Confidence Interval Calculator

A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter (such as a mean or proportion). For example, a 95% confidence interval means that if you repeated the study 100 times, approximately 95 of those intervals would contain the true population value. It does not mean there is a 95% probability that the true value lies within any single computed interval.

Use a Z-interval when you know the population standard deviation (σ) or when your sample size is very large (n > 30 is a common rule of thumb). Use a t-interval when the population standard deviation is unknown and you are estimating it from the sample standard deviation (s). The t-distribution has heavier tails than the normal distribution, producing wider intervals for smaller samples — this reflects additional uncertainty.

The confidence level (e.g. 90%, 95%, 99%) reflects the long-run reliability of the estimation procedure. A 95% confidence level means that the method used to construct the interval will capture the true parameter 95% of the time across repeated samples. A higher confidence level produces a wider interval because you need to cast a wider net to be more certain of capturing the true value.

Larger sample sizes produce narrower confidence intervals because the standard error (SE = σ/√n) decreases as n increases. This means more data leads to more precise estimates of the population parameter. Doubling precision requires quadrupling the sample size, since SE is proportional to 1/√n.

The margin of error (E) is calculated as E = critical value × standard error. For a mean with known σ, SE = σ/√n. For a proportion, SE = √(p̂(1−p̂)/n). The critical value comes from the Z or t distribution depending on which type of interval you are computing. The confidence interval is then estimate ± E.

A proportion confidence interval estimates the true population proportion based on observed successes in a sample. For example, if 45 out of 100 surveyed people prefer a product, p̂ = 0.45. The confidence interval tells you the plausible range for the true proportion in the population. This calculator uses the standard Wald interval formula: p̂ ± z* × √(p̂(1−p̂)/n).

Key assumptions include: the sample is randomly selected from the population, observations are independent, and for the Z or t interval, the population is approximately normally distributed (or n is large enough by the Central Limit Theorem, typically n ≥ 30). For proportion intervals, both np̂ and n(1−p̂) should be at least 5 to ensure the normal approximation is valid.

Degrees of freedom (df) for a one-sample t-interval equals n − 1, where n is the sample size. The t-distribution with df degrees of freedom is used to find the critical value t*. As df increases (i.e., larger sample sizes), the t-distribution approaches the standard normal distribution, and the t* value converges toward the corresponding z* value.