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Re: [Phys-l] Basic statistics



When I said "carefully stated" I was trying to exclude puns. Tim makes a pun on the phrase "applies to." While his statement is correct, it doesn't contradict mine, namely, the chi-square distribution does not become normal in the large N limit. The Poisson distribution, by the way, does.
The texts tell us, as I understand them, that the means of vary large samples taken from tinite variance distributions are normally distributed about a central value. I do not know whether there is a proof that relaxes the limiting conditions.
Note, by the way, that the central limit theorem, as quoted, makes not statement about the distribution of variances.
Regards,
Jack




On Fri, 10 Nov 2006, Folkerts, Timothy J wrote:

Jack replied to one of my comments

In my mind, "standard deviation of the mean" is a perfectly fine, unmixed term.

But you seem not to agree. See below:...
ok, so who, among "NIST, CDC, professional statisticians, etc." uses it?

I'm not saying that it is the best terminology, nor even that it is common terminology. I was merely trying to point out that it is perfectly possible and reasonable to calculate "the standard deviation of ______________" where "___________" could be anything - "heights of students in a class", or "measurements of 'g'", or "the results of rolling one die" or " the mean value when rolling 10 dice". In other words, if you have a set of values that happen to be the means of some sets of data, you can calculate the standard deviation of the means.

From my own experience, "standard error" is indeed used more commonly when talking about the standard deviation when applied to a value (like the mean) calculated from a sample.


As already pointed out, the central limit theorem does not apply to all
distributions. You must state the theorem carefully. In the case of the
chi-square distribution it is easy to demonstrate by example with MathCad
or a similar program that the large N limit is not Gaussian.

But the Central Limit Theorem DOES apply to the chi-square distribution. Certainly the large X limit of the chi-square distribution with X degrees of freedom does not approach the normal distribution. However, the large Y limit of the sum of Y random variables drawn from a chi-square distribution with X degrees of freedom does aproach a normal distribution. And that is how the CLT would be applied here.



Also, Ludwik said:
Would it be OK to say that the CLT applies to any distribution of x
that is confined to a region between x1 and x2? By this I mean that the
probability density function is zero a all t x<x1 and at all x>x2. That
would cover nearly any practical distribution I can think off.

That should be correct. If the distribution HAS a mean and standard deviation, then the CLT applies (as I understand it). I think it is obvious that any bounded distribution must have a finite mean and standrad deviation.


Tim F


--
"Trust me. I have a lot of experience at this."
General Custer's unremembered message to his men,
just before leading them into the Little Big Horn Valley