Why is the mean of value of gaussian white noise not zero?

Helge (view profile)

on 8 Aug 2019
Latest activity Commented on by Helge

Helge (view profile)

on 8 Aug 2019
Accepted Answer by Jos (10584)

Jos (10584) (view profile)

I create a vector of randomly selected values from a normal distribution of mean = 0 and std = 1. I calculate the mean and expect as a result zero. And I calculate the mean of the absolute value and I expect as a result but instead I get a non-zero mean and a result of for the mean of the absolute value.
How is that?
>> noise=randn(1,1000000);
>> mean(noise)
ans =
0.0013
>> mean(abs(noise))
ans =
0.7985

Renato SL

Renato SL (view profile)

on 8 Aug 2019
For what you expect to be the value of the mean of the absolute value, I think the discussion here explains it well.
Helge

Helge (view profile)

on 8 Aug 2019
Thanks a lot, that was the discussion, I couldn't find this morning!

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Answer by Jos (10584)

Jos (10584) (view profile)

on 8 Aug 2019

The numbers are randomly drawn from a normal distribution. Although this underlyin distribution has a mean of 0 and a standard deviation of 1, this does not mean that your selected numbers should have this mean and standard deviation! Think about it when you draw only a few numbers from this distribution. Would you expect the mean to be 0 all the time?
To make the mean and std of your selection of numbers (almost) equal to 0 and 1, respectively, you can do:
R = randn(1,100) ;
R = (R - mean(R))./ std(R) ;
mean(R), std(R)

Walter Roberson

Walter Roberson (view profile)

on 8 Aug 2019
Flip a fair coin many times. The statistical mean is that half come out Heads and half come out Tails, so statistically for N throws, number of heads = N/2 and number of tails = N/2 . Okay, now flip the coin exactly once, N = 1. The expected number of heads is then 1/2 and the expected number of tails is 1/2. But the single coin flip was either all heads or all tails, not 1/2 a head and 1/2 a tail. Why wasn't the result of a single flip 1/2 a head and 1/2 a tail ?
Helge

Helge (view profile)

on 8 Aug 2019
I get your points. I just thought, that if I would do the experiment 1 mio times, I would get closer to zero than 0.0013, but of course this depends also on the standard deviation, which I set to 1.