Overview

Point estimates (like a sample average) are rarely exactly right. Confidence intervals (CIs) acknowledge this uncertainty by providing a range—an interval—that we are “confident” covers the true value.

Core Idea

Instead of saying “The average height is 175cm,” we say “We are 95% confident the true average height is between 173cm and 177cm.” It turns a single guess into a net that catches the truth with a specified probability.

Formal Definition (if applicable)

A $100(1-\alpha)%$ confidence interval is an interval computed from sample data such that, if the sampling were repeated many times, $100(1-\alpha)%$ of the generated intervals would contain the true population parameter.

$$ P(L \leq \theta \leq U) = 1 - \alpha $$

Intuition

Imagine throwing a ring at a stick in the ground (the true parameter). You can’t see the stick perfectly. A confidence interval is the size of the ring. A wider ring (higher confidence) is more likely to catch the stick, but gives you less precision about where the stick is.

Examples

  • Election Polling: “Candidate A is polling at 48% with a margin of error of +/- 3%.” This is a confidence interval of [45%, 51%].
  • Product Testing: “The average battery life is 10 hours +/- 0.5 hours.”

Common Misconceptions

  • “95% chance the true value is in this interval”: This is technically incorrect in frequentist statistics. The true value is fixed; the interval is random. Once calculated, the interval either contains the value or it doesn’t. The “95%” refers to the reliability of the method, not the specific interval.
  • “We capture 95% of the data”: No, CIs are about the mean (parameter), not the individual data points.

Applications

Used in reporting scientific results to convey precision. A narrow CI implies a precise estimate; a wide CI implies high uncertainty.

Criticism / Limitations

Often misinterpreted by the public and even researchers. Bayesian credible intervals are an alternative that actually allows for probability statements about the parameter.

Further Reading

  • “Statistical Intervals” by Hahn and Meeker
  • Explanations of Frequentist vs Bayesian intervals