One example of the Central Limit Theorem (CLT) is the rolling of dice. Consider rolling a single fair six-sided die, which has an equal chance of landing on any number between 1 and 6. The distribution of the outcomes is uniform, meaning that each outcome has the same probability of occurring.
Now, suppose we roll n independent dice and calculate their average. The Central Limit Theorem tells us that as n gets larger, the distribution of the sample mean of the dice rolls will become increasingly normal. In particular, the sample mean will be approximately normally distributed with mean 3.5 (the average of the numbers on a die) and standard deviation 1.70/sqrt(n) as n gets larger.
For example, suppose we roll 10 dice and calculate their average. The distribution of the sample mean of the dice rolls will be approximately normal with mean 3.5 and standard deviation 0.54. As we increase the number of dice rolled, the distribution of the sample mean will become increasingly concentrated around the mean of 3.5 and the standard deviation will decrease proportionally to the square root of the sample size.
This example illustrates how the Central Limit Theorem allows us to use the normal distribution to approximate the distribution of the sample mean, even when the underlying distribution of the individual variables is not normal.