Confidence Interval Calculator

Confidence intervals: A complete guide

Confidence intervals provide a range of plausible values for a population parameter based on sample data. Instead of a single point estimate, a confidence interval expresses uncertainty by combining variability and a chosen confidence level. Wider intervals reflect higher confidence or more variability; narrower intervals reflect lower confidence or larger, more precise samples.

Confidence interval for a mean

For a mean with known population standard deviation, the \((1-\alpha)\times 100\%\) confidence interval is \[ \left[\,\bar{x} - z_{\alpha/2}\cdot \frac{\sigma}{\sqrt{n}},\ \bar{x} + z_{\alpha/2}\cdot \frac{\sigma}{\sqrt{n}}\,\right]. \] When \(\sigma\) is unknown and the sample size is small, use the t-distribution: \[ \left[\,\bar{x} - t_{\alpha/2,\,df}\cdot \frac{s}{\sqrt{n}},\ \bar{x} + t_{\alpha/2,\,df}\cdot \frac{s}{\sqrt{n}}\,\right], \] where \(df=n-1\) and \(s\) is the sample standard deviation.

Confidence interval for a proportion

For a proportion with \(\hat{p}=x/n\), a normal-approximation interval is \[ \left[\,\hat{p} - z_{\alpha/2}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\ \hat{p} + z_{\alpha/2}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\,\right]. \] This works best for reasonably large \(n\) and \(\hat{p}\) not near 0 or 1; otherwise, consider Wilson or exact methods.

Choosing confidence levels

• 90%: Narrower interval, more precision, less confidence.
• 95%: Standard in research and reporting.
• 99%: Wider interval, more conservative inference.

Best practices

• Match method to assumptions: Use t for unknown \(\sigma\) with small samples; z for known \(\sigma\) or large \(n\).
• Check sample size: Larger \(n\) reduces margin of error and tightens intervals.
• Report clearly: Provide estimate, margin of error, bounds, and confidence level.

Worked examples