In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a bell curve) even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.

• In the App, we explore the CLT for independent uniform distributed random variables in the (0,1) interval
• We generate a sample of values with size equal to the size chosen by the user and calculate the their mean
• We repeat the last step 10,000 times
• We plot the empirical distribution

`x <- rep(0,10000)
 for (i in 1:10000) {
   x[i] <- mean(runif(n = input$sample_size,min = 0, max = 1))
 }
 
 mean_norm <- 0.5
 sd_norm <- sqrt(1/(12*input$sample_size))
 y <- dnorm(x = seq(from=0,to=1,length.out= 10000),
            mean = mean_norm,sd = sd_norm)
 
 myData <- data.frame(n=seq(from=0,to=1,length.out= 10000),
                      x=x,y=y)
 
 ggplot(myData) +
   geom_density(aes(x=x),size=1) +
   geom_line(aes(x=n,y=y),col="red",size=2, alpha=0.3) +
   ggtitle("Empirical Uniform(0,1) vs Normal Distribution")`