Standard Error
Standard error of a mean or a proportion, with the margin of error and interval it produces.
From s = 15 and n = 36 · the spread of individual values is 15, 6× larger
The standard error shrinks with √n, so halving it takes four times the data. Quadrupling this sample to 144 would bring it to 1.25; to halve it again you would need 576.
The standard error measures how much a sample statistic would bounce around if you repeated the study. For a mean it is s ÷ √n: a sample of 36 with a standard deviation of 15 gives SE = 2.5, and a 95% confidence interval of roughly ±5 around the sample mean.
The spread of a statistic, not of the data
This is the distinction that causes most of the confusion. The standard deviation describes how widely individual observations scatter — how different one person is from another. The standard error describes how widely the sample mean would scatter if you drew the sample again and again. It is always smaller, because averaging cancels out some of the noise.
In the example, individual values spread by 15 but the mean of 36 of them spreads by only 2.5 — six times tighter. That factor is √36, and it is the whole mechanism: each observation contributes its own random error, and those errors partly cancel rather than accumulating.
Diminishing returns from more data
Because the divisor is √n rather than n, precision improves slowly. Halving the standard error takes four times the sample; halving it again takes sixteen times the original. This is why survey sample sizes cluster around a thousand or so — going further costs a great deal and buys very little. It is also why a study reporting a suspiciously tight interval from a small sample deserves a second look.
s the sample standard deviation, p̂ the sample proportion, n the sample size
- 1 Take the sample standard deviation. Here s = 15, describing how much individual values differ from one another.
- 2 Take the square root of the sample size. √36 = 6.
- 3 Divide. 15 ÷ 6 = 2.5, the standard error of the mean.
- 4 Pick a critical value for the confidence level. For a mean estimated from a sample, use t with n − 1 degrees of freedom: t(95%, df = 35) = 2.0301.
- 5 Multiply to get the margin of error. 2.0301 × 2.5 = 5.08, so a sample mean of 100 gives a 95% interval of 94.92 to 105.08.
How the standard error shrinks with sample size
With a standard deviation of 15 throughout. Each halving of the error costs four times the data.
| Sample size n | √n | Standard error |
|---|---|---|
| 9 | 3 | 5.00 |
| 36 | 6 | 2.50 |
| 100 | 10 | 1.50 |
| 144 | 12 | 1.25 |
| 400 | 20 | 0.75 |
| 900 | 30 | 0.50 |
Where it turns up
Almost everywhere in inferential statistics, because it is the denominator of nearly every test statistic. A t-statistic is (estimate − hypothesised value) ÷ standard error; a confidence interval is estimate ± critical value × standard error. Recognising that shape means one idea covers means, proportions, differences and regression coefficients alike — only the formula for the standard error itself changes.
Two cautions. Error bars on a chart may show standard deviation, standard error or a confidence interval, and these are very different sizes — a caption that does not say which is not interpretable. And the formulas here assume independent observations from a random sample; clustered or repeated-measures data have correlated errors that do not cancel in the same way, so a naive standard error computed on them is too small.