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About this tool
Population or sample standard deviation, variance and mean.
The Standard Deviation Calculator takes a list of numbers and returns the standard deviation, variance, mean and count, dividing the sum of squared deviations by n for a population or by n−1 (Bessel's correction) for a sample. Paste values separated by commas, spaces or line breaks — negatives and decimals are picked up automatically — and switch between the two modes to see how much the divisor changes the answer. It is aimed at students checking homework and at anyone summarising the spread of a small dataset without opening a spreadsheet.
Open Standard Deviation Calculator on AltFTool — it loads instantly in your browser.
Type or paste your data into the Numbers box, which is prefilled with the set 2, 4, 4, 4, 5, 5, 7, 9. Values can be separated by commas, spaces or new lines, and at least two are needed before a result appears.
Set Type to Population or Sample, depending on whether the values cover the whole group or only a subset; Sample divides by one less than the count.
The Result panel updates as you type with σ to four decimal places, the chosen type named beneath it, and Variance, Mean and Count tiles; press Download to save that summary as standard-deviation-calculator.txt or Copy to put it on the clipboard.
Toggling between the n and n−1 divisors recalculates instantly, so you can see the size of the correction on your own data rather than trusting a rule of thumb.
Numbers are extracted with a pattern match, so commas, spaces, tabs and newlines all work and you can paste a column straight out of a spreadsheet.
Variance, mean and the count of parsed numbers are returned alongside σ, which makes it easy to spot when a value was mistyped and dropped from the set.
Use sample (n−1) when your numbers are a subset drawn from a larger group you want to make inferences about, and population (n) only when the list is the entire group. Sample is the more common choice in coursework and research, and it always produces the larger figure — for the dataset 2, 4, 4, 4, 5, 5, 7, 9 the population value is exactly 2 while the sample value is about 2.138.
Bessel's correction is the use of n−1 instead of n in the denominator, and it exists because a sample's own mean sits closer to its data than the true population mean does. Dividing by n would therefore understate the spread; the smaller divisor corrects that bias, and its effect shrinks as the sample grows — noticeable at n = 5, negligible past a few hundred values.
Standard deviation is the square root of variance. Variance is expressed in squared units, which makes it awkward to interpret, so the square root is reported to bring the figure back into the same units as the original data — a variance of 4 corresponds to a standard deviation of 2.
At least two — with a single value there is no deviation to measure and the calculator will ask for more input. Note that sample standard deviation with exactly two values divides by 1, which makes the result unusually sensitive to each number.
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