 Welcome back for our third video here dealing with the problem of apportionment. We've discussed four methods so far, the methods proposed by Hamilton, Jefferson, Adams, and Webster. And we're going to end up with the method that is in use today by Congress to apportion representatives. It's called the Huntington Hill method and it's been in use since the 1940 census. It's sometimes also called the method of equal proportions, but the reason it's called the Huntington Hill method is because it gives credit to the two mathematicians who came up with this method. So how does it work? It's similar to Adams, Jefferson, and Webster's method in that we have a modified divisor. It's just one last new way of rounding. We incorporate rounding according to the value of the geometric mean of the lower and upper quotas for any particular standard quota. And that's a mouthful of mathematical terms. So I'm going to try to break that down for you. First of all, if the lower quota were in and the upper quota would then be in plus one because it would be one larger. If the lower quota were two, the upper quota would be three, two plus one. Then we find the geometric mean of those two numbers by multiplying them together, two times three, and then take the square root of that. That value is called the geometric mean. And what Huntington and Hill suggested is that if, for any particular standard quota, if it were less than the value of the geometric mean, we round down. And if it's larger than the value of the geometric mean, we round up. So it helps to have a table of some of these geometric means so you can get an idea of what we're dealing with. So if you had a number between one and two, the geometric mean of one and two, I've multiplied one times two and taken the square root, you get 1.414. So if a state, for instance, had a quota of 1.40, they would stay at one because it's less than 1.414. But if they had a quota of 1.52, that's bigger than 1.414 so it would go up to two. Now what you notice is that the decimal portion of this geometric mean gets steadily larger as the size of the lower quota integer gets larger and larger. So this actually ends up helping smaller states slightly more than larger states because these smaller states only have to have a decimal larger than 4.1 to get rounded up. Well, by the time you deserve 14, you've got to have a decimal larger than 4.9 to go up to the next level. So let's do a simple example of this. We're not going to go back to the 18.20 data. It's much more difficult to do this on a larger data set. Suppose a country wants to apportion 25 members to their national legislature according to the populations of their respective states. So it's a small country. They have four states. We'll call them A, B, C, and D. So these are their respective populations and that's the total population of the country, a little more than half a million. So we're going to calculate the standard divisor as a good starting point. We take the total population, divide by the number of items we want to apportion, and we get 22,995. That's how many people each state would need in order to deserve one representative. So I've got the standard divisor up here for reference. I've got all our populations again, and I've actually just started off by dividing these populations by the standard divisor. We get 13.15, 2.86, 3.58, 5.41. And what I want to do is then round each one of these values according to the associated geometric mean. Now, if you recall back to that chart, all the geometric means started with a decimal of 0.4 something. So if it's already less than 0.4, it's going to go down. It's not going to go up. So we know that 13.15 is going down to 13, and we know that one. Also, all of them, since they were in the 40s as the decimals, anything bigger than that obviously is going up. So we know that state B is going to go up to 3 and state C is going to go up to 4. This one you may not remember though. 5.41, we need to consult the table and see what the geometric mean actually is for 5. So I'm going to try to go back to that slide where we can calculate that. Here we go. With 5. So the geometric mean is 5.477. So you would need to have a decimal larger than 477 in order to go up. We go back and reference our particular country example. We have 4.1. That's not larger than 477, so it goes down. We get 5. I add all those up. Oh, and it works out to be perfectly 25 in this case. If it had not equaled 25, then just like Jefferson's method or Adam's method or Webster's method, we would pick a modified divisor, divide again, and add up and see if we get 25 after we've rounded according to the geometric mean. So it involves guess and check somewhat too, just like Jefferson, Adam's, and Webster's method did. We just happened to get lucky on this nicely chosen example I provided for us. So, which method is best? We know that Hamilton's method never violates the quota rule, but it's been subject to three weird paradoxes in its enactment. On the other hand, we know that all the adjusted divisor methods have the potential to violate the quota rule, but they don't, you can show this mathematically, don't encounter the paradoxes. So which do you want to choose? A couple of mathematicians, Belinsky and Young, did some research on all of these methods, not counting Hamilton's method because they threw out the one with the paradoxes, and they added in one more method that we haven't talked about called Dean's method, and looked at the bias that these different methods have. The way this is read, the further above zero is the cumulative percent bias that you are towards smaller states, and the further below zero you are is the cumulative percent bias you are to larger states. So you can see that Jefferson's method, they redid all the actual census data for each of these 10-year periods starting in 1790, and you can see that Jefferson's method has gotten more and more bias towards larger states as the size of our house of representatives has gotten larger and the size of our population of the country has gotten larger. Adam's method, if we would have used it, you can see is getting more and more bias towards the smaller states. It turns out that Webster's method is actually doing the best job here being least bias towards either smaller large states, and we've been using the Huntington Hill method, here it's just called Hill, and you can see that it's also pretty unbiased, it's slightly biased towards the smaller states, maybe if we estimate it's about 3% if we went over there. So it still has a little bias towards the smaller states, but it's a pretty balanced approach as well. And because of that, that's why up to this point today we tend to keep using the Huntington Hill method. But you may still ask, well, is there a better one out there that we haven't found yet? Is there some method that never encounters paradoxes and yet always satisfies the quota rule? So nobody gets more than their upper quota or less than their lower quota. And Mr. Belinsky and Mr. Young were actually able to answer that question. It's called their impossibility theorem. Michael Belinsky, H. Peyton Young, they are able to prove that there exists no perfect apportionment method that always satisfies the quota rule while encountering no paradoxes. And you may not actually grasp the importance of this statement in mathematics, but it's very difficult to prove something is impossible. And they were actually able to show this on a level that's much above this video, but we'll take their word for it that it's actually impossible to produce a perfect apportionment method. So I'll leave it up to you which method you feel is best. Currently, our country feels the Huntington Hill method is best, but there is continual debate. In fact, different states have brought lawsuits in recent years over the method of apportionment. So this will continue to be in the news for the coming years. Thank you.