Confidence Interval Calculator
Calculate confidence intervals for population means and proportions using z or t distributions.
| Confidence | z* | Interpretation |
|---|---|---|
| 90% | 1.645 | 90% of such intervals contain μ |
| 95% | 1.960 | 95% of such intervals contain μ |
| 99% | 2.576 | 99% of such intervals contain μ |
| 99.9% | 3.291 | 99.9% of such intervals contain μ |
Build a confidence interval around a sample mean (using a known or estimated standard deviation) or a sample proportion, at 90%, 95%, or 99% confidence.
How Confidence Interval Calculator Works
When the population standard deviation σ is known, the interval uses the z-distribution: margin of error = z* × (σ/√n), and the final interval is the sample mean plus or minus that margin of error.
When only the sample standard deviation s is available — far more common in real research — the calculator instead uses the t-distribution, whose critical value depends on both the confidence level and the degrees of freedom (n−1). The t-critical value is always slightly larger than the equivalent z-critical value, especially for small samples, which widens the interval to account for the extra uncertainty of estimating the true population standard deviation from a limited sample.
For a sample proportion, the standard error formula changes to √(p̂(1−p̂)/n), reflecting that variability is greatest when the proportion is close to 0.5 and shrinks as it approaches 0 or 1 (since there's naturally less room for variation near the extremes).
See It In Action
Who Uses Confidence Interval Calculator and Why
- Reporting a margin of error around a survey's average result for a research summary.
- Estimating a range for the true population proportion from a sample poll's results.
- Deciding whether a small-sample study should use the t-distribution instead of the z-distribution.
- Comparing how interval width changes across 90%, 95%, and 99% confidence levels for the same data.
Mistakes to Avoid
- Using the z-distribution when the population standard deviation is actually unknown and only estimated from the sample — the t-distribution is the correct choice in that far more common situation.
- Interpreting '95% confidence' as a 95% probability that this one specific interval contains the true mean, rather than its actual meaning about the long-run behavior of repeated sampling.
- Forgetting that degrees of freedom (n−1) shrink as sample size drops, which makes the t-critical value noticeably larger — and the interval wider — for small samples.
Tips for Best Results
- Default to the t-distribution unless you genuinely know the population standard deviation ahead of time, rather than estimating it from your sample.
- Remember a higher confidence level always widens the interval — pick the level based on what precision your specific report actually needs.
Fixing Common Problems
My interval seems unexpectedly wide for a small sample. — This is expected behavior — small sample sizes have fewer degrees of freedom (n−1), which pushes the t-critical value higher than the equivalent z-critical value, widening the interval.
Terms Explained
Degrees of freedom: n−1 for a sample-based confidence interval, used to select the correct t-critical value — smaller samples have fewer degrees of freedom and wider intervals.
Standard error: A measure of how much a sample statistic is expected to vary from the true population value, shrinking as sample size grows.