Right it depends
The 30 rule is not a good one.
There are distributions where the Central Limit Theorem does not hold.
What is important is the frequency and size of extreme values.
Basically the more likely and the larger these values are the more
cases you need to assume normality.
On the other hand if a distribution has fixed limits and a small range
of values it actually very quickly tends to normality, try this with
the following applet
http://www.chem.uoa.gr/applets/appletcentrallimit/appl_centrallimit2.html
and selecting number 4 which is about the worst option available or
number 8 which is where there is normally problems, both with 10 are
producing pretty similar to normal values.
Jean
On 29 August 2012 13:32:13, Margaret MacDougall wrote:
> Hello
>
> I refer in particularly to the following statement of the Central
> Limit Theorem:
>
> The Central Limit Theorem is an important theorem used in mathematical
> statistics used to make inferences about populations based on limited
> amounts of information.
> The principle is that if you have n random variables, Y1, Y2,...,Yn
> each with mean (expected value) u; and each with some variance s^2, then
> U = sqrt(n)*((Y - u)/s^2), where Y is the average of the realised
> value of these n random variables
> will converge to the standard normal distribution as n approaches
> infinity. (Source: http://everything2.com/title/Central+Limit+Theorem)
>
> Clearly the term 'converge' is of some relevance here, both in terms of
> a) the size of the samples used to generate the "realised" values of
> any given variable *in the event that this variable pertains to a
> non-Normal population*;
>
> and
>
> b) the number of random variables under consideration.
>
> I understand that where there is departure from Normality under 1),
> one rule of thumb is to require that:
>
> the sample size be at least 30 ... (1)
>
> and that more generally one would require
>
> the number of variables under b) to be at least 30...(2)
>
> I would be most interested to learn whether there is a tried and
> tested procedure in place to estimate just how much U ought to deviate
> from the Standard Normal distribution when at least 1 of conditions
> (1) and (2) above are violated.
>
> One reason for asking is that I have seen the Central Limit theorem in
> the above form appealed to in a case where the samples under a) are
> considerably smaller than 30 and insufficient justification has been
> provided that the corresponding populations are Normally distributed.
>
> Many thanks
>
> Best wishes
>
> Margaret
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