The Central Limit Theorem

Question

I’m in grad school and taking statistics because I have to. I’m a total right-brained person. Art is what I do for fun. I am having a terrible time trying to understand this theorem and finding the mean and standard deviation of the distribution of sample means. Everything I’ve found so far is not helpful. Please help😭

In: Mathematics

Answer 1

Imagine you have a big bag of different colored candies, and you want to know the most common color in the bag. But instead of looking at all the candies at once (because that would take a long time), you decide to take small handfuls of candies and see what color is most common in each handful.

Now, if you take just one handful, it might not show the true most common color because you could just get a handful with mostly one color by chance. But if you take lots and lots of handfuls and look at the most common color in each one, you’ll start to notice a pattern. The more handfuls you take, the more the pattern you see will look like the true most common color in the whole bag.

The Central Limit Theorem is a bit like that. It tells us that if we take lots of small samples (handfuls) from a big group (the candy bag) and find the average (most common color) of each sample, those averages will form a pattern. And this pattern will look like a smooth hill, with the top of the hill being the true average of the whole big group.

So, even if we don’t see all the candies, by taking lots of small samples, we can still get a pretty good idea of what the most common color is in the whole bag!