Standard error of a mean
Describe typical sample-mean variation across repeated samples.
Standard error of a mean is one of 2 sampling formulas in the statistics section of this library, and it is used at ap · university level.
Why standard error of a mean works
Averaging lets random highs and random lows cancel each other out. The more observations you average, the more cancelling happens, and the spread of the sample mean shrinks with the square root of the sample size rather than with the sample size itself.
What each symbol means
$n$ is sample size and $\sigma$ or $s$ is spread.
Standard error of a mean: when it holds
Observations should be independent; using $s$ estimates the unknown population standard deviation.
When it stops applying
The formula assumes the observations are independent. If your 36 students were recruited in 6 friendship groups, the effective amount of independent information is far less than 36 and the true standard error is larger. Sampling a large fraction of a small finite population has the opposite effect, and needs a correction factor built from the population size.
Standard error of a mean: a worked example
If $s=12,n=36$, then $SE=2$.
The mistake to avoid
What people do: Quoting the standard deviation of the individuals as the uncertainty in the average.
Why it goes wrong: Those measure different things. With a spread of 12 and 36 observations, individuals vary by about 12 while the sample mean varies by only 2, so using 12 overstates the uncertainty six times over.
Do this instead: Divide by the square root of the sample size, and describe the result as the wobble in the average rather than the spread of the data.
Standard error of a mean: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ is sample size and $\sigma$ or $s$ is spread.
- Check the conditions before substituting. Observations should be independent; using $s$ estimates the unknown population standard deviation.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Sampling
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where standard error of a mean comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Standard Deviation — the lesson behind this formula: measure typical distance from the mean.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about standard error of a mean
How much more data do I need to halve the standard error?
Four times as much, because of the square root. Going from 36 observations to 144 takes a standard error of 2 down to 1, and halving it again would take 576.
Is the standard error the same thing as the standard deviation?
No. The standard deviation describes how much individual observations differ from each other; the standard error describes how much the average would move if you repeated the whole study.
Does the population have to be normal for this formula?
The formula itself needs only independence and a finite spread. Normality matters for what you build on top of it, and for large samples the central limit theorem makes the mean approximately normal anyway.
What does the standard error mean in plain language?
It is the typical distance between your sample mean and the true population mean. It answers how far off your estimate is likely to be, purely from the luck of which cases you drew.