Sampling distributions and standard error

A sampling distribution describes how a statistic varies from sample to sample; its standard deviation is the standard error.

\SE(\xbar) = \frac{\sigma}{\sqrt{n}} \approx \frac{s}{\sqrt{n}}

Standard error of a sample mean

$\sigma$: population standard deviation
$s$: sample standard deviation (divisor $n - 1$ ), used when σ is unknown
$n$: sample size

\SE(\phat) = \sqrt{\frac{p(1-p)}{n}} \approx \sqrt{\frac{\phat(1-\phat)}{n}}

Standard error of a sample proportion

$p$: population proportion
$\phat$: sample proportion $x/n$

When to use

Every time you report an estimate: the SE says how far the estimate typically lands from the parameter. It is the building block of confidence intervals and test statistics.

Assumptions

Independent observations (a random sample, or a sample under about 10% of a finite population; otherwise apply the FPC). The formulas measure random sampling error only, not bias.

How to interpret

SD vs SE: the SD describes the spread of individual values; the SE describes the spread of a statistic across repeated samples.
$\SE \propto 1/\sqrt{n}$ : quadrupling the sample size halves the SE (diminishing returns).
The mean of the sampling distribution of $\xbar$ is μ (unbiased); of $\phat$ it is $p$ .

Common mistakes

Reporting the SD where the SE is needed, or describing individual variability with the SE.
Expecting a bigger sample to fix a biased sampling method.
Dividing by $n$ instead of $\sqrt{n}$ .

Example

Sample: 12, 15, 11, 14, 13, 16, 12, 15. $\xbar = 13.5$ , $s = 1.7728$ , $\SE = 1.7728/\sqrt{8} = 0.6268$ . A survey with $\phat = 0.30$ from $n = 60$ has $\SE = \sqrt{0.3 \times 0.7/60} = 0.0592$ .