Standard Deviation Calculator
This free standard deviation calculator measures how spread out your numbers are — it computes both the population (σ) and sample (s) standard deviation, plus variance, mean, min, max and range, instantly in your browser. Just paste your data and read off the results.
Enter your numbers
Your results
Which to use: report σ when your data is the whole population; report s when your data is a sample used to estimate a larger population. Results update instantly as you type — all calculations happen in your browser and your numbers are never sent anywhere.
How to use this calculator
Type or paste your numbers into the box above. Commas, spaces, semicolons and line breaks all work as separators, and decimals or negative numbers are fine. You need at least two valid numbers — the standard deviation of a single value is not defined.
The results update automatically as you type, so there is nothing to submit. The headline panel shows the two standard deviations — population σ and sample s — and the table below adds count, mean, both variances, minimum, maximum and range.
How it works
Standard deviation measures the typical distance of values from their mean. First the calculator finds the mean: add all the values together (Σx) and divide by how many there are (n). Then it averages the squared differences from the mean — that average is the variance — and takes its square root.
Population standard deviation (σ) — use when your data is the entire group you care about:
σ = √(Σ(x − μ)² / N)
Sample standard deviation (s) — use when your data is a subset standing in for a larger group; the n−1 (Bessel's correction) keeps the estimate unbiased:
s = √(Σ(x − x̄)² / (n − 1))
Worked check with the default data (2, 4, 4, 4, 5, 5, 7, 9): mean = 40 ÷ 8 = 5; squared differences sum to 32; population variance = 32 ÷ 8 = 4, so σ = 2; sample variance = 32 ÷ 7 ≈ 4.5714, so s ≈ 2.138. Assumptions: values are ordinary decimal numbers with equal weight; results are rounded to at most 4 decimal places. These are exact statistics of your data — not estimates.
Frequently asked questions
When should I use population vs. sample standard deviation?
Use the population formula (σ, dividing by N) when your numbers cover the entire group you care about — for example, the ages of every employee in a department. Use the sample formula (s, dividing by n−1) when your numbers are a subset and you want to estimate the spread of a larger group — for example, the test scores of 30 students standing in for a whole school. When in doubt, sample standard deviation is the safer default.
Why do you divide by n−1 for the sample standard deviation?
This is Bessel's correction. A sample's variance measured around its own sample mean tends to come out smaller than the true population variance, so dividing by n−1 instead of n removes that downward bias and gives a better estimate of the population's spread.
What does a standard deviation of 2 actually mean?
It means a typical value sits about 2 units away from the mean. In the default example on this page, the mean is 5 and the population standard deviation is 2, so values mostly fall within roughly 3 to 7. A bigger standard deviation means the numbers are more spread out.
Can the standard deviation be negative?
No. The standard deviation is the square root of an average of squared differences, and squared values can never be negative — so the result is always zero or greater. A standard deviation of zero means every value in the data set is identical.
Can I enter decimals or negative numbers?
Yes. This calculator accepts decimals and negative numbers, separated by commas, spaces, semicolons or line breaks. You need at least two valid numbers — standard deviation of a single value is not defined.