Standard error and confidence intervals

When we measure the mean of a sample, we obtain one estimate — different from the one we would get with another sample drawn from the same population. The key question is: how precise is this estimate?

Property — Standard error of the mean

If the population has standard deviation $\sigma$ and we draw $n$ independent values, the sample mean $\bar X$ is a random variable with $\text{(standard deviation of }\bar X\text{)} = \frac{\sigma}{\sqrt{n}}.$ This number is called the standard error of the mean ($\text{SE}$).

The practical consequence is the famous $\sqrt{n}$ rule: to halve the uncertainty you need to quadruple the sample, not simply double it.

Property — 95% confidence interval (Gaussian approximation)

When $n$ is large enough (usually $n\ge 30$), by the central limit theorem, $\bar X$ is approximately Gaussian with mean $\mu$ and standard deviation $\sigma/\sqrt{n}$. The 95% confidence interval is $\bar x \pm 1,96\cdot\frac{s}{\sqrt{n}},$ in the sense that $95\%$ of the intervals built this way contain the true value $\mu$.

Example — Election poll

$n=1000$ people interviewed, of whom $520$ declare a vote for party $A$. Estimate of the percentage: $\hat p=0,52$. For a proportion, $s\approx\sqrt{\hat p(1-\hat p)}=\sqrt{0,52\cdot 0,48}\approx 0,4996$. Standard error: $0,4996/\sqrt{1000}\approx 0,0158$. $95\%$ interval: $0,52\pm 1,96\cdot 0,0158 \approx 0,52\pm 0,031 = (48,9\%;\,55,1\%).$ The margin $\pm 3,1\%$ is the famous “margin of error” that we read next to every poll. It is compatible both with a win for party $A$ and with a substantial tie.

Example — How much to sample

An institute wants to estimate a proportion with error $\le 2\%$ at $95\%$. Setting $\hat p(1-\hat p)\le 0,25$ (maximum at $\hat p=0,5$): $1,96\cdot\frac{0,5}{\sqrt n}\le 0,02 \iff \sqrt n\ge \frac{1,96\cdot 0,5}{0,02}=49 \iff n\ge 2401.$ At least $2401$ people interviewed are needed: one sees why serious polls invest in twice as many respondents to halve the margin.

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Topics: Probabilita
Concepts: Errore standard · Intervallo di confidenza · Margine di errore · Teorema del limite centrale
Methods: Intervallo confidenza
Skills: Calcolare · Stimare