# Random sampling distribution question

I have been reading this article on the random sampling distribution (RSD) and non-normal distributions. Basically, if I understand it correctly, the article proposes that the RSD of the mean of a sample of different size can be compared against the normal distribution using a comparison of kurtosis and skewness.

The logic of the author is that

as the sample size increases, the distribution of the means gets more and more normal. So with non-normal distributions, the sample size needed to detect the change in the average, which we are looking for, also has to be large enough so that the RSD is reasonably approximated by the normal distribution.

In figure 3 row 1, for example, I read that taking the RSD of the mean of 15,000 samples of 5 has a skewness of 0.207, kurtosis of 0.107 etc.

What I don't understand is how the number of samples the author draws in Figure 3 and Figure 4 under column 'n' (15,000 and 1,000) is chosen. It seems that they are chosen for convenience (i.e. 15,000 doesn't work so the author uses 1,000) instead.

I must be missing something. Can anybody explain how to choose n (i.e., the second column in the two tables, not the Variable column / row names)?

• Is there any way of pasting into your question more of the context necessary to understand & answer it? People may not want to navigate elsewhere & read an article just to answer your question for you. In addition, we want this thread to remain valuable even if the link goes dead. – gung Apr 27 '16 at 1:38
• It is noteworthy that you are reading material written by a non-statistician. Have you considered consulting related threads on our site for comparable information? That might be a little less frustrating than trying to guess what that article was trying to say. – whuber Apr 27 '16 at 2:05
• n denotes at the same time the number of variables and the sample size? Ouch... – dv_bn Apr 27 '16 at 4:32

The number of simulations 1000 and 15000 are arbitrary. The author is using Monte-Carlo simulation to approximate the distribution of the sample mean. He wants the number of simulations to be large enough so that the approximation is reasonable, and he probably picked a number just big enough so that the curves looked sorta smooth if you don't zoom in too much.

### Point of the article...

1. The sample mean itself is a random variable.
2. The Central Limit Theorem proves that (under certain regularity conditions) the sample mean converges in distribution to the normal distribution as the sample size goes to infinity.
3. We can do some simple, ad-hoc simulation and observe results consistent with the Central Limit Theorem:
• The distribution of the sample mean looks more normal as sample size increases.
• As sample size increases, skewness and kurtosis of the sample mean converge to what you would expect if the sample mean were normally distributed.

When learning probability/statistics stuff, you should be aware that there are tons of practitioners (possibly me included!) that say things that are imprecise or sometimes plain wrong. This can be horribly confusing.

I'd try to stick to sources from well tested and established books, teachers, professors, lectures etc...