Central Limit Theorem

Author

Russell Almond

Published

September 8, 2020

Pick a distribution: * Uniform – platykurtic * Binomial – symmetric and mesokurtic * Exponential – highly skewed * Gamma (shape = 3) – skewed * T (df =3) – high kurtosis

Slide the sample size up and down, notice how the empirical distribution function and histogram coverge to the normal distribution function and density.

#| standalone: true
#| viewerHeight: 750
library(shiny)
nmax <- 1000
mmax <- 100
rdist <- list(Uniform=runif,
              Binomial= function(n) rbinom(n,1,.5),
              Exponential = rexp, 
              Gamma = function(n) rgamma(n,3),
              "T" = function(n) rt(n,3))

ui <- fluidPage(
inputPanel(
  selectInput("dist",label="Distribution Type",
              choices=c("Uniform","Binomial","Exponential","Gamma","T"),
              selected="Unifor"),
   sliderInput("NN", label = "Number of Samples:",
              min = 25, max=nmax, value=nmax, step=5),
  sliderInput("MM",label="Size of each sample:", min=1, max=mmax,value=1,step=1)

),
mainPanel(
  plotOutput("convplot")))

server <- function (input,output) {

  XX <- reactive(matrix(do.call(rdist[[input$dist]],list(nmax*mmax)),nmax,mmax))

  output$convplot <- renderPlot({
  layout(matrix(1:4,2,2))
  X1 <- XX()[1:input$NN,1]
  Xmean <- rowMeans(XX()[1:input$NN,1:input$MM,drop=FALSE])

  hist(X1,main="Average of sample of size 1",probability=TRUE)
  curve(dnorm(x,mean(X1),sd(X1)),add=TRUE,lty=2,col="red")

  qqnorm(X1,main="Average of sample of size 1")
  qqline(X1)

  hist(Xmean,
       main=paste("Average of sample of Size",input$MM),probability=TRUE)
  curve(dnorm(x,mean(Xmean),sd(Xmean)),add=TRUE,lty=2,col="red")

  qqnorm(Xmean,main=paste("Average of sample of Size",input$MM))
  qqline(Xmean)
  
  
  
})
}
shinyApp(ui=ui,server=server)

The left column shows the original distribution. (I call that the black hat in my CLT demo.)

The right column shows the distribution of means of size \(M\) (adjusted with the second slider). (This is the white hat distribuiton, and \(M\) is the number of cards averaged to get the white hat value.)

The top row shows histograms with a normal curve on top.

The bottom row shows a QQ-plot. This shows how much the sample is different from a normal distribution. A normal distribution should be right on top of the diagonal line.

A U-shaped curve indicates skewness (and an upside down U is negatively skewed). Try the exponential distribution.
An S-shaped curve indicates high kurtosis (a backwards S, low kurtosis) Try the Student-t and Uniform distributions.

As \(M\) (the number of cards averages to get to the white hat) gets bigger, the distribution should get closer and closer to the normal distribution.

Take home

Even if the underlying data aren’t normal, the distribution of the means of various groups should be close to normal.
Close depends on the sample size.
A bigger sample is needed if the data are highly skewed (expontential and gamma) or leptokurtic (exponential and Student t).