What is the "Mean?"

The "mean" is a type of average value. It is a summary statistic obtained by adding up all of the data values and then dividing by the number of data points. For example; the mean of 1, 2, 3, & 4 is 2.5 [ (1+2+3+4)/4=> 10/4=> 5/2 ]. But what does this number tell us about the data?

The "Mean" as Expected Payoff

The "mean" is sometimes referred to as the expected payoff (or expected value). This gives a good hint as to the meaning of "mean." Many games of chance, such as the raffle or lottery, combine a small chance of winning with a huge prize. So what is the most logical "value" to assign to our raffle ticket? The answer is the mean (or expected payoff). For example, if we had a one-in-a-million chance of winning the $10,000 prize the expected payoff is only $0.01 ($10,000 / 1,000,000=1/100). So if we are charged more than a cent for this raffle ticket we would be better off passing up the offer to play (unless we get enough enjoyment out of the game to offset the costs).

The "mean" might also tell us how to fairly divide something up. For example, let's say you brought 2 cans of soda pop over to your brother's house. He already had 2 cans, and your sister brought 5 more cans. How might you fairly divide them up? If you take the "mean" you discover that each person can drink 3 cans of pop [ (2 + 2 + 5)/3=> 9/3=> 3 ]. The "mean" has the very nice property that all you need to know is the number of data points and the total value (sum) of these data points. So, with our soda pop example, all you need to know is that there are 9 cans of pop and 3 people (it doesn't really matter who brought what). Often we don't know exactly "who brought what", but we usually know the total value and the number of people who contributed to that value. Hence, the "mean" is often the only summary statistic we can realistically calculate...