Standard Deviation Calculator guide
Paste a list of numbers and get sample and population standard deviation side by side, with variance, mean, count, sum, min, and max. Built for homework, lab data, and quick sanity checks.
What you get from one paste
Drop in numbers separated by commas, spaces, or line breaks. The calculator returns count, sum, mean, minimum, maximum, population standard deviation, sample standard deviation, and both variances. Results show up to 4 decimal places, which is more than any intro stats class asks for.
The box loads with a sample dataset, 10, 12, 23, 23, 16, 23, 21, 16, so you can see every output before typing your own. The walkthrough below uses those same eight numbers, so you can check each step against the screen.
Step by step with the sample data
Step 1, the mean. The sum is 144 and the count is 8, so the mean is 144 / 8 = 18.
Step 2, deviations. Subtract 18 from each value: -8, -6, 5, 5, -2, 5, 3, -2. They always add to zero, which is exactly why you cannot just average them.
Step 3, square and add. The squares are 64, 36, 25, 25, 4, 25, 9, 4. Their total, the sum of squares, is 192.
Step 4, divide. Population variance is 192 / 8 = 24. Sample variance is 192 / 7 = 27.4286.
Step 5, square root. Population SD is the square root of 24 = 4.8990. Sample SD is the square root of 27.4286 = 5.2372. Min is 10, max is 23. If your screen matches these, you are reading the tool correctly.
The two formulas
Population standard deviation: sigma = sqrt( sum of (x - mu)^2 / N ). Sample standard deviation: s = sqrt( sum of (x - x-bar)^2 / (N - 1) ). The only difference is the denominator.
That N - 1 is Bessel's correction. A sample hugs its own mean more tightly than it hugs the true population mean, so dividing by N makes the spread look smaller than it really is. Subtracting one from the denominator fixes the bias in the variance. With 8 values the gap is noticeable, 4.90 versus 5.24. With 1,000 values it almost disappears.
Rule of thumb: if you measured everyone you care about, use population. If you measured some to learn about all, use sample. When a textbook problem does not say, it almost always wants sample.
Matching Excel, Google Sheets, and your calculator
In Excel and Google Sheets, STDEV.S (or plain STDEV) is the sample version and STDEV.P is the population version. VAR.S and VAR.P do the same for variance. On a TI-84, 1-Var Stats shows Sx for sample and a sigma-x for population.
If your answer disagrees with a classmate's, check this first. Nine times out of ten one of you used N and the other used N - 1. The second most common cause is a typo in the data, so compare count and sum first: if those match, the inputs match.
Mistakes to watch for
Rounding the mean too early. If the mean is 17.6667 and you round it to 17.7 before squaring deviations, the final SD drifts. Keep full precision until the end, which the calculator does automatically.
Forgetting the square root. Variance and SD are different numbers in different units. A variance of 24 is not a standard deviation of 24.
Stray characters. A dollar sign, a percent sign, or a header like "Score" pasted from a spreadsheet can break parsing. Paste only the numbers.
Reading SD as a range. An SD of 5 does not mean every value is within 5 of the mean. In the sample data, 10 is 8 below the mean, more than 1.5 standard deviations away.
Where it shows up
IQ tests are scaled to a mean of 100 and a standard deviation of 15, so a score of 130 sits 2 SD above average. Manufacturers use SD to monitor whether parts stay inside tolerance. Investors use it as a basic measure of volatility. Teachers use it to see whether a test separated students or bunched everyone together. Same math every time.
How we calculate: sources
Frequently asked questions
Should I use sample or population standard deviation?
Use population SD when your data is every member of the group you care about, like all 25 students in one class. Use sample SD when your data is a subset used to estimate a larger group, which covers most surveys, experiments, and homework problems.
What is the formula for standard deviation?
Find the mean, subtract it from each value, square the differences, and add them up. Divide by N for population variance or N - 1 for sample variance, then take the square root.
Why divide by N - 1 for a sample?
A sample tends to sit closer to its own mean than to the true population mean, so dividing by N underestimates spread. Dividing by N - 1, called Bessel's correction, removes that bias from the variance.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean. Standard deviation is its square root, which puts it back in the original units. If your data is in inches, SD is in inches and variance is in square inches.
What does a standard deviation value actually mean?
It is the typical distance of a value from the mean. For roughly bell-shaped data, about 68% of values fall within 1 SD of the mean and about 95% within 2 SD.
How do I enter my numbers?
Separate them with commas, spaces, or new lines, in any mix. You can paste a column straight from Excel or Google Sheets.
Is my data kept private?
Everything runs in your browser. Nothing you enter is uploaded to a server or stored by us.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so the smallest possible value is 0, which happens only when every number is identical.
What is the standard deviation of 10, 12, 23, 23, 16, 23, 21, 16?
The mean is 18. Population SD is 4.8990 and sample SD is 5.2372.