Introduction

In this vignette, we demonstrate the Central Limit Theorem (CLT) using the demo_clt() function. The CLT states that, regardless of the original population distribution, the distribution of the sample mean approaches a normal distribution as the sample size increases.

Here, we use the Gamma distribution with varying skewness, which is controlled by the shape parameter:

Shape = 0.5: Highly skewed distribution
Shape = 1: Exponential distribution (moderately skewed)
Shape = 2: Less skewed distribution

We explore the behavior of the sample mean for different sample sizes (n = 5, 10, 20, 40) to illustrate the convergence towards the normal distribution.

Load Required Packages

Gamma Distribution with Shape = 0.5

The Gamma distribution with shape = 0.5 is highly skewed. We expect to see the sample mean distribution becoming more normal as the sample size increases.

demo_clt(rgamma, n = c(5, 10, 20, 40), nrep = 1000,
         shape = 0.5, rate = 1, pmean = 0.5, psd = sqrt(0.5))

Gamma Distribution with Shape = 1

The Gamma distribution with shape = 1 is equivalent to the Exponential distribution. This example has moderate skewness, and we observe the effect of increasing the sample size.

demo_clt(rgamma, n = c(5, 10, 20, 40), nrep = 1000,
         shape = 1, rate = 1, pmean = 1, psd = sqrt(1))

Gamma Distribution with Shape = 2

The Gamma distribution with shape = 2 has less skewness. Here, the sample mean distribution converges more quickly to a normal distribution even for smaller sample sizes.

demo_clt(rgamma, n = c(5, 10, 20, 40), nrep = 1000,
         shape = 2, rate = 1, pmean = 2, psd = sqrt(2))

A General Distribution Constructed from a Mixture

When pmean and psd are not supplied, demo_clt() estimates them from 100,000 simulated observations. For instance, consider sampling from the following mixture distribution.

mymix_rng <- function(n, rate = 0.5) {
    ifelse(runif(n) < rate,
           rgamma(n, shape = 0.5), rgamma(n, shape = 4))
}

demo_clt(mymix_rng, n = c(5, 10, 20, 40), nrep = 1000)

Conclusion

The plots above demonstrate the Central Limit Theorem in action. As the sample size increases, the distribution of the sample mean approaches a normal distribution, even for highly skewed underlying distributions like the Gamma distribution with shape = 0.5.

This vignette illustrates the robustness of the CLT and its importance in statistical analysis, especially when dealing with non-normal data in biostatistical contexts.