 Another important task is to determine whether two populations have different means. For example, do trees grown using fertilizer A produce more fruit than trees grown using fertilizer B? Or do children who've gone through Head Start do better in school than children who don't? Or do people who've purchased a warranty actually save money? To answer this question, we need to find a probability distribution for the difference between means. Now to find a formula for the probability distribution for a difference of means requires some multivariable calculus. So here's a quick version of that derivation. OK, well don't worry about the quick version. With somewhat more work, which we omit, we can show that if x and y are independent and normally distributed, the difference will also be normally distributed. Suppose x and y are independent and normally distributed with mean mu x, and standard deviation sigma x, sigma y. Then the difference will be normally distributed with mean equal to the difference of the means, and standard deviation equal to the square root of the sum of the squares of the standard deviations. So let's take an example. Suppose our heights are normally distributed with a given mean and standard deviation, and the height of a two-year-old tree is also normally distributed with a given mean and standard deviation. Let's find the probability a one-year-old tree is taller than a two-year-old tree. Under the assumption, the heights are independent. So if x is the height of a one-year-old tree and y is the height of a two-year-old tree, we want to find the probability that x is greater than y, or equivalently the probability that the difference is greater than zero. So the difference will be normally distributed where our mean is the difference of the means in the same order, and our standard deviation will be the square root of the sum of the squares of the standard deviations. And so we compute and find. So the difference will be normally distributed with mean and standard deviation, and we can use whatever our favorite device for computing normal distributions to calculate the probability that x minus y is greater than zero will be about 20%.