# Normality for population and sample [closed]

I'm new to statistics and got to the point normal distribution. can we define normality for samples too or they're just for population?

• Normality applies to a distribution, be it the sample or the population. My guess is you could get information that's more useful to you if you add details (e.g. why do you want to know this? what prompted this question?) Commented May 27, 2021 at 1:11
• I am reading a statistics book and got to the THE DISTRIBUTION OF MEANS and the book said "If random samples of size n are drawn from a normal population, the means of these samples will conform to normal distribution" and means of samples is not a population for sure. that's the reason. sorry for poor English
– vhd
Commented May 27, 2021 at 15:54
• @Dave The CLT is not involved in the quoted statement: it follows from the normality of linear combinations of multivariate normal variables.
– whuber
Commented May 27, 2021 at 19:47

A sample cannot be normal because it is finite, so it has a minimum and maximum. A true normal distribution has no such bounds and ranges from $$-\infty$$ to $$\infty$$.

(There are other arguments for why a sample cannot be normal, and my explanation is a bit superficial, but that is the most straightforward way to describe it, I believe.)

While the sample itself cannot be normal, it can be drawn from a normal population. In that case, it is common to use a bit of slang and call the sample "normal".

• so what about The Central Limit Theorem that says if the sample size is 30 or more the sample mean is "approximately normally distributed"? in here it's claiming normality for sample.
– vhd
Commented May 26, 2021 at 17:51
• @shahabshahroodi That is a case of the slang I mentioned.
– Dave
Commented May 26, 2021 at 17:57
• thanks. just one more question. what if our population is a sample of a bigger population itself?
– vhd
Commented May 26, 2021 at 17:59
• @shahabshahroodi That will depend on what you mean, but a finite population cannot be normal, either, for the same reason I posted.
– Dave
Commented May 26, 2021 at 18:03