What is the correct interpretation of a 95% confidence interval?

How to think about it

Confidence intervals are reported in nearly every analysis, yet the correct interpretation trips up experienced practitioners. The subtlety is frequentist: the parameter is fixed and unknown; it is the interval that is random.

The precise frequentist statement

Construct the same kind of 95% CI from 1000 independent samples of the same size. Approximately 950 of those intervals will bracket the true parameter. The remaining 50 will miss it entirely — purely by sampling chance.

For any single interval you have computed, that interval either contains the true value or it does not. There is no probability to assign to this specific interval; the true parameter is not a random variable (in the frequentist framework).

What a 95% CI is NOT

• It is not “there is a 95% chance the parameter lies in this interval.” The parameter is fixed — probability statements about a fixed unknown require a Bayesian credible interval, which needs a prior.
• It is not a claim that 95% of the data falls in the interval. That would be a prediction interval.
• Wider is not always worse — a wider interval from a small sample correctly conveys more uncertainty.

Width and what drives it

The half-width (margin of error) for a mean is approximately z * sigma / sqrt(n) where z = 1.96 for 95%. To halve the width, you need four times the sample size.

Relationship to hypothesis testing

A two-sided 95% CI is the set of all null hypothesis values that would NOT be rejected at alpha = 0.05. If the CI for a difference excludes zero, the two-tailed test at alpha = 0.05 is also significant — the two approaches are mathematically equivalent.

Reporting best practice

Report the interval alongside the point estimate and sample size: “The estimated mean is 42.3 (95% CI: 39.1 to 45.5, n = 200).” This lets readers judge both statistical and practical significance.