Point and interval estimation
A point estimate is a single number computed from a sample to stand in for an unknown population parameter, but it is almost never exactly right; an interval estimate reports a range around it — built from the sample's own variability — that would contain the true parameter in a stated proportion of repeated samples, which is the honest way to communicate how much a point estimate can be trusted.
Practise this conceptA point estimate — a sample mean, a sample proportion — is the best single guess at an unknown population parameter, but a single number carries no sense of how much to trust it. An interval estimate fixes that by reporting a range built from the point estimate plus and minus a margin of error, E = z·s/√n: the sample's own standard deviation s and sample size n set how much the estimate could plausibly wander, and the quantile z sets how much of that plausible range to actually cover.
The interpretation is the part that trips almost everyone up on first contact: a 95% confidence interval does not mean "there is a 95% chance the true value is in this particular interval" — the true value either is or isn't in it, with no probability left over once the sample is drawn. It means that if you repeated the whole sampling procedure many times, about 95% of the resulting intervals would contain the true value. The randomness lives in the procedure, not in the fixed number you're trying to pin down.
Two knobs control the interval's width, and they trade off against each other in opposite directions: raising the confidence level (say from 95% to 99%) widens the interval, because covering the true value more often demands more room; increasing the sample size narrows it, but only as 1/√n — quadrupling n, not doubling it, is what actually halves the width.
Key ideas
Requires: Common probability distributions
Unlocks: Hypothesis testing
Formulas
| Symbol | Name | Unit |
|---|---|---|
| quantile | 1 (dimensionless) | |
| standard deviation | 1 (dimensionless) | |
| sample size | 1 (dimensionless) | |
| confidence interval half-width | 1 (dimensionless) |
Common misconceptions
- Believes a 95% confidence interval means there is a 95% probability that the true population parameter lies inside this particular computed interval, rather than that 95% of intervals built this way from repeated samples would contain it.
- Believes a wider confidence interval is always worse than a narrower one, without recognising the trade-off against confidence level — a narrower interval at the same sample size only comes from accepting a lower confidence level.
- Believes doubling the sample size halves the width of the confidence interval, missing that the half-width scales with 1/√n, so the sample size must be quadrupled to halve it.
Resources
Part of these UT courses
This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.