The formula
How it works
Standard deviation measures how spread out a set of numbers is around their average. A small value means the data clusters tightly around the mean; a large one means it is widely scattered. Enter your numbers to find it.
FAQ
Population or sample — which do I choose?
Use population if your numbers are the entire group you care about. Use sample if they are a subset you are using to estimate a larger group; the sample version divides by n − 1, giving a slightly larger, less biased result.
How does it relate to variance?
Variance is the average of the squared distances from the mean, and standard deviation is just its square root. Taking the root brings the figure back into the same units as your original data, which is why it is easier to interpret.
What counts as a “high” standard deviation?
There is no universal cutoff — it depends on the scale of your data and what you are comparing it to. A useful check is the coefficient of variation (standard deviation divided by the mean), which lets you compare spread across data sets with different units or sizes.
Can standard deviation be negative?
No. Because it comes from squaring distances and then taking a square root, the result is always zero or positive. A value of zero means every number in the set is identical.
How does an outlier affect the result?
A lot — because each distance from the mean is squared, a single extreme value can pull the standard deviation up sharply, far more than it would affect the mean. It is worth checking your data for typos or entry errors if the number looks surprisingly large.
What does one standard deviation from the mean actually tell me?
For data that roughly follows a normal distribution, about 68% of values fall within one standard deviation of the mean, and about 95% fall within two. It gives you a quick sense of how typical or unusual a given value is.
Why do some calculators give a slightly different answer for the same numbers?
The most common reason is the population versus sample choice — dividing by n instead of n − 1 gives a different result. Always check which convention a tool or textbook is using before comparing figures.
About the standard deviation calculator
This calculator finds the standard deviation of a set of numbers — a measure of how much the values spread out around their average. Two data sets can share the same mean yet look completely different: one tightly bunched, the other widely scattered. Standard deviation captures that difference in a single number, which is why it is one of the most important tools in statistics for describing and comparing data.
How to use it
Type or paste your numbers, separated by commas or spaces, and choose whether they are a whole population or a sample. The calculator returns the mean, the variance and the standard deviation. For example, the set 2, 4, 4, 4, 5, 5, 7, 9 has a mean of 5 and a population standard deviation of 2. Pick “sample” if your numbers are only part of a larger group you want to draw conclusions about.
The formula
The population standard deviation is : find the mean , measure how far each value is from it, square those distances, average them to get the variance, and take the square root. The sample version divides by instead of , a correction that makes it a better estimate of a larger population’s spread from a limited sample.
Where it is used
Standard deviation is everywhere data is analysed. In finance it measures the volatility and risk of an investment; in manufacturing it drives quality control, flagging when a process drifts; and in science it reports the uncertainty in measurements. Exam boards use it to scale grades, and it underpins the normal distribution, confidence intervals and almost every statistical test that follows.