Why is the mean of a standard normal distribution zero? Supposedly I have a set of values, normally distributed, if i find their arithmetic mean (which i am supposing is the mean in context for a normal distribution) there must be a value that will be an average of all the values throughout the distribution. 
Then why is the mean of SND = 0?
 A: The sample mean is never truly zero, it's close though.
If you could measure the mean of an infinite sample from a Standard Normal Distribution, that would be zero, by definition.
This is what you get if you simulate 10 values from a standard distribution:
[1]  1.2240818  0.3598138  0.4007715  0.1106827 -0.5558411  1.7869131  0.4978505 -1.9666172  0.7013559 -0.4727914

and the mean is: -0.4245589
If n = 100, then the mean is: 0.02161711
For n = 10000 , then: -0.002404524
As n gets bigger, you get closer to the truth (ie: mean = 0).

there must be a value that will be an average of all the values throughout the distribution

Zero is a value, the same principle will hold if you simulate from a distribustion with mean = 3, for example.
[1] 4.208902 3.431187 2.652505 1.998890 3.677987 3.498629 2.734247 4.564173 1.229492 2.982534

And the mean is infact close to the expected (3): 2.93444 (with a relative small sample)
A: You may be confusing the expected value of the standard normal distribution, which is indeed zero, with the mean of a sample from the standard normal, which can take (literally) any value. However, note that if you take larger and larger samples, their means will converge towards the expectation.
