Standard Deviation Calculator
Find population and sample standard deviation, variance and mean from your numbers.
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Step-by-step working
Each value, its difference from the mean, and the squared difference.
How standard deviation works
Standard deviation is the most common way to describe how spread out a set of numbers is, and this calculator works it out for you in seconds. It is free, with no sign-up, and it runs entirely in your browser. It is made for students checking homework, teachers preparing lessons, and researchers who want a quick summary of a data set.
What is standard deviation?
Standard deviation measures how far, on average, values sit from the mean. A small value means the numbers cluster tightly around the mean, while a large value means they are widely spread. It is always in the same units as the original data, which makes it easier to understand than variance, which is measured in squared units.
What is the difference between population and sample standard deviation?
Population standard deviation uses all the numbers you care about, and it divides the sum of squared differences by the count, N. Sample standard deviation treats your numbers as a sample from a larger group, and it divides by N minus 1 instead. The sample value is a little larger, because dividing by a smaller number gives a larger result. Use the population version for a complete set, and the sample version when you are estimating from a sample.
How do you calculate standard deviation step by step?
First, work out the mean by adding all the numbers and dividing by how many there are. Next, subtract the mean from each value, square each difference, and add the squares together. That total is the sum of squared differences. Finally, divide by N for the population variance, or by N minus 1 for the sample variance, and take the square root. The step-by-step table above shows each of these stages for your numbers.
Why do we square the differences?
Differences from the mean are squared for two reasons: to stop positive and negative differences cancelling each other out, and to give bigger gaps more weight. A set with one value far from the mean will have a larger standard deviation than a set where every value is a little way off, even if the average distance is the same. That is why standard deviation is a good measure of spread, and it is also why it is sensitive to outliers.
What is variance?
Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of variance. Variance is useful in statistics, because it has handy mathematical properties, but standard deviation is easier to describe in plain language, because it is back in the original units.
Worked example
Take the numbers 2, 4, 4, 4, 5, 5, 7, 9. Their mean is 5, and the sum of squared differences is 32. The population variance is 32 divided by 8, which is 4, so the population standard deviation is the square root of 4, which is 2. The sample variance is 32 divided by 7, which is about 4.5714, so the sample standard deviation is about 2.1381. Try it in the calculator above to see each step.
What is a good standard deviation?
There is no good or bad standard deviation on its own. A small value means the data is consistent, which is often what you want in a measurement, while a large value means it varies a lot, which may be natural for the thing being measured. The right way to judge it is to compare it with the mean, for example by looking at the coefficient of variation, which is the standard deviation divided by the mean. The OpenStax statistics textbook explains this in its chapter on sampling and data.
Frequently asked questions
Can standard deviation be negative?
No. Standard deviation is the square root of a sum of squared values, and squared values are never negative, so the result is always zero or more. A value of zero means every number in the set is identical.
What does a standard deviation of zero mean?
It means there is no variation at all: every value is the same as the mean. In practice this is rare, unless the data really is constant, or the set only has one value.
Should I use population or sample standard deviation?
Use population standard deviation when your numbers are the whole group you care about, for example every student in a class. Use sample standard deviation when your numbers are a sample from a larger group, for example 50 people surveyed out of a town of 10,000. When in doubt, most calculators and spreadsheets default to the sample version.
How is standard deviation different from range?
Range is just the largest value minus the smallest, so it only looks at two numbers and ignores the rest. Standard deviation uses every value, so it gives a much better picture of how spread out the whole set is.
What is the difference between standard deviation and standard error?
Standard deviation describes the spread of individual values, while standard error describes how uncertain the mean is as an estimate of the population mean. Standard error is the sample standard deviation divided by the square root of the sample size, so it gets smaller as you collect more data.
Why divide by N minus 1 for a sample?
Dividing by N minus 1 instead of N corrects a bias: a sample tends to underestimate the spread of the full population, because the sample mean is fitted to the sample itself. The correction, called Bessel’s correction, makes the sample variance an unbiased estimate of the population variance on average.
Does this calculator store my data?
No. The calculator runs entirely in your browser. Nothing you type is uploaded, stored or shared, and closing the page clears it.
More tools you may need
Comparing one value with a group? Use our Z-Score Calculator. Testing whether a result is significant? Try our P-Value Calculator.
