What is the difference between standard deviation and standard error?

Question

Hi, I was wondering what the main difference is between standard deviation and standard error and how do I interpret them on excel graphs?

Answer 1

They answer two different questions.

• Standard deviation (s) — how spread out your individual measurements are.
• Standard error (SE) — how precisely you know the mean of those measurements.

They are related by a single factor:

SE = s / √n

The behaviour that separates them

This is the clearest way to keep them straight: ask what happens when you collect more data.

• s does not shrink. It converges toward the true variability of whatever you are measuring. If your balance is noisy, or the objects genuinely differ, more measurements will not change that — they will just characterise it better.
• SE does shrink, as 1/√n. Averaging more measurements really does pin down the mean more tightly.

So s is a property of the thing and your method; SE is a property of your confidence in a number you computed.

How to interpret each in a lab report

Use s when describing variability: "the tablets weighed 250 mg with a standard deviation of 8 mg" tells the reader how much tablet-to-tablet variation there is, or how repeatable your technique was.

Use SE when reporting a measured value: "the concentration was 0.168 ± 0.004 M" tells the reader how well the average itself is known. A result you are asserting as the answer should carry SE (or a confidence interval), not s.

For a rough 95% confidence interval, use mean ± 2 × SE (more precisely, mean ± t × SE, with t from a table for your n).

Error bars — always say which you plotted

SE bars are always smaller than SD bars, by a factor of √n. Because of that, error bars are genuinely ambiguous unless labelled, and choosing SE purely because it looks tighter is a known bad habit in published figures. State it in the caption.

A worked feel for the numbers

Suppose five measurements give a mean of 1.063 g with s = 0.056 g.

SE = 0.056 / √5 = 0.025 g

So an individual item typically falls within about 0.056 g of the mean, but the mean is known to about 0.025 g. Report the mass as 1.063 ± 0.025 g.

The practical consequence of that square root: to halve your uncertainty in the mean you need four times as many measurements, not twice. That is why there are sharply diminishing returns on repeating a run — and why, past a point, improving your instrument (reducing s) beats collecting more data.