An Estimate With Its Uncertainty Attached
A sample mean is a single number that will differ from the true population value by some unknown amount. A confidence interval expresses how large that gap is likely to be:
Interval = estimate ± (critical value × standard error)
The standard error is the standard deviation of the estimate itself — how much the sample mean would bounce around if you repeated the sampling.
Confidence Level Chart
| Confidence level | Critical value (z) | Interval width relative to 95% |
|---|---|---|
| 80% | 1.282 | 65% |
| 90% | 1.645 | 84% |
| 95% | 1.960 | 100% |
| 98% | 2.326 | 119% |
| 99% | 2.576 | 131% |
Higher confidence buys a wider interval. A 99% interval is more likely to contain the true value precisely because it claims less about where that value is.
What "95% Confident" Actually Means
It refers to the procedure, not to this particular interval. If the sampling and interval construction were repeated many times, 95% of the resulting intervals would contain the true population value. This one either does or does not — the true value is fixed, and it is the interval that varies from sample to sample.
The everyday reading — "a 95% chance the true value is in this range" — is the Bayesian credible interval, which is a different construction and often gives a similar answer for uninformative priors.
Sample Size Drives Precision
The standard error contains √n in the denominator, so the interval narrows with the square root of the sample. Quadrupling the sample halves the margin of error. This is the fundamental economics of survey research: precision gets expensive quickly.
Intervals for Proportions
For a percentage, the standard error is √(p(1−p)/n). It is largest at p = 0.5, which is why poll margins of error are quoted for a 50/50 split — that is the worst case. A result of 90/10 has a noticeably tighter interval from the same sample.
Published poll margins also cover sampling error only. Non-response bias, question wording, and turnout modelling are usually larger sources of error and are not in the number.
Frequently Asked Questions
Should I use z or t?
Use t when the population standard deviation is unknown and the sample is small, which is most real situations. Above about 30 observations the two are close enough that it rarely changes a conclusion.
Can a confidence interval include impossible values?
Yes — a proportion interval can extend below 0 or above 1 with small samples or extreme proportions. Methods such as the Wilson score interval exist specifically to avoid this.
If two intervals overlap, are the groups the same?
Not necessarily. Overlapping intervals can still correspond to a statistically significant difference. Testing the difference directly is the correct approach.