Index of dissimilarity

The index of dissimilarity is a demographic measure of the evenness with which two groups are distributed across component geographic areas that make up a larger area.

A group is evenly distributed when each geographic unit has the same percentage of group members as the total population.

The index score can also be interpreted as the percentage of one of the two groups included in the calculation that would have to move to different geographic areas in order to produce a distribution that matches that of the larger area.

The index of dissimilarity can be used as a measure of segregation.

In terms of black–white segregation, a score of .60 means that 60 percent of blacks would have to exchange places with whites in other units to achieve an even geographic distribution.

[1][2][3][4] Index of dissimilarity is invariant to relative size of group.

The basic formula for the index of dissimilarity is: where (comparing a black and white population, for example): The index of dissimilarity is applicable to any categorical variable (whether demographic or not) and because of its simple properties is useful for input into multidimensional scaling and clustering programs.

It has been used extensively in the study of social mobility to compare distributions of origin (or destination) occupational categories.

Consider the following distribution of white and black population across neighborhoods.

The formula for the Index of Dissimilarity can be made much more compact and meaningful by considering it from the perspective of Linear algebra.

Suppose we are studying the distribution of rich and poor people in a city (e.g. London).

which shows the number of rich people in each block of our city:

which shows the number of poor people in each block of our city:

as the total number of rich people in our city, than a compact way to calculate

as the total number of poor people in our city, then:

We finally return to the formula for the Index of Dissimilarity (

Consider a city consisting of four blocks of 2 people each.

One block consists of 2 rich people.

One block consists of 2 poor people.

Two blocks consist of 1 rich and 1 poor person.

Firstly, let's find the rich vector

Next, let's normalize the rich and poor vectors:

Finally, let's find the index of dissimilarity (

We can prove that the Linear Algebraic formula for

Let's start with the Linear Algebraic formula:

When the Index of Dissimilarity is zero, this means that the community we are studying has zero segregation.

For example, if we are studying the segregation of rich and poor people in a city, then if

in the linear algebraic formula, we get the necessary condition for having zero segregation:

Then, the total number of rich people is

, and the total number of poor people is