1/🧵 Welcome to this thread on the Central Limit Theorem (CLT), a key concept in statistics! We'll cover what the CLT is, why it's essential, and how to demonstrate it using R. Grab a cup of coffee and let's dive in! ☕️ #statistics #datascience #rstats

2/📚 The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size (n) increases, given that the population has a finite mean and variance. It's a cornerstone of inferential statistics! #CLT #DataScience #RStats

3/🔑 Why is the CLT important? It allows us to make inferences about population parameters using sample data. Since many statistical tests assume normality, CLT gives us the foundation to apply those tests even when the underlying population is not normally distributed. #RStats

4/🧪 Let's demonstrate the CLT using R! We'll start by creating a function that generates random samples from any distribution, calculates the sample means, and returns the distribution of these means. We'll also set a seed for reproducible results. #RStats #DataScience

library(ggplot2)
library(gridExtra)
set.seed(42)
sample_means <- function(dist_function, n,
sample_size, iterations) {
means <- replicate(iterations, mean(dist_function(n)))
data.frame(SampleMeans = means)
}

5/🎲 Next, we'll use our sample_means function to simulate the sampling distribution of the mean for different sample sizes. We'll do this for a uniform distribution, but you can try it with any distribution you like! #RStats #DataScience

uniform_dist <- function(n) {runif(n, min = 1, max = 6)}
n_values <- c(10, 30, 50, 100)
iterations <- 1000
simulated_means <- lapply(n_values,
function(n) sample_means(uniform_dist, n, 1, iterations))

6/📈 Time to visualize the results! We'll use ggplot2 to create histograms for each sample size and compare the distributions. As the sample size increases, we should see the histograms become more "bell-shaped." #RStats #DataScience

plot_sample_means <- function(simulated_means, n_values) {
plots <- lapply(1:length(n_values), function(i) {
ggplot(simulated_means[[i]], aes(x = SampleMeans)) +
geom_histogram(aes(y = after_stat(density)),
bins = 30, fill = "dodgerblue", alpha = 0.7) +

geom_density(color = "red", size = 1) +
labs(title = paste("n =", n_values[i]),
x = "Sample Means", y = "Density") +
theme_minimal()
})
do.call("grid.arrange", c(plots, ncol = 2))
}
plot_sample_means(simulated_means, n_values)

7/🔍 As you can see, the distribution of sample means becomes more normal as the sample size increases, just as the Central Limit Theorem predicts. This holds true even though we started with a uniform distribution! #CLT #RStats #DataScience

8/💡 So, there you have it! We've demonstrated the Central Limit Theorem using R. Remember, the CLT is a powerful tool for making inferences about population parameters from sample. #CLT #RStats #DataScience