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In the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalised sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The Question is how could this be proven as it looks very intimidating at times. Why is 30 considered the minimum sample size in some forms of statistical analysis? Is there any rationale for this.

Dear Ransingh, There is a very good animation that you can see at http://onlinestatbook.com/stat_sim/sampling_dist/ You can change the distribution type of independent random variables from different distributions, and see how the averages (or sums) become normally distributed. As the sample size approaches 30, the curve becomes quite normal. Of course, beyond 30, it will become even more beautifully normal but the normality test passes very well at 30. The differences you see in the match of the obtained curve with the perfect bell curve between sample sizes of 5 and 6 are much bigger than what you see between 30 and 31. As the perfection towards normality grows very slowly with an addition of every single number after 30, the number 30 is considered as a reasonably good minimum size. Of course, the number 30 is just a rule of thumb and one could always take more to be safer.