Alright?

So, how do I do it?

So, I'm going to do this a minute.

Let b be the number of people who live in a block,

little b, and let big B be the number of people total.

Then, if I take little b over big B, that's going to tell me the percentage of

blue people in that block right, relative to the total number of blue people.

So it's just going to be the proportion of

the total number of blue people in that block.

And similarly little wise or big Y yellow people in that block.

Now why do I want to do that?

Why do I want to look at those two numbers?

Because, if I take the difference between big, b over B and y over

Y, that's going to tell me how distorted

the distribution is in that particular block.

But I need to be more precise.

Suppose I have a district that has five blue and

three yellow, and I want to have a perfectly representative district.

What that would mean is that 5 over 150 of the, there's 150

blue people and five of those blue people live in this particular block.

So 5 over 150 equals 1 over 30.

So one out of every 30 blue people lies inside that block.

Now there's 90 yellow people in three out of the 90 yellow people live in that block

so one out of 30 yellow people live inside

that block or poor people live inside that block.

So, 1 over 30 minus 1 over 30 equals 0.

So what we get is that, if you had a perfectly representative block between

rich and poor, what I'm calling blue and yellow, we'd have a difference of 0.

But if we've got relatively more blue, or relatively more yellow, since

I'm taking the absolute value, that's what these two lines mean, right here.

The absolute value.

It means that I'm going to get a positive number.

So I'm going to have more, I'm just going to represent more segregation.

So, let's look at our particular example.

So these are, this block right here is all blue, right?

So, there and there's ten blue people in there.

Now, there's 150 blue people, total.

So ten out of 150 blue people lie in that block.

There is no, no yellow people, no poor people in that block.

so I have 10 over 150 minus 0 over 90.

So that equals 10 over 150, I can get rid of the zeroes, it equals 115th.

So in every one of these blocks, my index is going to be one fifteenth.

Now in these yellow blocks, right here, there's no blue

people, there's no rich people, so that's 0 over 150.

But there's 10, yellow people are poor people so that's 10 over 90.

So there's way too many yellow people than there should be proportionally

and so take 0 minus 10 over 90 I get 1 9th right?

got these absolute value signs here so everything becomes positive.

So these districts, these blocks are 1 9th.

And finally I've got these green districts,

now remember these have 5 blue, so,

that's 5 over 150 and, they've got 5 yellow, so that's 5 over 90.

Right, and I take the absolute value.

What do I get there?

Well, that's 1 over 30 minus 1 over 18.

So, that's, this is complicated.

We're going to find out that this is equal to 1 over 45.

Okay?

So this is 1 over 45.

What we get then is every one of those ten

blue districts, the index of the assembly is 1 over 15.

Every one of the yellow districts, the index of similarity is 1 9th, and

every on in the districts that's 5 blue and 5 yellow is 1 over 45.

Okay.

So, how do we figure out how segregated this whole region is?

What we do is we say, we've got 6 districts, or blocks here that

have a dissimilar of 1 over 45, so we get 6 times 1 over 45.

And we get 6 here that are dissimilar to 1 9th, so we're going to add

6 times 1 9th, and then we've got 12 that have a dissimilarity of 115th.

So we get 12 times 1 over 15.

And if we add all that up, we get 72 over 45.

So 72 over 45 is, it's a tentative

measure, we're going to change this a little bit

because, what does that mean, what does 72 over 45 mean, is that bad, is that good?

So, let's, let's go through and let's sort of put our measure through the paces.

So whenever you construct a measure, what you try

and do, is do some extreme cases, to see how

well it works, so, let's start out with a simpler

case, to see if this measure sort of makes sense.

And I've got 4 blocks, that are 4 blue

4 are yellow, and here's another case, where I've got

all eight of them are 50 50 and let's

compute our index of similarity in each of these cases.

So, let's start with this one.

Well, each one of these blocks is going to be five blue, right?

And five yellow.

The total number of blue and yellow, right?

Since I've got 8 blocks, I've got 80 people.

So that means there's going to be 40 blue, and 40 yellow.

So, for each one of these blocks, I get 5 over 40 minus 5 over 40, which is 0.

So every single block contributes zero and

my total index of dissimilarities, dissimilarity is 0.

So that's great, right, because that means that if I,

if everyone is perfectly mixed, my index would be 0.

So it seems like it's a pretty good index.

But let's go back and look at this other case.

So now I've got this case where I've got, you know,

4 that are all yellow, and 4 that are all blue.

So, once again, I've got 40 yellow, and 40 blue.

But now we've gotta think, for each one of these yellow districts, what do I have?

I've got 0 over 40 blue minus 5 over 40, yellows.

Right?

I'm sorry, 10.

10.

So we've got yellows, so 10 over 40 yellows.

So what that means is that going to be equal to 1 4th.

And since all these are the same I'm going to get a fourth, a

fourth, a fourth, a fourth and so on and also for the blues.

Right?

By the same logic.

So every single one of these is going to give me is a viable fourth.

When I add all those up I get two.