Confidence Interval Calculator 2026
Calculate the confidence interval, margin of error, and standard error for any sample. Supports both Z-test and T-test distributions with optional finite population correction.
Sample Data & Settings
Enter your sample statistics and choose a confidence level
The average value observed in your sample
Number of observations in your sample (must be ≥ 2)
Population standard deviation (σ) if known, or sample standard deviation (s) if estimated from data
Total population size. Used to apply finite population correction when sampling a large fraction of a known population.
Higher confidence levels produce wider intervals
Confidence Interval Results
Bounds, margin of error, and standard error
Enter your sample statistics above and click Calculate Confidence Interval to see the bounds, margin of error, and full statistical breakdown.
Critical Values at a Glance
Common Z-scores (standard normal) and T-scores (df = 30) for popular confidence levels. Use these to manually verify calculator results or construct intervals by hand.
| Confidence Level | α (significance) | Z-score (two-tailed) | T-score (df = 30) |
|---|---|---|---|
| 80% | 0.20 | 1.282 | 1.310 |
| 85% | 0.15 | 1.440 | 1.476 |
| 90% | 0.10 | 1.645 | 1.697 |
| 95% | 0.05 | 1.960 | 2.042 |
| 98% | 0.02 | 2.326 | 2.457 |
| 99% | 0.01 | 2.576 | 2.750 |
| 99.5% | 0.005 | 2.807 | 3.030 |
| 99.9% | 0.001 | 3.291 | 3.646 |
Confidence Interval FAQ
Everything you need to understand confidence intervals, from the underlying statistics to practical interpretation and common pitfalls.
A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter (such as the mean) with a specified level of confidence. For example, a 95% confidence interval means that if you were to repeat the sampling process many times, approximately 95% of the calculated intervals would contain the true population mean.
The margin of error (MOE) is the half-width of the confidence interval. It represents the maximum expected difference between the observed sample statistic and the true population parameter. It is calculated as: MOE = Critical Value × Standard Error. A smaller margin of error indicates greater precision in the estimate.
Use a Z-test when the population standard deviation is known or when the sample size is large (typically n ≥ 30), because the sampling distribution approaches normality. Use a T-test when the population standard deviation is unknown and must be estimated from the sample, especially with smaller sample sizes (n < 30). The T-distribution has heavier tails, providing a more conservative interval for small samples.
Sample size has an inverse-square-root relationship with the margin of error. As sample size (n) increases, the standard error decreases, producing a narrower and more precise confidence interval. To halve the margin of error, you need to quadruple the sample size. This is why larger surveys yield more reliable estimates.
It does NOT mean there is a 95% probability that the true parameter lies within your specific interval. Rather, it means that if you were to draw many random samples and compute a 95% confidence interval from each, about 95% of those intervals would contain the true population parameter. For any single interval, the parameter either is or isn't inside it — we just don't know which.
The finite population correction (FPC) is applied when sampling a significant portion (typically more than 5%) of a known, finite population. It reduces the standard error because sampling without replacement from a limited pool provides more information than sampling from an infinite population. The FPC factor is √((N − n) / (N − 1)), where N is population size and n is sample size.