Simulating the central limit theorem, ending up with plots side-by-side showing the bell-curve distribution of an originalNormal distribution with chosen mean and SD, which will be broad-based, as compared to the distribution of the sample means, also bell-shaped, but narrower at the base.

```
set.seed(2)
CentralLimit <- function(n){
means_of_samples <<-numeric()
for (i in 1:n){
means_of_samples <- c(means_of_samples,mean(rnorm(sampsize,mean,sd)))
}
samples_means <<- means_of_samples ##The << makes the object available outside the function.
sd(samples_means) ## although only the sd of the samples_means will be returned
}
```

Now were are going to decide on a sample size, mean and standard deviation:

```
sampsize <- 100
mean <- 0
sd <- 1000
CentralLimit(1000)
```

`## [1] 105.7034`

```
# [1] 105.7034 Very close to theoretical: sd/sqrt(n) 1000/sqrt(100) = 100.
head(samples_means)
```

`## [1] -30.69816 29.21465 142.93250 314.10070 -147.08809 252.46291`

```
# [1] -30.69816 29.21465 142.93250 314.10070 -147.08809 252.46291
# Building up the population distribution using the dnorm, which will give the pdf value for every quantile fed into it:
x <- seq(-(5 * sd),(5 * sd), length = 1000)
y <- dnorm(x, mean, sd)
```

Plotting,

```
par(mfrow=c(1,2),oma=c(0,0,1,0))
plot(x,y, type="l",ylab="pdf",xlab="population",
frame = FALSE, col="dark blue", lwd=2, yaxt="n")
# And now the histogram of the sample means distribution:
hist(samples_means, freq = F, yaxt="n", main = "",
xlim=c(-(5 * sd(samples_means)),(5 * sd(samples_means))))
x <- seq(-(5 * sd(samples_means)),(5 * sd(samples_means)),
length=1000)
curve(dnorm(x,mean=mean(samples_means),
sd=sd(samples_means)),col="red",lwd=2,add=T)
title("CENTRAL LIMIT THEOREM - MONTE CARLO",outer=T)
```

For different distributions:

Distribution of sample means of 30 coin tosses with qqnorm plotting for normality and Shapiro-Wilk normality test:

```
tosses <- 30
samsize <- 1000
vector <- NULL
for (i in 1:1000){vector[i] = mean(rbinom(samsize,tosses,0.7))}
head(vector)
```

`## [1] 20.953 20.948 21.136 21.047 21.003 20.975`

`plot(density(vector))`

`shapiro.test(vector)`

```
##
## Shapiro-Wilk normality test
##
## data: vector
## W = 0.99819, p-value = 0.3731
```

```
# We cannot exclude normality.
qqnorm(vector);qqline(vector)
```

Distribution of sample means of \(1000\) samples from an Exponential with qqnorm plotting for normality and Shapiro-Wilk normality test:

```
vector <- NULL
for (i in 1:1000){vector[i] = mean(rexp(1000,5))}
head(vector)
```

`## [1] 0.2013455 0.2057055 0.1892531 0.2037769 0.1904333 0.2068772`

`plot(density(vector))`

`shapiro.test(vector)`

```
##
## Shapiro-Wilk normality test
##
## data: vector
## W = 0.99767, p-value = 0.1713
```

```
# We cannot exclude normality.
qqnorm(vector);qqline(vector)
```

Distribution of sample means of \(5\) observation of a Uniform distr. with qqnorm plotting for normality and Shapiro-Wilk normality test:

```
vector1 <- NULL
for (i in 1:1000){vector1[i] = mean(runif(5))}
head(vector1)
```

`## [1] 0.5280128 0.6665951 0.5537420 0.2750550 0.3204791 0.6542897`

`plot(density(vector1))`

`shapiro.test(vector1)`

```
##
## Shapiro-Wilk normality test
##
## data: vector1
## W = 0.99732, p-value = 0.09689
```

```
# We cannot exclude normality.
qqnorm(vector1);qqline(vector1)
```