Standard error of the mean:
how far your sample average may sit from the truth

Spread of the data versus precision of the average

Weigh 36 people and you get 36 different numbers, and their spread is a fact about people. Now imagine repeating the whole exercise with another random 36 people. The average would change a little from sample to sample, and that wobble in the average is a different kind of uncertainty.

The standard deviation measures the first, the scatter of individual values. The standard error of the mean measures the second, how much a sample average would move if you could repeat the sampling. Authors often quote the standard error because it is smaller and looks more impressive, so readers need to know which one they are looking at.

The formula

The pattern is simple. The numerator is how variable individuals are, which you cannot change, and the denominator grows with sample size, which you can. Because the root is involved, effort and reward are lopsided: to halve the standard error, you need four times as many observations, and to cut it to a tenth you need a hundred times as many.

Strictly, the formula uses the population standard deviation, which is unknown in practice and replaced by the sample standard deviation. That substitution is why small samples are handled with the t distribution, not the normal one, when building intervals.

Worked example: 36 measurements

A sample of 36 weights has a mean of 72 kg and a sample standard deviation of 15 kg.

An approximate 95% interval for the true mean uses about 1.96 standard errors, which gives 72 ± 4.9 kg, from 67.1 to 76.9 kg. With only 36 observations the more accurate multiplier from the t distribution at 35 degrees of freedom is about 2.03, giving 72 ± 5.1 kg. The difference is small but grows if the sample is smaller.

What changing the sample size does

The table shows the diminishing returns. Going from 9 to 36 observations gains 2.5 kg of precision, while going from 144 to 900 gains only 0.75 kg for six times the data. This is why a pilot study followed by a power calculation is better value than collecting as much data as possible.

Remember that the formula assumes a random sample whose observations are independent. A large biased sample or clustered, repeated measurements from the same people do not shrink the error the way the formula implies.

Keep in mind that none of this protects against bias. If the sample was drawn only from volunteers or only from one clinic, the standard error describes how consistent that biased average is, not how close it is to the population truth. Precision and accuracy are separate, and the formula only speaks to the first.

Which one should appear in a report

• Use the standard deviation to describe how variable the individuals are.
• Use the standard error or a confidence interval to describe how precisely the mean is estimated.
• Say which you are reporting, for example mean ± SD or mean ± SE, since the two differ by a factor of √n.
• Prefer confidence intervals, which many journals now ask for in place of bare SE.

A common trap is a bar chart whose error bars show the SE without saying so; the bars look short while the real spread of the data is six times larger in the case above.

Comparing two groups and planning sample size

When comparing the means of two independent groups, the standard errors combine as the square root of the sum of their squares. If one group has an SE of 2.5 and the other 3.0, the standard error of the difference between their means is √(2.5² + 3.0²) = √15.25 = 3.9. A difference smaller than about twice that figure is easily produced by chance alone.

The formula can also be run in reverse to plan a study. Rearranging SE = s ÷ √n gives n = (s ÷ SE)². If the standard deviation is expected to be 15 and you want the mean pinned down to a standard error of 1.0, you need (15 ÷ 1.0)² = 225 observations. Halving the target to 0.5 would need 900.

Pilot data usually provide the SD estimate. Because that estimate is itself uncertain, planners build in a margin of extra participants, and they allow for people who drop out, which raises the real number to recruit above the formula's answer.

Checking the number and using it

A rough check: the standard error must be smaller than the standard deviation whenever n is above 1, and equal to it when n = 1. If your SE exceeds the SD, you divided by something other than √n.

The sample standard deviation itself should use n − 1 in its denominator when computed from a sample. A dedicated sample standard deviation tool will do this, and a mean tool completes the picture. Those provide the inputs; deciding whether the sample is representative remains a judgement the formula cannot make.

When you read a published result, it is worth working backwards. Given a reported mean, an SE and a sample size, multiply the SE by √n to recover the SD, and judge whether the spread of individual values is plausible for the thing measured. A paper that reports a tiny SE for a measurement that obviously varies a lot between people is probably quoting SE where a reader would expect SD.