📊Standard Deviation Calculator
Paste your numbers and instantly get population and sample standard deviation, mean, variance, count, and sum — with the full step-by-step working shown.
Free Standard Deviation Calculator Online
Need to find the standard deviation of a data set fast? Our free Standard Deviation Calculator does it the moment you paste your numbers. As a fast and reliable standard deviation calculator, it handles both population and sample standard deviation, and returns the full picture — mean, variance, count, and sum — so you get every related statistic in one place.
But it does more than a single number. See the complete step-by-step working: how the mean is found, how each deviation is squared and summed, and how the result is divided and rooted — so you understand exactly how the answer is reached, not just what it is. Paste numbers separated by commas, spaces, or new lines and it just works. Whether you're checking homework, analysing results, or learning statistics, the method is laid out clearly.
Simply paste your data to see the results instantly. No sign-up, no downloads, no limits — just clear statistics whenever you need them.
Population vs Sample Standard Deviation
Population standard deviation (σ) divides the sum of squared deviations by N — the total count of values. Use this when your data set is the entire group you care about (e.g. the heights of every student in one specific class).
Sample standard deviation (s) divides by N − 1 (Bessel's correction). Use this when your data is a sample drawn from a larger population and you want to estimate the population's standard deviation. The − 1 correction makes the estimate unbiased. When in doubt for inferential statistics, sample SD is the common choice.
See the FAQ for a deeper explanation, or the About page for details on how the working is computed.
About the Standard Deviation Calculator
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.
Frequently Asked Questions
How does the standard deviation calculator work?
Paste or type your numbers into the data field and the tool instantly calculates the standard deviation, along with the mean, variance (population and sample), count, and sum — no submit button or page refresh needed. The results and working update live as you type or paste.
Is this calculator free to use?
Yes, it's completely free with no sign-up, no downloads, and no limits. Analyse as many data sets as you like. Everything runs entirely in your browser and your data is never sent to a server.
What's the difference between population and sample standard deviation?
Population standard deviation (σ) divides the sum of squared deviations by N — the total number of values. It describes the spread of the entire group you've measured.
Sample standard deviation (s) divides by N − 1 (Bessel's correction). It estimates the spread of the larger population from which your data was sampled. The − 1 correction compensates for the fact that a sample tends to underestimate the true population spread. The tool calculates and displays both simultaneously.
Which one should I use — population or sample standard deviation?
Use population standard deviation when your data is the entire group you care about — for example, the test scores of every student in one particular class, not a subset.
Use sample standard deviation when your data is a sample drawn from a larger population and you're trying to estimate the population's spread — for example, a survey of 200 people out of millions. When in doubt, or when working with inferential statistics, the sample version (÷ N − 1) is the standard choice used by spreadsheets and statistics software.
How should I enter my data?
Enter your numbers separated by commas, spaces, new lines, or tabs — or any mix of these. You can paste a column directly from a spreadsheet (each value on its own line), a CSV row, or type values by hand. Negative numbers and decimals are fully supported. Any non-numeric token (such as a header row or a label) will trigger a clear error message so you can remove it.
Does it show the steps?
Yes. The Step-by-Step Working section beneath the results walks through the full calculation: the count, the sum, the mean, a table showing each value's deviation from the mean and its squared value, the sum of those squared deviations, and finally how that total is divided and square-rooted for both the population and sample versions. Every intermediate value is displayed so you can follow or verify each step.
What else does it calculate?
Alongside the two standard deviations, the calculator reports the mean (x̄), population variance (σ²), sample variance (s²), the count of values (N), and the sum of all values (Σx). All seven statistics appear simultaneously and update instantly whenever your input changes.
Is my data stored anywhere?
No. Your data is processed entirely in your browser using JavaScript. Nothing is sent to a server, stored in a database, or logged. Close the tab and everything is gone — there is no account, no history, and nothing persisted beyond the current session.
Does it work on mobile devices?
Yes, the tool is fully responsive and works on smartphones, tablets, and desktops. On narrow screens the input and results stack vertically so everything stays accessible without horizontal scrolling. The step-by-step working table scrolls horizontally within its container if it is wider than the screen, so the page itself never overflows.
How accurate are the results?
The calculator uses the standard statistical formulas applied in double-precision floating-point arithmetic — the same precision used by spreadsheets and scientific calculators. Results match those produced by Excel's STDEV (sample) and STDEVP (population) functions. Decimals, negative numbers, and very large or very small values are all handled correctly.