Confidence interval example: twenty samples, one miss

What it shows

Twenty 95% confidence intervals for a population mean, each from a different random sample of the same population. It shows what the confidence level refers to: in repeated sampling, about 95% of such intervals contain the true mean.

In this draw one interval of the twenty misses, shown in orange. The random seed was chosen to give exactly one miss, the expected number, with every interval end clearly to one side of the true mean, so the figure is a teaching illustration rather than a typical run: another seed could give none, or two.

How to read it

1. 1

   The dashed vertical line is the true mean, 170 cm.

2. 2

   Each horizontal line is one interval, with a dot at its sample mean.

3. 3

   The orange interval, sample 3, runs from 170.90 to 177.61 cm, so it does not contain 170.

4. 4

   The 95% describes the method, not any single interval: each interval either contains the true mean or does not.

The numbers behind it

• Population: adult height, normal with mean 170 cm and standard deviation 10 cm (simulated).
• Each sample has n = 25; each interval is the sample mean ± t × s / √25, with t = 2.0639 (24 degrees of freedom).
• Misses: 1 of 20 (sample 3, sample mean 174.26 cm). Expected misses: 20 × 0.05 = 1.

Ways to use it in teaching

• Ask students what the 95% refers to before showing the figure, and compare their answers afterward.

• Each student draws a sample from a known population and computes an interval; pooling the class’s intervals reproduces the figure.

• Discuss why one miss in twenty is no sign that anything went wrong.

For how intervals relate to significance tests, see our explainer on what a p-value means.

Sources

• Confidence interval. (n.d.). In Wikipedia. Retrieved October 18, 2026, from https://en.wikipedia.org/wiki/Confidence_interval
• Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/details/books/introductory-statistics-2e