# Script for generating sampling variance of means

# Wipes any pictures you generated
close.screen(,all.screens=T)

# Makes 4 Panels for plotting
split.screen(c(2,2))

#Assigns to Screen 1 (of the two by two panel set)
screen(1)

# Helper function to make R simulate X observations and calculate the mean

mean_rand<-function(X){mean(rnorm(X,0,1))}

# We are going to explore how the number of samples affects the estimate of mean

# I use the sapply function, which is part of the apply family
# apply is a very powerful tool which can take the place of making loops in R
# Loops are terrible in R--they are very very slooooooooow
# Basic principle of apply is that R applies the function you specify to vector or table you specify
# For more information on apply/sapply ?apply and ?sapply

# We will plot the output of this function: sapply(rep(10,500),mean_rand)
# The rep function is telling R to replicate the value 10, 500 times 
# which looks like: [10 10 10 10 .... 10]

# sapply is then taking that vector and applying the function mean_rand to each element
# So we have 500 elements of value 10, so R will generate mean_rand(10) 500 times



# Plot.density plots a density object. Density evaluates your data (similar to histograms)
# rnorm randomly creates data following a normal distribution with default mean and stand. dev.
# the only argument is the size of simulation 10 in this case, then increasing

plot(density(sapply(rep(10,500),mean_rand)),xlab="Estimate of Mean",ylab="Frequency",main="10 sample size mean estimate",xlim=c(-1,1))

screen(2)
plot(density(sapply(rep(100,500),mean_rand)),xlab="Estimate of Mean",ylab="Frequency",main="100 sample size mean estimate",xlim=c(-1,1))

screen(3)
plot(density(sapply(rep(1000,500),mean_rand)),xlab="Estimate of Mean",ylab="Frequency",main="1,000 sample size mean estimate",xlim=c(-1,1))

screen(4)
plot(density(sapply(rep(10000,500),mean_rand)),xlab="Estimate of Mean",ylab="Frequency",main="10,000 sample size mean estimate",xlim=c(-1,1))