CLT Explorer

Interactive Shiny app for visualising the central limit theorem across distributions, sample sizes, and statistics

Purpose

The CLT Explorer gives a visual answer to the question every introductory statistics student eventually asks: why does the sampling distribution of the mean end up looking normal even when the raw data do not? The app lets a reader pick a source distribution (uniform, exponential, bimodal, Cauchy, and others), select a sample size, and watch the sampling distribution of the chosen statistic build up as thousands of samples are drawn in real time.

User inputs

• Source distribution with parameters (mean, variance, skewness, kurtosis where applicable)
• Sample size \(n\) from 1 to 1000
• Number of replicate samples from 100 to 100000
• Statistic of interest (mean, median, trimmed mean, standardised mean)
• Overlay: theoretical normal curve implied by the CLT

Outputs

• Histogram of the raw source distribution (top panel)
• Histogram of the sampling distribution of the chosen statistic (middle panel)
• Q-Q plot of the sampling distribution against the normal (bottom panel)
• Summary table: simulated mean and SE vs. theoretical mean and SE

Didactic value

The app makes visible what chapters of textbook prose describe in words: that the CLT is about the sampling distribution of a statistic, not the distribution of the data, and that its approach to normality depends on both \(n\) and the shape of the source. The Cauchy distribution is included deliberately, so that readers can see that the CLT does not apply when the assumptions (finite variance) fail.

Embedded in

• statistical-foundations/central-limit-theorem.md
• statistical-foundations/sampling-distributions.md
• inference/one-sample-t-test.md

Source code

Local: apps/01-clt-explorer/

Run with:

shiny::runApp("apps/01-clt-explorer")