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Standard Deviation Calculator

Calculate mean, variance and standard deviation for a data set.

What is the Standard Deviation Calculator?

A standard deviation calculator measures how spread out a set of numbers is around its average (mean) — a low standard deviation means values cluster tightly around the mean, a high one means they're widely scattered. It works by first calculating the mean, then finding the squared difference of every value from that mean, averaging those squared differences (this is the variance), and taking the square root of the variance to get the standard deviation. To calculate it manually: find the mean, subtract it from each value and square the result, sum and divide by n (population) or n−1 (sample) for the variance, then take the square root — sample variance divides by n−1 (Bessel's correction) because it gives a less biased estimate when your data is a sample rather than the entire population.

How it works

Standard deviation measures how spread out a set of numbers is around its average (mean) — a low standard deviation means values cluster tightly around the mean, while a high standard deviation means they're more spread out. It's calculated by finding the mean, working out how far each value differs from the mean (squared, so negative and positive differences don't cancel out), averaging those squared differences (this is the variance), and then taking the square root of the variance to get back to the original units.

UK context

There are two versions of the formula: population standard deviation (used when your data represents the entire group you're interested in) divides by the number of data points, n. Sample standard deviation (used when your data is a sample taken to estimate a wider population) divides by n − 1 instead — a small but important distinction called Bessel's correction, which makes the sample version a slightly less biased estimate of the true population spread.

Tips

Check whether your task specifically asks for population or sample standard deviation — using the wrong formula is a very common exam and coursework mistake.
Standard deviation is in the same units as your original data (e.g. £, cm, kg), while variance is in squared units — standard deviation is usually the more intuitive figure to report.
A standard deviation of zero means every value in the data set is identical to the mean.

Frequently asked questions

What's the difference between population and sample standard deviation?

Population standard deviation divides by n (the number of data points) and is used when your data is the entire group you care about. Sample standard deviation divides by n − 1 and is used when your data is a sample used to estimate a larger population — this correction (Bessel's correction) makes the estimate less biased.

What's the difference between variance and standard deviation?

Variance is the average of the squared differences from the mean. Standard deviation is the square root of variance, which brings the measure back into the same units as the original data — making it easier to interpret directly.

What does a low vs high standard deviation mean?

A low standard deviation means the data points tend to be close to the mean (low spread/variability). A high standard deviation means the data points are spread out over a wider range of values.