mean statistics from 4 samples out of a population normally distributed N(0,1) behaves as N(0,0.5)

mean.statistics.R

library("ggplot2")

# calculate a 1000 iterations for a selection of four sample from 
# a normal population N(0,1)

mean.statistics= sapply(1:1000, function(x) {sum(rnorm(4,0,1))/4})

#base plot, but no functions

p = ggplot(data.frame(mean.statistics), aes(mean.statistics)) +  xlim(-2,2)

# The statistics of the sample mean (four samples),  behaves as N(0,0.5)

p +   geom_histogram(aes(y=..density..), binwidth=0.125, colour='darkgreen') + 
      geom_density(colour='red', aes(mean.statistics)) +
      stat_function(fun=function(x) {dnorm(x,0,0.5)}, colour='yellow')