What does a 95% confidence interval mean?

If you repeated your study 100 times and computed a 95% CI each time, about 95 of those intervals would contain the true population parameter. It does NOT mean there's a 95% chance the true value is in your specific interval.

When should I use a t-distribution instead of Z?

Use the t-distribution when the population standard deviation is unknown and the sample is small (n < 30). The t-distribution has heavier tails to account for extra uncertainty. For n ≥ 30, Z and t give nearly identical results.

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Confidence Interval Calculator

Calculate confidence intervals for population means using Z or t critical values.

Inputs

CI = x̄ ± (critical value) × (σ or s) / √n

Sample Mean (x̄)

Standard Deviation Type

Sample Std Dev (s)

Sample Size (n)

Confidence Level

About Confidence Intervals

A confidence interval provides a range of plausible values for an unknown population parameter. A 95% confidence interval means that if you repeated the sampling process many times, 95% of the calculated intervals would contain the true population mean.

Z vs. t Distribution

Use the Z-distribution when the population standard deviation (σ) is known or when the sample size is very large (n > 30). Use the t-distribution when σ is unknown and you're estimating it from the sample. The t-distribution has heavier tails, producing wider intervals for small samples.

Factors Affecting Width

The margin of error decreases as sample size increases (inversely proportional to √n), and increases with higher confidence levels. To halve the margin of error, you need to quadruple the sample size.

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Take this calculation further

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For a related calculation, use P-Value Calculator to calculate p-values from Z-test or T-test statistics. Determine significance at α=0.05 and α=0.01. Alternatively, use Z-Score Calculator to convert values to z-scores and percentiles, or find the percentile for any z-score using the standard normal distribution.

For the underlying method, read A Practical Guide to Statistical Sample Size: How Many People Do You Actually Need to Survey?.