A 95% confidence interval does not mean there is a 95% probability that the true parameter lies within that interval. This is one of the most persistent misstatements in applied statistics — appearing in journal articles, news reports, and textbook exercise solutions. The correct interpretation is subtler. This guide explains it without pretending the subtlety does not exist.
The setup
You have a sample. From that sample, you calculate a statistic — say, the sample mean x̄. You want to make a statement about the population mean μ, which you cannot observe directly.
A confidence interval is a range of values, calculated from the sample, that is constructed by a procedure which — over many repetitions of the same sampling process — captures the true parameter a specified percentage of the time.
The correct interpretation
"We are 95% confident that the interval [a, b] contains the true population mean."
What this means: the interval was produced by a method that would include the true value in 95% of samples drawn from this population. It does not mean: the probability that μ lies between a and b is 95%. After the interval is computed, μ either is or is not in the interval. There is no probability involved — the uncertainty is in the procedure, not in the parameter.
Why the wrong interpretation is so tempting
The Bayesian interpretation — "given this data, the probability the parameter lies in this range is 95%" — is what most people actually want to say. It is also technically what a credible interval (the Bayesian equivalent) provides. The frequentist confidence interval does not provide this, but it looks like it does, and the distinction requires careful language to preserve.
In practice, for symmetric, well-behaved distributions, the numerical answers from frequentist confidence intervals and Bayesian credible intervals are often close. But the interpretation is different, and conflating them causes errors in legal, medical, and policy contexts.
Construction of a confidence interval for a mean
Given: sample mean x̄, sample standard deviation s, sample size n, desired confidence level (1−α):
CI = x̄ ± t*(α/2, n−1) · (s / √n)
where t*(α/2, n−1) is the critical value from the t-distribution with n−1 degrees of freedom at significance level α/2.
For large n (typically n > 30), the t-distribution approaches the normal distribution, and you can use z* (1.96 for 95%) instead.
Worked example
A teacher measures reading times for 16 students. Sample mean = 45 minutes. Sample standard deviation = 8 minutes.
Standard error = 8 / √16 = 8 / 4 = 2.
For a 95% CI with 15 degrees of freedom, the t* value is approximately 2.131.
CI = 45 ± 2.131 × 2 = 45 ± 4.26 = (40.74, 49.26).
Interpretation: "We are 95% confident that the interval (40.74, 49.26) contains the true mean reading time for the population from which this class was drawn."
What changes when you change the confidence level
A 99% CI is wider than a 95% CI. More confidence requires more room. A 90% CI is narrower. The tradeoff between width and confidence level is unavoidable — to be more certain, you must sacrifice precision.
If a very wide interval is useless for your purpose, the only fix is more data (larger n). Doubling your sample size reduces the margin of error by a factor of √2 ≈ 1.41.
Frequently Asked Questions
Can the true parameter fall outside a 95% confidence interval?
Yes. In 5% of samples, the procedure will generate an interval that does not contain the true value. That is what the 5% means — it is the long-run error rate of the method, not a guarantee for any single interval.
How is a confidence interval different from a prediction interval?
A confidence interval captures where the population parameter (mean) is likely to be. A prediction interval captures where a single new observation is likely to fall. Prediction intervals are wider because they account for both parameter uncertainty and individual variation.