An Error Bar That Shrinks Is Not the Standard Deviation

Why the division is by five and not by six

The squared deviations here sum to 17.5, and the standard deviation divides that by five — one less than the count. The reason is that the deviations are measured from the sample's own mean, which sits by construction as close to the data as any number can, so the spread comes out slightly too small. Dividing by n − 1 corrects for that. It is not a rounding nicety at this size: dividing by six instead gives 1.71, about ten per cent lower, and the correction only becomes negligible once the sample is large. Use n − 1 whenever the data is a sample of something bigger, which in practice is nearly always.

The formula assumes the observations are independent

Dividing by √n only works if each observation carries new information. Six measurements of the same object with the same instrument, or six responses from members of the same household, are not six independent facts, and treating them as such makes the standard error too small and the resulting confidence too high. This is the single most common way a study reports a precision it does not have — the sample looks large because it counts rows, while the number of independent units is far smaller. Count the independent units, not the rows, before dividing.

Frequently asked questions

Which should error bars on a chart show?

It depends on the claim being made. If the point is how much individuals vary — how much patients differ, how much parts differ — draw the standard deviation, because that is the variation a reader will encounter. If the point is whether two group means differ, draw the standard error or, better, a confidence interval, because that is what the comparison rests on. Whichever you pick, say so in the caption: an unlabelled bar is a number the reader cannot use.

Does a small standard error mean the result is right?

No — it means the result is precise, which is a different property from being correct. A biased measurement repeated ten thousand times produces a tiny standard error around the wrong value, and the arithmetic will report that precision with complete confidence. Precision comes from sample size, accuracy comes from how the sample was drawn and how the instrument behaves, and no amount of the first repairs a failure of the second.