Standard Deviation Calculator | Mean, Variance & SD Free

Standard Deviation Calculator

This calculator takes a list of numbers, separated by commas, spaces, line breaks, or semicolons (or auto-detected from mixed formatting), and returns the count, sum, and mean together with both the sample and population variance and standard deviation. The sample standard deviation is shown as the main result, since most real datasets are a sample rather than a complete population. Data can also be pasted from the clipboard; it is appended to whatever is already in the field and the result recalculates automatically.

How the calculation works

The mean is the sum of the values divided by how many there are. From the mean, the calculator finds each value's squared difference and adds them together to get the sum of squared differences (SSD). Dividing the SSD by the count gives the population variance; dividing by the count minus one gives the sample variance, the standard choice when the numbers are a sample drawn from a larger group rather than the whole group itself. That minus-one adjustment offsets the tendency of a sample to underestimate the true spread of the data it came from. Standard deviation is the square root of variance, which puts the result back into the original units instead of squared units.

Mean = Σx / n
SSD = Σ(x - Mean)²
Population variance = SSD / n
Sample variance = SSD / (n - 1)
Standard deviation = √Variance

Worked example

Take five quiz scores out of 100: 72, 85, 90, 78, and 95. The sum is 420, so the mean is 420 / 5 = 84. Each score's squared difference from the mean is 144, 1, 36, 36, and 121, giving a sum of squared differences of 338. Dividing 338 by the count of 5 gives a population variance of 67.6, so the population standard deviation is the square root of 67.6, about 8.2219. Dividing the same 338 by 4 (5 minus 1) gives a sample variance of 84.5, so the sample standard deviation is about 9.1924, larger than the population figure because it corrects for sampling uncertainty.

Reference values for common spreads

The table below shows how the sample and population standard deviation change on small datasets (n = 4 or 5) at three levels of spread, from tightly clustered to widely scattered.