Standard Deviation Calculator
Calculate population and sample standard deviation
Standard Deviation Calculator tool
Results
Standard Deviation
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Variance
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Mean
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Count
0
Sum
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Sum of Squared Differences
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Step-by-step:
Standard Deviation Calculator: key facts
- What it does
- Calculate population and sample standard deviation
- Category
- Math Calculators
- Cost
- Free, with no account, sign-up, or install.
- Your data
- Runs entirely in your browser — the files and text you enter are never uploaded to a server.
- Last reviewed
- . Report an incorrect result.
What this tool is for
Standard deviation measures how spread out a set of numbers is — whether they cluster tightly around the average or scatter widely. This calculator computes it from a list of values and, crucially, lets you choose between the population and sample formulas, which give different answers and are easy to confuse.
Students check statistics homework with it; analysts use it to quantify variability in data; anyone summarizing measurements uses it to describe consistency. Beyond the standard deviation itself, it shows the variance, mean, count, sum, and the sum of squared differences, with the steps laid out so you can follow exactly how the result was built.
How to use the Standard Deviation Calculator
- Paste or type your numbers into the box, separated by commas, spaces, or new lines.
- Choose whether your data is a full population or a sample.
- Read the standard deviation and variance.
- Review the supporting figures — mean, count, sum, and sum of squared differences — and the step-by-step breakdown.
Population and sample are different divisors
Standard deviation measures how far values typically sit from their mean. The calculation finds the mean, squares each deviation from it, averages those squares to get the variance, and takes the square root to return to the original units.
The mode selector changes one thing: what that averaging divides by. Population mode divides by n, and is correct when your data is the entire group you care about. Sample mode divides by n − 1, and is correct when your data is a sample from a larger population you are trying to describe.
That n − 1 is Bessel's correction, and it exists because a sample's own mean sits closer to its own values than the true population mean does. Dividing by n would systematically underestimate the real spread, so the smaller divisor compensates. The difference matters most on small samples and becomes negligible as n grows.
- The default set — 2, 4, 4, 4, 5, 5, 7, 9 — has a mean of 5 and a population standard deviation of exactly 2.00, from a variance of 4.00.
- Treated as a sample instead, the same eight values give a variance of 4.57 and a standard deviation of 2.14.
- That is a 7% difference from the divisor alone, on identical data — which is why the mode selector is not a cosmetic choice.
What the Standard Deviation Calculator gets right
- It offers both population and sample standard deviation, dividing by n or by n−1 respectively, so you get the statistically correct figure for your situation.
- It shows the full working — mean, variance, and the sum of squared differences — turning a single number into a transparent calculation you can verify or learn from.
- It accepts numbers separated however you have them, by commas, spaces, or lines.
- Everything is computed in your browser, so your data is never uploaded.
Interpreting the spread
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. That empirical rule is what makes the figure intuitive, and it is also the basis of the two-sigma convention behind most reported confidence intervals.
The rule depends on the distribution being approximately normal, and plenty of real data is not. Income, city sizes, and web page traffic are heavily skewed, and for those a standard deviation can exceed the mean and describe the data poorly. Plotting the values is worth more than any single summary statistic.
Two related points. Standard deviation shares the units of the original data, which makes it interpretable, whereas variance is in squared units and mainly useful as an intermediate step. And comparing spread between datasets of different scale calls for the coefficient of variation — standard deviation divided by the mean — rather than the raw figure.
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population standard deviation divides by the number of values (n) and is used when your data covers the entire group. Sample standard deviation divides by n−1 and is used when your data is a sample meant to represent a larger population. The sample version is slightly larger and corrects for sampling bias.
Which one should I use?
Use population standard deviation only when your numbers represent every member of the group you care about. If they are a subset drawn to estimate a wider population — which is most real-world cases — use the sample version.
What does standard deviation actually tell me?
It describes spread. A small standard deviation means the values sit close to the mean; a large one means they are widely dispersed. It is expressed in the same units as your data, which makes it easy to interpret.
What is variance, and how does it relate?
Variance is the average of the squared differences from the mean, and standard deviation is its square root. Variance is shown here too, but standard deviation is usually more intuitive because it shares the units of your original data.
