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

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]

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

[Helge]

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

Answer 1

[Jos (10584)]

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]

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]

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.