# Central Limit Theorem: Sample Size or Number of Samples?

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.

• 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. Feb 19, 2021 at 6:40
• 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. Feb 19, 2021 at 6:41
• 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). Feb 19, 2021 at 7:54
• @seanv507: That seems like a full answer, can you expand it to an answer? Feb 19, 2021 at 12:29