Now let's go back to the online game, which we considered earlier where you win a small prize with probability 0.2. We gamble n times and count a number of small prizes, and we called this a random variable X. At that time we saw that X has a binomial distribution with n, and p = 0.2. Here are some probability histograms for that binomial distribution. If n = 1, we see that it's a very much skewed histogram, and for n = 10 it still has a long right tail. But for n = 50, the whole thing looks pretty much like a normal curve. This is an example of the famous central limit theorem. It says that when we sample with replacement and n is large, then the sampling distribution of the sample average, or of the sum, or of the percentage approximately follows the normal curve. That means we can use normal approximation to compute probabilities. To standardize, we subtract off the expected value of this statistic, and then divide by its standard error. The name central limit theorem comes from the fact that it has a central place in statistical theory. The reason why the central limit theorem is so important is that it shows that the statistic has a normal distribution no matter what the population histogram is. Let's look back at the distribution of annual household incomes in the United States. We saw that this distribution is very much skewed to the right. So it's very far from normal. The mean of the household incomes was $67,000, and the standard deviation was $38,000. So, if we sample n incomes at random among all the households, then we know that the sample average follows the normal curve, even though the histogram of the incomes itself is far from normal. And to do normal approximation, we need to subtract off the expected value of the statistic, which in this case is the average of all incomes, namely $67,000. And then we have to divide by the standard error of the statistic and we know by the square root law that one is $38,000 over the square root of the sample size. For example, let's take 100 incomes. If n = 100, then square root n is 10, and so the standard error is equal to $3,800. The central limit theorem says that a sample mean follows the normal curve centered at $67,000, with a standard error of $3,800. So the empirical rule says, if we go one standard error above the average, then there's a 16% chance to fall above that number. And one standard error above average is precisely that $70,800 we were looking at. Let's go back to our example about online gambling. We did n gambles, and looked at the number of small prizes which we called X. Remember when counting things, we use labels which have 0s and 1s on them. In that case, every time a small prize comes up, we give it a label 1, and 0 goes to everything else. That means that the number of small prizes, simply equals the sum of these labels. And since we are now looking at the sum, we can use the central limit theorem. To apply normal approximation, we need to find the mean and the standard error of x. So remember when we looked at simulations, we had formulas for the expected value and the standard error. And if you plug in, you'll find that in this case mu = p and sigma is square root p times 1 - p. That was the formula for one gamble. Now we are looking at the sum of n gambles and we had formulas for the expected value and standard error, and these formulas show that the expected value of the sum X = n times p and the standard error is square root n times p times 1 - p. Those are the numbers we use when we standardize x. And remember, we found the same thing when we talked about the binomial distribution.
