• If you have a population with mean \(\mu\) and standard deviation \(\sigma\) and take sufficiently large random samples (with sample size n) from the population with replacement, then the distribution of the sample means will be approximately normally distributed, with mean \(\mu\) and standard deviation \(\sigma/\sqrt{n}\).

• For more details on Central Limit Theorem, please referring to the Wikipedia link below: https://en.wikipedia.org/wiki/Central_limit_theorem

• In my Shiny Application “Validate Central Limit Theorem using Simulation”, users will be able to see how does the CLT work.

• No matter which distribution the users choose to start with, the simulated sample means will be normally distributed.

• The effect will be more obvious with large number of simulation (even with a small sample size).

• Please refer to the link below to my Github Repo, which includes all the coding files to my Shiny app and this presentation.
  https://github.com/ZhaoZ-2020/DevelopDataProducts

Users can choose the original distribution (where the sample will be drawn) from the four distributions below:

- Standard Normal Distribution 
- Poisson Distribution (with lambda = 1)
- Standard Uniform Distribution
- Binomial Distribution (with n=10 and p=0.5)

Users will need to specifiy the sample size and the number of simulation they want to perform using the sliders provided.

In addition, they can also indicate if they want to show the extra lines below in the output plots:

- sample mean in the first density plot
- average of the sample mean in the second density plot
- density plot from a normal distribution (for comparison purpose)
  

See example below from standard Uniform distribution with n=5 and B=1000:

set.seed(123); par(mfrow=c(1,2)); n<-5; B<-1000
x<-matrix(runif(n*B),nrow=B); xbar<-apply(x,1,mean)
hist(x[1,],prob=TRUE,main="Density plot of one sample")
hist(xbar,col='lightblue',prob=TRUE,main="Density plot of B sample means")