Standard Deviation Calculator

What It Does

This calculator reads one data set — numbers separated by commas, spaces or line breaks — and returns the mean, the count and both readings of the spread: the population standard deviation σ = √(Σ(x − x̄)² / n) and the sample standard deviation s = √(Σ(x − x̄)² / (n − 1)), each with its variance beside it. Nothing is rounded in the arithmetic and no unit is assumed: a set of times comes back in seconds, a set of prices in your own currency. A set of fewer than two values, or one holding a token that is not a number, is refused and reported rather than guessed at.

Real-World Usage

The spread answers questions the average cannot: whether a delivery time is reliably 30 minutes or swings by twenty, whether two machines hold the same temperature, whether an exam's marks are bunched or spread, whether a production batch is consistent. In quality work this same figure — usually at three or six standard deviations from the mean — sets a control limit; in finance it is what volatility means. Because it is in the data's own unit, it reads directly against the mean: a mean of 100 with σ = 15 says that a value of 85 sits one spread below the middle.

Local Context

The calculation is scale-free and country-independent: the same formula runs on every page this site serves, and no currency, unit or regional convention changes it — a reader in Madrid, Milan or Detroit pastes the same list and reads the same σ. Only the display follows the page, where a German catalogue writes 2,1381 and an English one 2.1381 for the same number. The pair σ and s is offered together because which one applies is a property of your data and not of your country.

Is my list the whole population or a sample — which reading do I use?

The population reading σ divides the sum of the squared deviations by n, the sample reading s divides the same sum by n − 1. Use σ when your list is the whole of what you are describing — every mark in the class, every reading you took — and use s when the list is one sample drawn from something larger: the sample mean is itself an estimate, and the divisor n − 1 corrects the slight underestimate that dividing by n would give. Both figures are printed side by side so you never have to decide before seeing the numbers; they differ only in that divisor, and the gap narrows as the list grows.

Why does the sample figure come out larger than the population figure?

Because the divisor is smaller. With the same sum of squared deviations, dividing by n − 1 instead of by n can only give a larger number, and after the square root the sample standard deviation stays above the population one: this page's own default set of eight values gives σ = 2 and s = 2.1381. The two converge as the set grows — at n = 30 the difference is already under two per cent — and at n = 2 the sample reading is exactly 1.4142 times the population one, the √2 the arithmetic forces.

What does the standard deviation actually tell me?

It is the typical distance of a value from the mean, in the same unit as your data. A set of times with σ = 12 seconds sits about twelve seconds either side of its average, and a second set with σ = 3 seconds is tighter. It is a spread, not a range: a single wild value moves it a long way, because the deviations are squared before they are averaged, and a set whose values are all equal returns exactly 0 — that is the honest reading, not an error, because nothing in the set deviates from the mean.

Why does it refuse a set of one value, or a word inside the list?

The sample reading divides by n − 1, so a single value has no sample standard deviation at all, and because one result carries both readings the whole set is refused rather than printing a population figure beside a meaningless sample. A token that is not a number — a word, a stray currency symbol, a decimal comma inside a list separated by commas — refuses the set the same way. Empty entries, such as a trailing comma or a blank line, are simply skipped instead of being read as zero.