Is Central Limit Theorm true for other statistics

[noetsi]

Which reminds me of my doctoral program (management) in which there is signficant debate on all topics which never get resolved. Although no expert of course I am struck at just how often and how strongly statisticians disagree with each other.

The issue raised in the readings was less skew and kurtosis (nearly all likert data will be non-normal) but if there was close to zero responses in any one category when you just had 4. Because below 4 categories there was serious trouble with SEM. But of course looking for skew, and possibly transforming it I know is always a good idea.

Unfortunately when you have 40 plus variables running tests/plots for it is painful.

[ltleung]

the distribution of the mean is normal. sample variance follows chi-sq distribution hence, sample SD should follow chi distribution since the squared of a chi distributed rv follows chi-sq distribution. For the sample range, it is a lot more complicated. But it can be understood in the following way. Fisher–Tippett theorem says that the maximum follows gev distribution (generalised extreme distribution) n hence the minimum should also follow gev distribution since min(x)= -max(-x). Now the sample range is the difference between these two. If the shape parameters for these two distributions (the max n min) are zero (which is rarely the case) then the difference between them (the range) follows logistic distribution.

[Dason]

the distribution of the mean is normal. sample variance follows chi-sq distribution hence, sample SD should follow chi distribution since the squared of a chi distributed rv follows chi-sq distribution. For the sample range, it is a lot more complicated. But it can be understood in the following way. Fisher–Tippett theorem says that the maximum follows gev distribution (generalised extreme distribution) n hence the minimum should also follow gev distribution since min(x)= -max(-x). Now the sample range is the difference between these two. If the shape parameters for these two distributions (the max n min) are zero (which is rarely the case) then the difference between them (the range) follows logistic distribution.

You're making a lot of unstated assumptions in your paragraph there. Enough that I would advise most people to disregard it unless they already know what assumptions you're making.