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Good evening everyone,

I am currently doing a self-study on CLT and was working on an exercise which asks if the following statement is true or false

CLT guarantees that the population mean is normally distributed whenever sample size is sufficiently large.

I can understand that this statement is False, but I am pretty weak with the concept, thus will like to take this opportunity to validate my thoughts.

CLT does not guarantee that the population mean is normal, instead it allows us to treat a population as normal if a given population size is large enough. Also, the size of the sample will not affect the actual distribution of the population, but rather simply gives us a more accurate approximation.

Appreciate if seniors and exports here can give me some guidance.

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  • $\begingroup$ "if seniors" ... I'm not sure age confers any special value; "and exports here" ... and you probably mean 'experts', but you don't need an expert for basic material $\endgroup$
    – Glen_b
    May 4, 2014 at 3:38

2 Answers 2

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CLT does not guarantee that the population mean is normal, instead it allows us to treat a population as normal if a given population size is large enough.

No it doesn't. The population doesn't change shape when you take a larger sample

Also, the size of the sample will not affect the actual distribution of the population, but rather simply gives us a more accurate approximation.

So you understood that already, in which case your previous sentence is mystifying. How can you treat a population as normal when you now state that it's still as non-normal as ever! What did you mean to say?

Even more basically, what does the CLT actually say?

Write that out - not some handwavy approximation, but an actual, basic, statement of a simple version of theorem itself, such as one you might prove via the MGF - and then see what you can say.

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(UPDATED and corrected version after useful comment).

The first passage refers correctly to the "mean" which is a subject of investigation in the context of CLT, but totally incorrectly to the "population mean" -which is not what the CLT examines. The CLT examines, among other things, the sample mean, viewed as a random variable. So even if it said "the CLT guarantees that the population mean is not normal ..." it would still be a wrong statement because the CLT does not make statement about the population.

The second passage is a great example of devious writing: Each of its two sentences starts with a correct statement, and ends with a wrong statement (attempting to manipulate the perceptions of the reader so as he accepts both).

1st sentence

(NOT WRONG) "CLT does not guarantee that the population mean is normal (NOT WRONG)

(WRONG) instead it allows us to treat a population as normal if a given population size is large enough (WRONG)

Again, the CLT does not deal in populations.

2nd sentence

(CORRECT) Also, the size of the sample will not affect the actual distribution of the population (CORRECT),

(VAGUE-WRONG) but rather simply gives us a more accurate approximation.(VAGUE-WRONG)

Who is giving us this approximation? The CLT? or the large sample size? And "more accurate" compared to what other approximation? None other was mentioned. But if again, it implies "approximation of the population distribution" then again it is wrong - the Central Limit Theorem does not deal in populations.

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    $\begingroup$ Why is the first statement correct in essence? I thought that the population mean is supposed to be a constant whose value may be unknown to the statistician. How can this constant have a normal distribution? And what does the value of the population mean have to do with the size of the sample? $\endgroup$
    – Dilip Sarwate
    May 4, 2014 at 3:55
  • $\begingroup$ @DilipSarwate. Thank you. You are right, and I just corrected my answer. It was very late when I wrote it and I read "sample" where the statement wrote "population" -how more careless can one be?! $\endgroup$
    – Alecos Papadopoulos
    May 4, 2014 at 13:02

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