Central limit theorem example: skewed waiting times

What it shows

The central limit theorem with a strongly skewed population: waiting times with an exponential distribution, mean 5 minutes. Below it are the exact sampling distributions of the mean for samples of 2, 10 and 30 waiting times.

Nothing is simulated. The mean of n independent exponential values has a gamma distribution, which each panel draws, and a dashed normal curve with the same mean and standard error shows what the theorem says it approaches.

How to read it

1. 1

   Every panel is centered on the same mean, 5 minutes, marked by a dotted line.

2. 2

   As n grows the distribution narrows: the standard error falls from 3.54 minutes at n = 2 to 0.91 at n = 30.

3. 3

   At n = 2 the sampling distribution is still skewed and differs visibly from its normal curve; by n = 30 the two nearly coincide.

4. 4

   The n = 2 normal curve is cut off at zero: 7.9% of it lies below 0, where no waiting time can be.

The numbers behind it

• Population: exponential, mean 5 minutes, standard deviation 5 minutes.
• Standard error SE = 5/√n: 3.54 (n = 2), 1.58 (n = 10), 0.91 (n = 30).
• Sampling distribution of the mean of n values: gamma, shape n and scale 5/n.
• The time axis runs from 0 to 25 minutes; 0.67% of the population lies beyond it.

Ways to use it in teaching

• Students predict the shape of the n = 30 panel before it is revealed.

• Pair it with a simulation: repeated samples from any skewed variable, with the sample means plotted.

• Use it against a common confusion: the theorem concerns the distribution of the sample mean, while the data themselves stay skewed.

• n = 30 is not a universal threshold: how fast the sampling distribution approaches the normal depends on how skewed the population is.

For where sampling distributions lead next, see our guide to inferential statistics.

Sources

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

The population and sampling curves use the standard results for the exponential and gamma distributions.