[R] Using R to illustrate the Central Limit Theorem
(Ted Harding)
Ted.Harding at nessie.mcc.ac.uk
Fri May 13 15:00:00 CEST 2005
On 13-May-05 Bliese, Paul D LTC USAMH wrote:
> Interesting thread. The graphics are great, the only thing that
> might be worth doing for teaching purposes would be to illustrate
> the original distribution that is being averaged 1000 times.
>
> Below is one option based on Bill Venables code. Note that to do
> this I had to start with a k of 2.
>
> N <- 10000
> for(k in 2:20) {
> graphics.off()
> par(mfrow = c(2,2), pty = "s")
> hist(((runif(k))-0.5)*sqrt(12*k),main="Example Distribution 1")
> hist(((runif(k))-0.5)*sqrt(12*k),main="Example Distribution 2")
> m <- replicate(N, (mean(runif(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(3)
> }
>
> By the way, I should probably know this but what is the logic of
> the "sqrt(12*k)" part of the example? Obviously as k increases
> the mean will approach .5 in a uniform distribution, so
> runif(k)-.5 will be close to zero, and sqrt(12*k) increases as
> k increases. Why 12, though?
The reason is indeed simple! In demonstrating the convergence
of the distribution of mean(k X's) to a Normal distribution,
the reference (i.e. the limiting distribution) is N(0,1), which
has mean 0 and variance 1. Therefore, in comparing the distribution
of mean(k X's) with N(0,1) it needs to be standardised to itself
have mean 0 and variance 1. As you've already spotted, you
standardise for the mean by subtracting 0.5; to standardise
for the variance you need to divide by sqrt(variance(mean(k X's))).
This is sqrt(variance(X)/k). Finally (and this is where the "12"
comes in), the variance of an X uniformly distributed on (0,1)
is 1/12 (left as an exercise for the reader ... ). Hence 12*k.
Best wishes,
Ted.
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Date: 13-May-05 Time: 13:43:54
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