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The central limit theorem states that if we take a take a large enough sum of random variables, the sum will approach a normal distribution. I am confused about why we focus only on the sample size and not the number of samples. The way I'm thinking about it, when we take a sample of random variables then calculate the sum, that sum is just a singular number (we "compress" the sample into one statistic). If this is correct, then don't we need more samples to get more sums so we can actually plot the distribution? If we only calculate one sum (say with a sample size of 100,000 random variables), then we only have one number which is clearly not enough to create a distribution.

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  • $\begingroup$ Yes we need a collection of samples to get a lot of sums and hence the distribution. But is it really feasible to collect that many samples? We study sample because we have time constraint and we cannot study the whole population. As far as the point of constructing distribution, then we are proving in CLT that sum (a number) is coming from a normal distribution. It is just like drawing a sample point (one point) from a population. We have a population of sums and we draw only one and we know that will follow normal. $\endgroup$ – userNoOne Feb 19 at 6:40
  • $\begingroup$ To make the point clear, compare it with drawing a sample (one point) from a height of all the persons in your locality. We know that height follows a normal distribution and hence we will say that this one sample point you drew is coming from a normal population. $\endgroup$ – userNoOne Feb 19 at 6:41
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    $\begingroup$ the CLT is just telling you how the sample is distributed. Just as if I tell you the throws of a die are distributed equally with probability 1/6 - how you check this is a different issue (ie by throwing the die multiple times). $\endgroup$ – seanv507 Feb 19 at 7:54
  • $\begingroup$ @seanv507: That seems like a full answer, can you expand it to an answer? $\endgroup$ – kjetil b halvorsen Feb 19 at 12:29
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the CLT is just telling you how the average of the sample is distributed. Just as if I tell you the throws of a die are distributed equally with probability 1/6, or that a particular coin is fair (50% heads, 50% tails)

how you check this is a different issue eg by throwing the die multiple times and creating a histogram.

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Actually, in practice, when we conduct a survey/experiment. It is costly and takes time to reconstruct the same thing with the same controlled factors many times. Therefore, conducting one time with a large enough sample size will produce the desired results, where we can test the hypothesis under the CLT. Furthermore, some hypotheses based on observed data (eg economic and social data) cannot be reproduced. So, the sample size will give us more advantages than the number of samples.

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