Fundamental proof of signal averaging noise reduction

I am trying to experimentally show the mathematical proof of averaging n samples for a reduction of noise of sqrt(n)

https://en.wikipedia.org/wiki/Signal_averaging

my code is very simple, but the ratio of improvement does not converge towards a fixed ratio. I get random improvements even with very large numbers of averaging

In this example I am using a 16x sampling improvment, I was expecting a 4x reduction in noise according to the theory.

clear all

close all

clc

L1 = 1000

L2 = round(L1*16);

Noise = randn(1,L2);

PNoise_1 = mean(Noise(1:L1))

PNoise_2 = mean(Noise(1:L2))

Noisereduction = PNoise_1/PNoise_2

Answer 1

[Jeff Miller]

Your code doesn't really measure noise reduction in the right way. Noise reduction refers to variability across many repeated means, something like this (untested):

nIterations = 100;
sampleMns = zeros(nIterations,2);
for iIter=1:nIterations
    Noise = randn(1,L2);
    sampleMns(iIter,1) = mean(Noise(1:L1));
    sampleMns(iIter,2) = mean(Noise(1:L2));
end
sd1 = std(sampleMns(:,1));
sd2 = std(sampleMns(:,2));
Noisereduction = sd1/sd2;

[Christopher Meehan]

Thanks Jeff! I did test this code and it does converge towards the theoretical