# Why is the mean of a standard normal distribution zero? [closed]

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?

• By definition: it had to have some value! So what kind of answer are you hoping to get? – whuber Jul 3 '18 at 15:21
• If the mean of your data is not 0 then it is not distributed as a standard normal. The SND is defined as a Normal distribution with mean 0 and sd 1. – Peter Flom - Reinstate Monica Jul 3 '18 at 21:37

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

The more n tends to infinite, the more close you're from 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


An the mean is infact close to the expected (3): 2.93444 (with a relative small sample)

• So the standard normal distribution is somewhat hypothetical? – Asma Rahim Ali Jafri Jul 3 '18 at 15:58
• every distribution, in this terms, is hypothetical. You assume n to infinite. But you know that in the real world, you are working with something "close", the sample. – RLave Jul 3 '18 at 16:00
• +1, in response to these comments, I would add that the distribution (standard normal or another choice) describes the "population" while the observed data is a "sample". Thus the Expected Value is the "population mean" while the mean you are calculating (i.e. a Statistic) is the "sample mean". – knrumsey Jul 3 '18 at 16:07

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.

• I guess the expected value and the mean are taken as the same thing in this case, as far as Wiki says. – Asma Rahim Ali Jafri Jul 3 '18 at 15:59
• The expected value and the population mean are the same thing. The expected value and the sample mean are not. If you think Wikipedia says otherwise, please indicate where, precisely. – Glen_b -Reinstate Monica Jul 4 '18 at 0:58