• States that the distribution of the sample mean approximates a normal distribution
• This happens regardless of the population distribution.
• Is one of the most important concepts in probability statistics.

If \(X_1, X_2, ..., X_n\) are independent, identically distributed random variables with mean \(\mu\) and variance \(\sigma^2\), then:

\[ \bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \] approaches:

\[ \bar{X}_n \sim N\left(\mu, \frac{\sigma^2}{n}\right) \]

# Define population: exponential distribution (clearly not normal)
population <- rexp(100000, rate = 1)

# Function to sample and compute means
sample_means <- function(n, reps = 1000) {
  replicate(reps, mean(sample(population, n, replace = TRUE)))
}

n_values <- c(2, 5, 30)
means_list <- lapply(n_values, sample_means)
names(means_list) <- paste0("n=", n_values)

ggplot(data.frame(x = population), aes(x)) +
  geom_histogram(bins = 40, fill = "#8C1D40", color = "white") +
  labs(title = "Original Population: Exponential Distribution",
       x = "Value", y = "Frequency")

means_df <- data.frame(
  mean = c(means_list[[1]], means_list[[2]], means_list[[3]]),
  n = factor(rep(n_values, each = 1000))
)

ggplot(means_df, aes(x = mean, fill = n)) +
  geom_histogram(bins = 30, color = "white", alpha = 0.7, position = "identity") +
  labs(title = "Sampling Distributions for Different Sample Sizes",
       x = "Sample Mean", y = "Frequency") +
  facet_wrap(~n, scales = "free")

plot_ly(
  x = means_df$mean,
  y = as.numeric(means_df$n),
  z = rep(1:1000, length(n_values)),
  type = "scatter3d",
  mode = "markers",
  marker = list(size = 3, color = means_df$mean, colorscale = "Viridis")
) %>%
  layout(title = "Central Limit Theorem: 3D Visualization")

As \(n\) increases, variance decreases:

\[ Var(\bar{X}_n) = \frac{\sigma^2}{n} \]

and the distribution of \(\bar{X}_n\) becomes more concentrated around \(\mu\).

Thus, for large \(n\):

\[ Z = \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \sim N(0,1) \]

# Example of computing sample means for CLT visualization
sample_means <- function(n, reps = 1000) {
  replicate(reps, mean(sample(rexp(100000, 1), n, replace = TRUE)))
}
means_30 <- sample_means(30)
hist(means_30, main="Sampling Distribution (n=30)",
     col="#8C1D40", border="white")

• The CLT explains why the normal distribution appears so frequently in data analysis.
  
• Even non-normal populations produce approximately normal sample means when \(n\) is large.
  
• This forms the basis for many inferential statistical methods, including hypothesis testing and confidence intervals.