To see for yourself how the central limit theorem works, let\'s say we have a no

Question

To see for yourself how the central limit theorem works, let's say we have a normal distribution (with mean =100 and standard devation = 20). Let's generate some random samples of various sizes from this distribution. We can do this in excel using =norm.inv(rand(),100,20) and it will randomly generate numbers from this distirbution.

I generated four samples of size 5, 10, 20 and 30, and got the means of 124 (n=5); 91 (n=10); 105 (n=20); 103 (n=30). If I continue to increase the sample size, my average values should converge to the mean of 100.

Now you try. Pick a distribution and generate some sample sizes to prove this to yourself. Post and discuss your results.

Explanation / Answer

In R,

So as sample size increases the sample mean follows to population mean = 100

set.seed(5)
s1=rnorm(5,mean=100,sd=20)
mean(s1)
set.seed(5)
s2=rnorm(10,mean=100,sd=20)
mean(s2)
set.seed(5)
s3=rnorm(15,mean=100,sd=20)
mean(s3)
set.seed(5)
s4=rnorm(20,mean=100,sd=20)
mean(s4)
set.seed(5)
s5=rnorm(25,mean=100,sd=20)
mean(s5)
set.seed(5)
s6=rnorm(30,mean=100,sd=20)
mean(s6)
set.seed(5)
s7=rnorm(40,mean=100,sd=20)
mean(s7)
set.seed(5)
s8=rnorm(50,mean=100,sd=20)
mean(s8)
set.seed(5)
s9=rnorm(100,mean=100,sd=20)
mean(s9)
set.seed(5) > set.seed(5)
> s1=rnorm(5,mean=100,sd=20)
> mean(s1)
[1] 104.2784
> set.seed(5)
> s2=rnorm(10,mean=100,sd=20)
> mean(s2)
[1] 98.42297
> set.seed(5)
> s3=rnorm(15,mean=100,sd=20)
> mean(s3)
[1] 96.43686
> set.seed(5)
> s4=rnorm(20,mean=100,sd=20)
> mean(s4)
[1] 94.38884
> set.seed(5)
> s5=rnorm(25,mean=100,sd=20)
> mean(s5)
[1] 99.37997
> set.seed(5)
> s6=rnorm(30,mean=100,sd=20)
> mean(s6)
[1] 100.226
> set.seed(5)
> s7=rnorm(40,mean=100,sd=20)
> mean(s7)
[1] 101.3566
> set.seed(5)
> s8=rnorm(50,mean=100,sd=20)
> mean(s8)
[1] 101.2987
> set.seed(5)
> s9=rnorm(100,mean=100,sd=20)
> mean(s9)
[1] 100.6327
> set.seed(5)

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