Statistics and inference

Standard error of a mean

Attaches an error bar to any estimate, and sizes the sample an experiment needs.

\mathrm{se}(\bar{x}) = \frac{s}{\sqrt{n}}, \qquad \bar{x} \pm z_{\alpha/2}\,\mathrm{se}

Where

$z_{\alpha/2}$: 1.645 for 90%, 1.960 for 95%, 2.576 for 99%.
$n = (z\sigma/E)^2$: Rearranged to size a sample for a target margin $E$ .

Assumptions

Independent observations. Overlapping windows or autocorrelation shrink the effective $n$ without shrinking $n$ .
Use $t$ rather than $z$ when $n$ is small and the variance is estimated.

Sanity check. Halving the error bar takes four times the data. That fact sets the economics of research.

Where this is taught

Confidence intervals, and the interval you would trade · STAT · Confidence intervals
The law of large numbers and the central limit theorem · STAT · Limit theorems