How is the confidence interval calculated?

It uses the z-based formula mean ± z × (σ / √n), where z is the critical value for your chosen confidence level. For 95% confidence z is 1.96.

What does a 95% confidence interval mean?

If you repeated the sampling many times, about 95% of the intervals built this way would contain the true population mean. It is not the probability that this one interval contains the mean.

What is the z value for a 99% confidence interval?

About 2.576. The common critical values are 1.282 for 80%, 1.645 for 90%, 1.960 for 95%, 2.326 for 98% and 2.576 for 99%. A higher confidence level uses a larger z and produces a wider interval.

How does sample size affect the interval?

A larger sample size narrows the interval, because the standard error is σ ÷ √n. Quadrupling n halves the margin of error, so big samples give tighter, more precise estimates.

This tool uses the z critical value, which is appropriate when the population standard deviation is known or the sample is large. For small samples with an estimated standard deviation a t-interval is slightly wider.

Confidence Interval Calculator

A confidence interval is a range, built from sample data, that is likely to contain the true population mean. This calculator computes a z-based interval from your sample mean, standard deviation, sample size and chosen confidence level. It is a staple of statistics coursework, survey analysis and A/B testing, where you want to report not just an estimate but how precise it is.

How it works

The interval is the sample mean plus and minus a margin of error:

CI = mean ± z × (σ ÷ √n)

Here σ ÷ √n is the standard error of the mean, and z is the critical value for your confidence level. The tool uses these z values: 80% → 1.282, 90% → 1.645, 95% → 1.960, 98% → 2.326, 99% → 2.576. A higher confidence level uses a larger z, widening the interval; a larger sample size shrinks the standard error, narrowing it.

Example

Sample mean 100, standard deviation 15, sample size 36, at 95% confidence:

standard error = 15 ÷ √36 = 15 ÷ 6 = 2.5 margin = 1.96 × 2.5 = 4.9 interval = 100 ± 4.9 = 95.1 to 104.9

So you can be 95% confident the true mean lies between 95.1 and 104.9. All maths happens locally in your browser.