Standard Deviation Calculator
Paste a list of numbers to get the population and sample standard deviation, variance, mean, sum, count, min, max, and range - with the working shown.
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About the Standard Deviation Calculator
The arithmetic behind standard deviation is easy. The part people get wrong is picking which one they actually need, population or sample, and most calculators make it worse, they hand you a single number and never say which formula they ran. Grab the population figure when your stats professor wanted the sample one and your answer is quietly off, with nothing on screen to warn you.
This tool refuses to make you guess. Both answers sit side by side, the population figure (σ) next to the sample one (s), each labeled with the exact formula behind it so you can read the right one straight off. Paste your numbers separated by commas, spaces, tabs, or new lines and you also get the mean, count, sum, min, max, range, and both variances, live as you type. You can flip open the working to check every step. Nothing you paste leaves your browser.
Why the fuss over spread at all? Because two datasets can share an identical average and behave nothing alike. Scores of 70, 70, 70, 70 average 70, and so do 40, 60, 80, 100, but the mean can't tell them apart. Standard deviation is the number that can, it measures how far a typical value sits from that average. A small standard deviation means the numbers cluster tightly together, and a large one means they're spread out.
How to use
- Paste your numbers into the box. Commas, spaces, tabs, and new lines all count as separators, so a column copied straight out of a spreadsheet drops in with no cleanup. Anything that isn't a number gets ignored, so a stray label or a currency symbol won't break it.
- Read the summary row. Count, sum, mean, min, max, and range come first, the quick shape of your data before you get to spread.
- Read both spreads. Population variance and σ, then sample variance and s, each tagged with the formula it used so there's never a question which is which.
- Open the working. The "Show the working" link unfolds a table with every value, its distance from the mean, and that distance squared, totaled at the bottom. The by-hand steps, done for you.
- Edit anything. Change a single number and every figure recalculates on the spot. No submit button, no reload.
Population or sample: the one choice that matters
Everything up to the last step is the same for both. Add up the squared distances from the mean, then divide. Population standard deviation divides that total by n, the count of your numbers. Sample divides by n − 1. That's the whole difference, and it decides which number is correct for your situation.
Use the population version (σ) when your data is the entire group and there's nobody left out. Every student in the one class. All twelve months of last year's rent. You're describing exactly what you measured, so no adjustment is needed.
Use the sample version (s) when your numbers are a subset standing in for a larger group you didn't fully measure. You polled 500 of a city's residents. You pulled 30 parts off a line that makes thousands a day. You're estimating a spread you can't see directly, and a plain divide-by-n consistently comes out too small.
The reason is subtle but worth getting. To measure any deviation you first need a mean, and you don't have the true population mean, only your sample's mean, worked out from the same handful of numbers. That sample mean lands right in the middle of your sample by construction, so your values look a hair tighter around it than they would around the real mean sitting somewhere slightly off. Dividing by n − 1 instead of n bumps the answer up just enough to cancel that bias, and statisticians call that Bessel's correction. The n − 1 is the "degrees of freedom", once the mean is fixed only n − 1 of your values are actually free to move, the last one is pinned.
How much does it matter? On 8 numbers, dividing by 7 instead of 8 shifts the result noticeably. Stretch that out to a thousand numbers and the swap from 1000 to 999 is almost invisible. The choice matters most on small datasets, which is exactly when it's tempting to eyeball it. If you can't name a bigger group your numbers stand in for, you have a population, use σ. If you can, it's a sample, use s, and in most coursework and research the sample version is the one they're after.
Check the working, don't just trust the number
For a homework answer or a figure headed into a report, you want to check the number, not take it on faith. The "Show the working" panel exists for exactly that. Open it and you get the full by-hand calculation laid out, every value, how far it sits from the mean, and that gap squared, with the squared column summed at the bottom.
It helps in three real ways. If you're learning the formula, you watch the recipe run on your own numbers instead of a textbook's. If your answer looks wrong, the table shows you where, usually a value you fat-fingered on the paste. And if you're checking arithmetic you did by hand, you've got a second opinion sitting right next to yours.
Watch the middle column and you'll spot something. Add up the raw deviations, the plain distances from the mean before squaring, and they always total zero. That isn't a coincidence, the mean is the balance point, so the gaps above it exactly cancel the gaps below. Which is precisely why the formula squares them first. Squaring kills the signs so the spread doesn't vanish, and it weighs big gaps harder than small ones, so a couple of far-out values push the result the way your gut says they should. You can watch it happen in the panel, column by column.
Frequently asked questions
Why are the sample figures showing a dash?
Sample standard deviation needs at least two numbers. With a single value, n − 1 works out to zero, and dividing by zero is undefined, so there's no honest answer to show. Add one more number and the sample variance and s appear. The population figures still compute with one value, they just come out as zero, since a lone number has no spread from itself.
Can I paste a whole column from Excel or Google Sheets?
Yes. Copy the column and drop it straight in, the line breaks between cells read as separators. Commas, spaces, and tabs work too, so almost any messy paste parses. Negative numbers and decimals are fine. Non-numeric junk like a header word or a dollar sign gets skipped rather than counted, so you rarely need to tidy anything up first.
A thousands separator is the one thing that needed a rule, because the comma grouping 12,500 is the same character separating 4, 8, 15. It counts as grouping whenever your values are already broken apart some other way, a new line or a space or a tab, so a column of salaries keeps its rows. The line under the box says how many values came through, and that number is n. Both formulas divide by it, so a count that is off by two moves σ and s together and neither one looks wrong sitting there on its own.
What's the difference between variance and standard deviation?
Variance is the average of the squared distances from the mean. Standard deviation is its square root, nothing more. The reason people quote the standard deviation and not the variance is units. Square your dollars or centimeters and you get square dollars or square centimeters, which mean nothing physical, and the square root pulls the number back into the units you started with. Variance still matters for building models, where it combines more cleanly, but for reading a single dataset the standard deviation is the one that makes sense to a human.
How much should I round the answer?
Keep a couple more decimal places than your original data, then round only at the very end. Rounding partway through, especially the mean, quietly drags error into every later step. If your numbers are whole, two or three decimals on the standard deviation is plenty for most uses. For anything you'll run more math on, hang onto the extra digits until you're finished.
Can it give me a z-score or the median?
Not here. This page sticks to spread, so it stops at the mean, the two variances, and the two standard deviations. For median and mode, the sibling average calculator covers those. A z-score you can work out by hand once you have these numbers: subtract the mean from your value, then divide by the standard deviation.
Is it safe to paste salaries or medical readings?
Yes. Every calculation runs on your own device in plain JavaScript, so nothing is uploaded, stored, or sent anywhere. Once the page has loaded it keeps working with the connection switched off entirely, which means sensitive lists like payroll, grades, or lab measurements never leave your computer.