Standard Error
How much your estimate would wobble if you repeated the measurement on new data.
Also known as: SE, standard error of the mean, SEM
The standard error of a statistic is the standard deviation of that statistic across repeated samples of the same size. It answers “how much would this estimate wobble if I collected the data again?”, and it is the bridge between “this is the number I got” and “this is what I can say about the number I would get next time”.
For a mean, written SE or SEM:
SE(mean) = σ / sqrt(n) # if the population SD σ is known
SE(mean) = s / sqrt(n) # the usual case: σ unknown, s is the sample estimateThe σ/s distinction is the whole point. σ is the true spread of the population you draw from, and you rarely know it. s is what you can compute from the rows in front of you. Because s is estimated from the same data, tests built on it use a t test rather than a normal approximation, and the difference is largest when n is small.
Every statistic has its own standard error, and they do not shrink at the same rate:
| Statistic | Standard error | Note |
|---|---|---|
| Mean | s / sqrt(n) | the slowest to shrink: double precision needs 4× the data |
| Proportion | sqrt( p(1−p)/n ) | p near 0 or 1 gives a smaller SE for the same n |
| Regression slope | depends on residual spread and the spread of x | a narrow x-range inflates it |
| Difference of two means | sqrt( SE1² + SE2² ) | the two sample sizes need not be equal |
Note what the standard error leaves out. It measures sampling variability only. It says nothing about bias in the sample, about rows recorded wrongly, or about the metric being defined differently in the two groups you are comparing. A precise standard error around a biased estimate is a confident wrong answer.
The standard error is the multiplier inside a confidence interval, the half-width behind a margin of error, and the denominator of the statistic in a t test. None of those tell you how large an effect is — see effect size for that half of the answer.