📐 Confidence Interval Calculator
Build a confidence interval for a population mean or proportion from your sample statistics, with the margin of error and a plain-English interpretation.
95% confidence interval
94.3989 to 105.601
100 ± 5.60109
2.04523
Critical t (df=29)
2.73861
Standard error
5.60109
Margin of error
Critical values are exact — the t critical uses the inverse regularized incomplete beta function, not a rounded table. 🔒 Computed in your browser.
How the confidence interval calculator works
For a mean with known population SD, the interval is x̄ ± z·(σ/√n); with unknown SD it uses the t-distribution: x̄ ± t·(s/√n). For a proportion it is p̂ ± z·√(p̂(1−p̂)/n). Enter your sample statistics and confidence level, and the tool computes the critical value (exact z or t), the margin of error, and the interval.
The critical value is exact — the t critical uses the inverse of the regularized incomplete beta function, not a rounded table lookup — so the interval is precise for any degrees of freedom and confidence level.
Frequently asked questions
What is a confidence interval?
A range, computed from a sample, that is likely to contain the true population value. A 95% confidence interval means that if you repeated the sampling many times, about 95% of the intervals would contain the true parameter.
When do I use z versus t?
Use z when the population standard deviation is known (rare) or the sample is large. Use the t-distribution when the population SD is unknown and estimated from the sample — which is the usual case. This tool picks the right one based on your input.
What is the margin of error?
The half-width of the interval: the critical value times the standard error. The interval is the estimate plus or minus the margin of error, so a smaller margin means a more precise estimate.
How do I get a narrower interval?
Increase the sample size (the margin shrinks with √n), lower the confidence level (e.g. 90% instead of 99%), or reduce variability. The sample-size calculator works this backwards from a target margin.
What does 95% confidence actually mean?
It is a statement about the procedure, not a single interval: 95% of intervals built this way capture the true value. It does not mean there is a 95% probability the true value is in this particular interval.
Is my data uploaded?
No — the calculation runs entirely in your browser.
Related statistics tools
From the blog
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How Many People Do You Need to Survey? Sample Size Explained
The sample size for a proportion is n = z²·p(1−p) / E². At 95% confidence with a ±5% margin you need 385 responses — whether your population is 20,000 or 20 million. Why p = 0.5 is the safe choice, why halving the margin quadruples n, and when population size matters.
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Z-Score to P-Value: How to Read the Normal Distribution
A p-value is the tail area of the normal curve beyond your z-score. z = 1.96 gives a two-tailed p of 0.05; z = 1.645 a one-tailed p of 0.05. How the conversion works, one- versus two-tailed, and why a table is coarser than the exact erf calculation.