RE: [R] Using R to illustrate the Central Limit Theorem

From: <Bill.Venables_at_csiro.au>
Date: Fri 22 Apr 2005 - 08:49:26 EST


Here's a bit of a refinement on Ted's first suggestion.  

 N <- 10000
 graphics.off()
 par(mfrow = c(1,2), pty = "s")
 for(k in 1:20) {

    m <- (rowMeans(matrix(runif(M*k), N, k)) - 0.5)*sqrt(12*k)     hist(m, breaks = "FD", xlim = c(-4,4), main = k,

            prob = TRUE, ylim = c(0,0.5), col = "lemonchiffon")     pu <- par("usr")[1:2]
    x <- seq(pu[1], pu[2], len = 500)
    lines(x, dnorm(x), col = "red")
    qqnorm(m, ylim = c(-4,4), xlim = c(-4,4), pch = ".", col = "blue")     abline(0, 1, col = "red")
    Sys.sleep(1)
  }

-----Original Message-----
From: r-help-bounces@stat.math.ethz.ch
[mailto:r-help-bounces@stat.math.ethz.ch] On Behalf Of Ted.Harding@nessie.mcc.ac.uk
Sent: Friday, 22 April 2005 4:48 AM
To: Paul Smith
Cc: r-help@stat.math.ethz.ch
Subject: RE: [R] Using R to illustrate the Central Limit Theorem

On 21-Apr-05 Paul Smith wrote:
> Dear All
>
> I am totally new to R and I would like to know whether R is able and
> appropriate to illustrate to my students the Central Limit Theorem,
> using for instance 100 independent variables with uniform distribution
> and showing that their sum is a variable with an approximated normal
> distribution.
>
> Thanks in advance,
>
> Paul

Similar to Francisco's suggestion:

  m<-numeric(10000);
  for(k in (1:20)){
    for(i in(1:10000)){m[i]<-(mean(runif(k))-0.5)*sqrt(12*k)}     hist(m,breaks=0.3*(-15:15),xlim=c(-4,4),main=sprintf("%d",k))   }

(On my slowish laptop, this ticks over at a satidfactory rate, about 1 plot per second. If your mahine is much faster, then simply increase 10000 to a larger number.)

The real problem with demos like this, starting with the uniform distribution, is that the result is, to the eye, already approximately normal when k=3, and it's only out in the tails that the improvement shows for larger values of k.

This was in fact the way we used to simulate a normal distribution in the old days: look up 3 numbers in Kendall & Babington-Smith's "Tables of Random Sampling Numbers", which are in effect pages full of integers uniform on 00-99, and take their mean.

It's the one book I ever encountered which contained absolutely no information -- at least, none that I ever spotted.

A more dramatic illustration of the CLT effect might be obtained if, instead of runif(k), you used rbinom(k,1,p) for p > 0.5, say:

  m<-numeric(10000);
  p<-0.75; for(j in (1:50)){ k<-j*j
    for(i in(1:10000)){m[i]<-(mean(rbinom(k,1,p))-p)/sqrt(p*(1-p)/k)}     hist(m,breaks=41,xlim=c(-4,4),main=sprintf("%d",k))   }

Cheers,
Ted.



E-Mail: (Ted Harding) <Ted.Harding@nessie.mcc.ac.uk> Fax-to-email: +44 (0)870 094 0861
Date: 21-Apr-05                                       Time: 19:48:05
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