The square-root rule

The secret purpose of the coin flip experiment was to illustrate the "square root rule." It says that if you expect to get 100 heads, then don't be surprised if the number you get is different by this by approximately the square-root of 100, i.e. by 10. So any number between 90 and 110 is very probable. In fact, if enough people do it, you will probably get some results as far away as 80 and 120. (That is, twice the expected value.) But it becomes very unlikely that you will go as low as 70 or 130, because that is three times the expected variation.

If you expect 2 comets to hit the earth, then don't be too surprised if 1 does or if 3 does. (The square-root of 2 is about 1.4.) In fact, there is a reasonable chance that 4 might, or zero.

With 20000 coin tosses, you expect to get 10000 heads. The square root of 10000 is 100. So you expect to see 10,000 but you wouldn't be surprised if you saw 9900 or 10100.

Suppose half of the US voters intend to vote for Gore, and half intend to vote for Bush. But we don't know that yet, so we take a public opinion poll. We interview 200 people. This is just like flipping coins, because the people we pick are random, and we have an equal chance of getting a Gore voter or a Bush voter. In the end, we publish our results. But unlike the class (which did the experiment many times, one for each student), we do it just once -- i.e. just one poll. Then it is extremely likely that you will get numbers like 90/110 instead of 100/100. If we gete 90/110 we would say that Gore is getting 45% of the vote, and Bush is getting 55%. He has a "10 point lead." But it is nonsense.

A proper way to report the result is to say that he has a 10 point lead, but the uncertainty in the pole is 10 points.

How many people would we have to survey to be able to say that the poll has only a 2% chance of error? This is a good topic for a discussion section.