Question

I would like to see a proof of the Central Limit Theorem that applies to a simple probability dice scenerio, say rolling a 6 x amount of times. The goal is to help me understand the theorem with a simple example. Thanks!

Answer 1

The central limit theorem is exactly what the shape of the distribution of means will be when we draw repeated samples from a given population. Specifically, as the sample sizes get larger, the distribution of means calculated from repeated sampling will approach normality.

Dice are ideal for illustrating the central limit theorem. If you roll a six-sided die, the probability of rolling a one is 1/6, a two is 1/6, a three is also 1/6, etc. The probability of the die landing on any one side is equal to the probability of landing on any of the other five sides.

The population mean for a six-sided die is (1+2+3+4+5+6)/6 = 3.5 and the population standard deviation is 1.708.

We have randomly selected 30 samples that is in one sample a dice is thrown 50 times. Then, we have calculated mean for each sample.

Thus, if the theorem holds true, the mean of the thirty averages should be about 3.5 with standard deviation 1.708/ 30 = 0.31. Using the dice we “rolled” using R, the average of the thirty averages is 3.49 and the standard deviation is 0.30, which are very close to the calculated approximations