0.756.

## Assumptions

- Values are entered as a comma- or space-separated numeric list.
- Choose sample when the data is a subset of a larger population; choose population when it represents the whole population.
- At least two values are required.

## Common questions

### What is a standard deviation?

It measures how spread out a set of numbers is around their average. A small standard deviation means the values cluster tightly; a large one means they vary widely. It is the square root of the variance.

### What is the difference between sample and population standard deviation?

Population standard deviation (σ) uses every member of the group and divides by n. Sample standard deviation (s) uses a subset of a larger population and divides by n − 1 to correct a small-sample bias. Choose the one that matches your data.

### How do I calculate standard deviation by hand?

Find the mean, subtract it from each value and square the result, sum those squares, divide by n (population) or n − 1 (sample), and take the square root. The deviation table here shows each of those steps.

### What is the standard error of the mean?

It is the sample standard deviation divided by the square root of the sample size (s/√n). It measures how precisely the sample mean estimates the population mean and gets smaller as the sample gets larger.

### What does a 95% margin of error mean?

It is 1.96 times the standard error. The interval ± margin of error around your sample mean would contain the true population mean in about 95% of repeated samples of the same size.

## Related tools

- /mean-median-mode-calculator/
- /fraction-simplify-calculator/
- /random-number-generator/

## Sources

- [Standard deviation](https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm)
- [Sample vs population standard deviation](https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/a/population-and-sample-standard-deviation-review)

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