A confidence interval answers: if we repeated our survey many times, what range would contain the true average most of the time?
About this calculator
A confidence interval answers: if we repeated our survey many times, what range would contain the true average most of the time? Pollsters reporting \"52% ± 3%\" are showing a confidence interval around a proportion. Factory QC teams use the same idea for average batch weight or voltage.
Enter your sample mean, sample standard deviation, sample size n, and pick 90%, 95%, or 99% confidence. The calculator returns the lower bound, upper bound, and margin of error using the normal z critical value (1.645, 1.96, or 2.576). For small samples under 30, a t-distribution is technically more accurate, but the z approximation is fine for quick estimates.
Example: 40 students average 72 marks with SD 8. The 95% CI is roughly 72 ± 1.96 × (8/√40) = 72 ± 2.48, so we expect the true class mean to fall between 69.5 and 74.5.
Pair this with the [[sample-size]] calculator to decide how many responses you need before collecting data, the [[z-score]] calculator to standardise individual values, and [[mean-median-mode]] to compute mean and SD from raw numbers.
Common uses
- Report a range around average exam or survey scores instead of a single point
- Set acceptable limits for factory batch measurements with known variation
- Summarise A/B test results with a margin of error for conversion rate
- Present research findings with 95% confidence bounds in reports
- Check whether a claimed average falls inside your computed interval
Frequently asked questions
What does a 95% confidence interval mean?
It means that if you repeated the sampling process many times, about 95% of the intervals you build would contain the true population mean. It does not mean there is a 95% chance the mean is inside this one specific interval.
How do you calculate a confidence interval for the mean?
CI = sample mean ± z × (SD / √n). For 95% confidence, z = 1.96. Multiply the standard error (SD over root n) by z, then add and subtract from the mean.
Why does a larger sample size narrow the interval?
The standard error is SD divided by root n. As n grows, the standard error shrinks, so the margin of error gets smaller and the interval tightens around the sample mean.
When should I use 99% instead of 95% confidence?
Use 99% when missing the true value is costly — medical thresholds, safety specs, or legal claims. The interval will be wider. Use 95% for general reporting and surveys.
What is margin of error in a confidence interval?
Margin of error is half the width of the interval — the ± part added to and subtracted from the sample mean. It equals z times the standard error.