Confidence Interval Calculator

Confidence Interval Calculator: run confidence interval calculations online. Formula, assumptions, and interpretation guide.

A confidence interval (CI) gives a range of plausible values for a population parameter (such as a mean or proportion) estimated from a sample. A 95% CI means that if you repeated the study 100 times with different random samples, approximately 95 of those intervals would contain the true population value. It quantifies the uncertainty in your estimate — wide intervals indicate less precision, narrow intervals indicate more.

Confidence intervals are used in clinical trials, polling and election forecasting, quality control, A/B testing analysis, and published academic research to convey the precision of estimates.

Enter your sample mean (or proportion for percentage data).
Enter the sample standard deviation and sample size (n).
Select your confidence level — 95% is standard; 99% is used for high-stakes decisions.
The calculator returns the margin of error and the lower and upper bounds of the interval.
Report as: "estimate ± margin of error" or "[lower, upper]" with the confidence level stated.

Confidence interval formula

CI = x̄ ± z* × (s / √n)

Where x̄ = sample mean, z* = critical value (1.96 for 95% CI, 2.576 for 99% CI), s = sample standard deviation, n = sample size.

For proportions: CI = p ± z* × √(p(1−p)/n), where p is the sample proportion.

Interpreting your confidence interval

Width and precision

A narrow interval is more precise but requires a larger sample size. Doubling the sample size narrows the interval by a factor of √2 (≈1.41). If the 95% CI for a mean is [45, 55], you can say: "We are 95% confident the true population mean lies between 45 and 55." Critically, the CI does NOT mean there is a 95% chance the true value is in any particular interval — the true value is fixed; it is the interval that varies across samples.

Statistics tips and best practices

Always report the confidence level alongside the interval — "CI [45, 55]" is ambiguous without knowing if it is 90%, 95%, or 99%.
Larger samples produce narrower (more precise) intervals — plan your sample size before data collection.
If two CIs do not overlap, the difference is statistically significant at roughly the same confidence level.
For proportions near 0 or 1 (e.g. 2% defect rate), use the Wilson score interval rather than the standard formula for better accuracy.

Most published medical studies use 95% confidence intervals — the FDA requires CIs for primary efficacy outcomes in drug approval submissions.
Election polls typically report a "margin of error" — this is the half-width of a 95% CI. A ±3% margin with n≈1,000 respondents is standard for national polls.
To halve the margin of error, you need to quadruple the sample size — a fundamental trade-off in survey design.

Common mistakes to avoid

Saying "there is a 95% probability the true mean is in this interval" — the probability is either 0 or 1 (the parameter is fixed); the 95% refers to the long-run performance of the procedure.
Using the formula for small samples without checking normality — for n < 30, use the t-distribution (t-interval) rather than the z-distribution.
Ignoring that the CI assumes random sampling — biased samples produce biased intervals regardless of width.

Confidence intervals are statistical tools for inference from random samples. Results depend on assumptions of random sampling and approximate normality. For regulated industries (clinical trials, financial reporting), consult a qualified statistician.