Mean, median, mode and standard deviation

Paste your numbers in. Every average and every spread comes out at once, and where statistics has more than one answer — which it does, three times over — you get both, labelled, rather than whichever one the calculator happened to pick.

Why there are two standard deviations, and which you want

Every statistics calculator asks you to choose between sample and population, usually without saying what turns on it. The difference is one character: you divide the squared distances by n or by n − 1.

The reason is that a sample's own mean is closer to the sample than the true mean of the whole population would be, so the spread you measure is always a little too small. Dividing by n − 1 rather than n corrects for that. It is called Bessel's correction, and it matters most when you have few numbers: with five of them it changes the answer by about 12%, and with five hundred it is invisible.

If your numbers are everything there is — all twelve months of the year, every pupil in the class — you want the population figure. If they are a sample of something larger — forty customers standing in for all your customers — you want the sample figure, which is why it is the one most software gives you by default. This page shows both, because knowing which you need is the difficult part and hiding one of them does not help.

A mode can be none, one, or several

The mode is the value that occurs most often, which sounds simple until two values tie. In 1, 1, 2, 2, 3 both 1 and 2 appear twice, so there are two modes and a calculator that reports one of them is choosing arbitrarily.

And when every value appears exactly once there is no mode at all. Saying that every value is a mode is technically defensible and useless; this page says there is no mode, which is what you meant to ask.

Quartiles have several definitions and they disagree

There is no single agreed way to cut a list into quarters, and the common methods give different answers on the same data. For 1 to 9, the exclusive method puts the lower quartile at 2.5 and the inclusive one puts it at 3. Neither is wrong.

The difference is whether the median itself is included in each half before you take that half's median. Excel has both, as QUARTILE.EXC and QUARTILE.INC, and they disagree with each other for the same reason. This page defaults to the exclusive method, says so on every reading, and lets you switch.

When the mean and the median disagree, believe the median

The mean is dragged by outliers and the median is not. Nine people earning £25,000 and one earning £1,000,000 have a mean income of £122,500 and a median of £25,000, and only one of those numbers describes anybody in the room.

So the reading above says how far apart they are. A large gap is not an error; it is the data telling you it is skewed, and that the mean on its own will mislead.

Where to go next

If the numbers came from somewhere and are going somewhere:

• percentage calculator — for the single comparisons
• weighted grade calculator — an average where the parts are not equal

What this does not do

Commas are treated as separators, not thousands separators. Pasting 1,234 gives you two numbers rather than one, which is the right guess for a pasted list and the wrong one for a pasted total, so the reading warns you when your numbers look like they might have been the second kind.

There is no significance testing here, no confidence intervals and no correlation: those need a second question about what you are trying to find out, and this page only describes the numbers you gave it.