About the Standard Deviation Calculator
A free, instant calculator that computes population and sample standard deviation, mean, variance, count, and sum — with full step-by-step working shown from any paste-friendly data input.
What it is
The Standard Deviation Calculator accepts a list of numbers in any paste-friendly format and instantly computes the complete set of descriptive statistics: population standard deviation (σ), sample standard deviation (s), mean (x̄), population variance (σ²), sample variance (s²), count (N), and sum (Σx). It also renders a step-by-step breakdown of the calculation so the method is transparent alongside the result.
Everything runs in your browser — no data is sent to a server, no account is required, and there is no rate limit.
Key features
Both SD methods
Computes population standard deviation (÷ N) and sample standard deviation (÷ N−1) simultaneously, so you don't need to switch modes or run two separate calculations.
Step-by-step working
Shows every stage of the calculation: count, sum, mean, a table of each squared deviation, the sum of squared deviations, and the final division and square root for both methods.
Flexible paste-friendly input
Accepts numbers separated by commas, spaces, new lines, or tabs — or any mix. Paste directly from a spreadsheet column or CSV row without reformatting.
Seven statistics at once
Returns population SD, sample SD, mean, population variance, sample variance, count, and sum in one result panel — no need to run separate tools for each.
Clear error messages
Non-numeric tokens (such as a header row accidentally included in a paste) trigger a precise error message identifying the problem, so you can correct it quickly.
Fully responsive
On wide screens the input and results sit side by side. On mobile they stack vertically. The deviation table scrolls horizontally within its container rather than widening the page.
Population vs Sample standard deviation
Population standard deviation (σ) measures the spread of a complete population. It divides the sum of squared deviations from the mean by N. Use it when your data set contains every member of the group you're studying — for example, the weights of all widgets produced in one batch.
Sample standard deviation (s) estimates the spread of a larger population from a subset of it. It divides by N − 1 (Bessel's correction), which compensates for the bias introduced when a sample is used to estimate a population parameter. Use it when your data is a sample from a larger group — for example, survey responses from 500 people representing millions. Statistical software such as Excel, R, and Python's NumPy all default to sample SD when computing "STDEV" or "std" on a data array.
How the calculation works
The calculation follows four stages. First the mean x̄ is found as the sum of all values divided by N. Then each value's deviation from the mean (xᵢ − x̄) is squared, and those squared deviations are summed to give Σ(xᵢ − x̄)².
For population standard deviation, this sum is divided by N to give the variance σ², and the square root gives σ. For sample standard deviation, the sum is divided by N − 1 to give s², and the square root gives s. The step-by-step section renders every intermediate value — including the full deviation table — so the derivation is checkable against any worked example.
All arithmetic uses double-precision floating-point, matching the precision of spreadsheet functions. For very large or very small numbers the result is displayed in scientific notation to preserve accuracy.
Who it's for
Students
Verify homework answers and check the working step by step. Understand why population and sample SD give different results for the same data set.
Teachers
Generate worked examples instantly during class. Paste a data set from a textbook and project the step-by-step breakdown for students to follow.
Researchers & analysts
Quickly verify standard deviation and variance for a column of data. Paste from a spreadsheet without needing to open statistical software for a sanity check.
Exam candidates
Check answers during practice sessions. The working table lets you pinpoint exactly which squared deviation you may have computed differently.
Completely free
No sign-up, no rate limit, no premium tier. Run as many data sets as you like. If you find it useful, sharing it with a classmate or colleague is all we ask.