Confidence Interval Calculator

Estimate a population parameter with a range of plausible values. Calculate confidence intervals for means or proportions, complete with the margin of error for your chosen confidence level.

Confidence Interval Calculator

Calculate confidence intervals for means, proportions, and differences

Calculation Type

Confidence Level

Sample Mean (x̄)

Sample Standard Deviation (s)

Sample Size (n)

About Confidence Intervals

A confidence interval provides a range of plausible values for a population parameter. A 95% confidence level means that if we repeated the sampling process many times, 95% of the resulting intervals would contain the true parameter.

For means with small samples (n < 30), the t-distribution is used. For proportions and large samples, the normal (z) distribution is appropriate.

How the Confidence Interval Calculator Works

Choose whether you're estimating a population mean or a proportion. For means, enter sample mean, sample standard deviation, and sample size. For proportions, enter the number of successes and sample size (or enter the sample proportion directly).

Select your confidence level - typically 90%, 95%, or 99%. Higher confidence produces wider intervals. The calculator uses the t-distribution for means (when population SD is unknown) or the normal distribution for proportions with large samples.

Results show the confidence interval bounds, margin of error, and point estimate. The interval represents the range of plausible values for the population parameter. A visual display shows the interval with the point estimate marked.

When You'd Actually Use This

Survey result reporting

Report poll results with margin of error. "52% support (95% CI: 48%-56%)" shows the precision of your estimate from sample data.

Quality control specifications

Estimate average product dimension with confidence bounds. Verify if the entire confidence interval falls within specification limits.

Clinical trial outcomes

Report treatment effect with confidence interval. Shows both the estimated effect and the precision of that estimate for medical decision-making.

A/B test result analysis

Show conversion rate difference with confidence bounds. If the interval excludes zero, the difference is statistically significant.

Manufacturing process validation

Estimate process capability with confidence. Ensure the entire interval meets quality requirements, not just the point estimate.

Academic research reporting

Report study findings with confidence intervals. Modern journals prefer CIs over just p-values as they show effect size and precision.

What to Know Before Using

Confidence level is about the method, not the interval.95% confidence means 95% of intervals from repeated sampling would contain the true value. Your specific interval either does or doesn't contain it.

Larger samples give narrower intervals.Margin of error decreases with √n. Quadrupling sample size halves the margin of error. More data means more precise estimates.

Higher confidence means wider intervals.99% CI is wider than 95% CI. You gain confidence but lose precision. Choose based on how critical it is to capture the true value.

t-distribution accounts for SD uncertainty.When using sample SD (not population SD), t-distribution gives wider intervals, especially for small samples. Approaches normal for large n.

Pro tip: Confidence intervals that include the null value (0 for differences, 1 for ratios) indicate non-significant results at the corresponding alpha level. A 95% CI excluding 0 means p < 0.05.

Common Questions

What's the margin of error?

Margin of error is half the confidence interval width. If CI is 45% to 55%, the margin of error is ±5%. It's the maximum expected difference from the true value.

Why use t-distribution for means?

When population SD is unknown and estimated from sample, there's extra uncertainty. The t-distribution accounts for this, giving appropriately wider intervals.

Can confidence intervals be one-sided?

Yes, when you only care about an upper or lower bound. A one-sided 95% CI uses the same critical value as a two-sided 90% CI.

What if my sample is small?

For means with n < 30, ensure data is approximately normal. The t-distribution handles small samples but assumes normality. For proportions, need np ≥ 10 and n(1-p) ≥ 10.

How do I interpret overlapping CIs?

Overlapping CIs don't necessarily mean no significant difference. For comparing two means, check if the CI for the difference excludes zero.

What's the standard error?

Standard error = SD / √n. It's the SD of the sampling distribution. Margin of error = critical value × standard error.

Can I use this for medians?

Not directly. This calculator is for means and proportions. Median confidence intervals require bootstrapping or other non-parametric methods.

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