# [R] Physical or Statistical Explanation for the "Funnel" Plot?

Jason Rupert jasonkrupert at yahoo.com
Fri Mar 27 03:44:04 CET 2009

```The R code below produces (after running for a few minutes on a decent computer) the plot shown at the following location:

http://n2.nabble.com/Is-there-a-physical-and-quantitative-explanation-for-this-plot--td2542321.html

I'm just taking the mean of a given set of random variables, where the set size is increased.  There appears to be a quick convergence and then a pretty steady variance out to a set size of 10,0000.

I'm just wondering if there is a statistical explanation out there for this convergence and it has been explored further.  Thanks again.

# First case
N<-100000
X<-rnorm(N)
step_size<-1

# Groups
g<-rep(1:(N/step_size),each=step_size)

# The result
tmp_output<-tapply(X[1:length(g)],g,mean)

length_tmp_output<-length(tmp_output)
tmp_x_vals<-rep(step_size,length_tmp_output)
plot(tmp_x_vals, tmp_output, xlim=c(0,10000))
#points(tmp_x_vals, tmp_output)

for(ii in 1:10000)
{
step_size<-ii

# Groups
g<-rep(1:(N/step_size),each=step_size)

# The result
#tmp_output<-tapply(X,g,mean)
tmp_output<-tapply(X[1:length(g)],g,mean)

length_tmp_output<-length(tmp_output)
tmp_x_vals<-rep(step_size,length_tmp_output)
points(tmp_x_vals, tmp_output)
}

```