Confidence intervals

A single sample gives a single estimate — say, 52% support. But another sample would give something a little different. A confidence interval turns that one estimate into an honest range, and states how confident the method is.

Estimate, give or take

A confidence interval is your estimate plus and minus a margin of error:

$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

The margin is built from the sampling distribution's spread (σ/√n from the CLT) times a number z* that sets the confidence level — z* ≈ 1.96 for the usual 95%. "52% ± 3%" means the interval runs from 49% to 55%.

What "95% confident" actually means

This is the most misread idea in statistics:

95% confidence does not mean "95% chance the true value is in this interval." The true μ is fixed; your interval either caught it or didn't. It means: if you repeated the whole process many times, about 95% of the intervals you'd build would contain μ. The confidence is in the long-run reliability of the method, not in any one interval.

The levers of precision

A narrower (more precise) interval comes from:

• a larger sample (bigger n shrinks σ/√n), or
• accepting lower confidence (a smaller z* — but you're more often wrong).

You can't tighten the interval and raise confidence for free; more certainty about a narrower range always costs more data.

Why this matters

Every poll's "margin of error," every drug trial's effect range, every "the average is between X and Y" is a confidence interval. Reading them correctly — range plus reliability, not false precision — is basic statistical literacy, and it inoculates you against headlines that treat a noisy estimate as an exact fact.