StatReason

Standard Deviation Calculator

Enter a list of numbers to find how much they typically spread out around their mean.

Separate values with commas, spaces, tabs, or one per line.

Use sample when your data is a sample from a larger population (the common case). Use population only when your list is the entire population of interest.

What this answers

Standard deviation answers "how spread out is this data, in the original units?" Two datasets can share the exact same mean while one clusters tightly around it and the other scatters widely. The mean alone cannot tell them apart. Standard deviation reports that spread in the same units as your original data (dollars, seconds, points), which is what makes it more directly interpretable than variance, its squared-unit relative.

How it is calculated

Standard deviation is the square root of variance. Variance itself is the average squared distance of each value from the mean, using either an n minus 1 denominator (the sample convention, which corrects for the fact that a sample's own mean slightly understates true population variance) or an n denominator (the population convention, used only when your data genuinely is the entire population, not a sample of it). This calculator names whichever convention you selected directly in the result, since the two give different numbers on the same data and silently picking one for you would hide a choice that changes the answer.

Worked example

For the values 1, 2, and 3 under the sample convention: the mean is 2, the squared distances from the mean are 1, 0, and 1, summing to 2, divided by n minus 1 (which is 2), giving a variance of 1. The square root of 1 is 1, so the sample standard deviation is exactly 1. Try that input above with the sample convention selected and confirm the calculator returns 1 for both variance and standard deviation.

Assumption audit

Calculated from your data: the count of values, whether the sample convention has at least 2 values (required, since a single value has no spread to measure under that convention), and whether the data is constant (SD of exactly 0).
Evidence to review: whether your data is symmetric or skewed. Standard deviation describes spread well for roughly symmetric data; for strongly skewed data or data with extreme values, it can overstate the "typical" spread that most values actually show, and the interquartile range may describe it more usefully.
You must verify: which convention is actually correct for your situation, sample versus population, since the software cannot know whether your list represents a sample or a complete population. This choice is never guessed silently.

Common mistakes

Treating standard deviation as automatically meaning "plus or minus this much error" is one of the most common misreadings. It describes spread in the underlying data, not precision in an estimate; for that, see the Mean Confidence Interval Calculator, which uses standard deviation as an ingredient but answers a different question about how precisely the mean itself is known.

Limitations

Standard deviation assumes the mean is a meaningful center for your data. If your data is heavily skewed or has extreme values, both the mean and the standard deviation built on it can misrepresent what a typical observation actually looks like.