📐 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, whatever your population size.
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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.