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Hi.

Â In this lecture, we're going to introduce something called

Â a measure of dissimilarity, an index of dissimilan.

Â We're going to use that to look at different cities

Â and different regions and ask how segregated they are.

Â So remember in the last lecture when I did Shelling's

Â segregation law, and what we saw is that people had

Â fairly tolerant thresholds for you know, living with people with

Â different income groups or different races, still ending up being segregated.

Â And so the result is when look at the cities across United States, we see

Â substantial segregation by income, we see substantial segregation

Â by race, and we want to know, what we

Â want to do here is sort of figure out, can we construct some sort of measure, some

Â we've you know, categorizing numerically how segregated if

Â a particular city is along a particular dimension.

Â because when you have those measures, right, that

Â allows us to make, you know, better sense

Â of data, like to use and understand data

Â better, and that's one reason why you model.

Â So, to get started, let's remind ourselves

Â again of just what these patterns look like.

Â So this is the city of New York, and remember

Â that regions that are depicted in red are predominately Caucasian.

Â Regions that are depicted in blue are predominately

Â African American, yellow predominately Latino, and green predominately Asian.

Â So New York is interesting because it's just, like, these big

Â chunks of different racial groups spread out all over the city.

Â Not all cities look the same way.

Â Now here's Los Angeles.

Â Right, Los Angeles has this area called the Valley which is mostly white, South

Â Central which is mostly African American and

Â then over by Monterey park it's mostly Asian.

Â If you look at Houston, again you see all of these sort

Â of interesting patterns, and how people are racially distributed across the city.

Â And we look at DC, it's almost like there's a dividing line to

Â the East, most people are African American,

Â to the West most people are Caucasian.

Â So, different cities look different ways, what we'd like to do

Â is to have some sort of number for representing this racial disparity.

Â Okay?

Â Now, remember, the same is true for income,

Â so we can use the same for income disparity.

Â And if you look at a city like Chicago, what

Â you see is that there's red represents wealthy people here.

Â So there's wealthy people along this area known as the Gold Coast.

Â In the center of the city, it's mostly poor people.

Â And then, to the north and to the west, in the suburbs.

Â Right, something that's called Collar Counties, it actually looks like a collar.

Â These, again, are wealthy people.

Â Again, New York, remember the red dots here represent rich people.

Â People who make more than $200 thousand a year.

Â All around Central Park here, you see wealthy people, and as soon

Â as you move further out from the city, you see poor people.

Â So it's interesting, New York is sort of a

Â little bit different than Chicago, in that right in

Â the center of New York there's a lot of

Â wealth, and then as you move out, it gets poorer.

Â Chicago sort of looks the other way.

Â So what we want, is we want some measure for how segregated a city is.

Â So construct, we're going to construct a

Â very simple measure called the Index of dissimilarity.

Â And we're going to do it with just two

Â types of people, rich people and poor people.

Â So to represent rich people with blue dots and poor people with yellow dots.

Â Now, I'm going to place these people on a grid.

Â So in a 24 city block area here and

Â in each block, I'm going to put ten people, all right?

Â So, let's start out, and let suppose let's start 12

Â of these blocks right here I put all rich people.

Â 3:40

Alright?

Â So, how do I do it?

Â So, I'm going to do this a minute.

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

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

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

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

Â So it's just going to be the proportion of

Â the total number of blue people in that block.

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

Â Now why do I want to do that?

Â Why do I want to look at those two numbers?

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

Â Y, that's going to tell me how distorted

Â the distribution is in that particular block.

Â But I need to be more precise.

Â Suppose I have a district that has five blue and

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

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

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

Â So 5 over 150 equals 1 over 30.

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

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

Â so one out of 30 yellow people live inside

Â that block or poor people live inside that block.

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

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

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

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

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

Â The absolute value.

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

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

Â So, let's look at our particular example.

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

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

Â Now, there's 150 blue people, total.

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

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

Â so I have 10 over 150 minus 0 over 90.

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

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

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

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

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

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

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

Â got these absolute value signs here so everything becomes positive.

Â So these districts, these blocks are 1 9th.

Â And finally I've got these green districts,

Â now remember these have 5 blue, so,

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

Â Right, and I take the absolute value.

Â What do I get there?

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

Â So, that's, this is complicated.

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

Â Okay?

Â So this is 1 over 45.

Â What we get then is every one of those ten

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

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

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

Â Okay.

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

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

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

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

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

Â So we get 12 times 1 over 15.

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

Â So 72 over 45 is, it's a tentative

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

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

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

Â So whenever you construct a measure, what you try

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

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

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

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

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

Â all eight of them are 50 50 and let's

Â compute our index of similarity in each of these cases.

Â So, let's start with this one.

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

Â And five yellow.

Â The total number of blue and yellow, right?

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

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

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

Â So every single block contributes zero and

Â my total index of dissimilarities, dissimilarity is 0.

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

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

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

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

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

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

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

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

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

Â Right?

Â I'm sorry, 10.

Â 10.

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

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

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

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

Â Right?

Â By the same logic.

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

Â When I add all those up I get two.

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I don't get one I get two.

Â So now I've got a bit of an issue so if people are

Â perfectly segregated I get two and if they're perfectly mixed I get zero.

Â So this suggests I've got a pretty good measure here but what

Â I probably want to do is I want to divide it by two, right?

Â So if I divide it by two then if I get if you're perfectly mixed, you

Â get a score of zero and if you're

Â perfectly segregated you get a score of one.

Â All right?

Â So, if I go to this case where there's 40 rich

Â 40 poor and they're perfectly mixed my score is going to be zero.

Â Because I get five black and five yellow in

Â each districts oops this should be a five, right?

Â So I get a score of zero.

Â And when I do the other one I get a score of one.

Â Now when I look at my thing here when 72 over

Â 45 which didn't make any sense, now that's 72 over 90.

Â And if I divide this by nine that's going to be 0.8 so it's 80%.

Â So sometimes this is 80% segregation.

Â Which seems pretty segregated.

Â Now we can go back and we can look at our cities.

Â So now we can go back and we can look at our census data.

Â And we can look at a city like Philadelphia.

Â We can ask how segregated in it, is it.

Â And notice that it will hit 0.8 exactly the same as our example.

Â So this tells us the score in Philadelphia is 0.8.

Â Now we, if look at this map and say well how segregated is it, now we have a score.

Â And now we can do things that compare Philadelphia to Detroit.

Â Remember Detroit also looks segregated, but in Detroit

Â even though it looks segregated the scores only 0.6.

Â So Detroit is actually substantially less segregated.

Â In Philadelphia if, if you look at these two pictures, here's Philadelphia

Â and here's Detroit it's very hard to tell the difference between the two.

Â Okay so what have we learned in this lecture?

Â We've learned that [UNKNOWN] we can construct

Â a very simple measure called the index

Â of dissimilarity and by using that measure

Â we can compare how segregated different cities are.

Â And now once we've got this measure in our pocket, right, we can use

Â it to measure segregation by race, segregation

Â by income, and all sorts of segregation.

Â It's a really useful tool to help us sort of take data and understand the world.

Â All right.

Â Thank you.

Â