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. A score of zero (0%) reflects a fully integrated environment; a score of 1 (100%) reflects full 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. Index of dissimilarity is invariant to relative size of group.
Basic formula The basic formula for the index of dissimilarity is:
D = 1 2 ∑ i = 1 N | a i A − b i B | {\displaystyle D={\frac {1}{2}}\sum _{i=1}^{N}\left|{\frac {a_{i}}{A}}-{\frac {b_{i}}{B}}\right|}
where (comparing a black and white population, for example):
ai = the population of group A in the ith area, e.g. census tract A = the total population in group A in the large geographic entity for which the index of dissimilarity is being calculated. bi = the population of group B in the ith area B = the total population in group B in the large geographic entity for which the index of dissimilarity is being calculated. 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.
Numerical example Consider the following distribution of white and black population across neighborhoods.
Linear algebra perspective 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). Suppose our city contains N {\displaystyle N} blocks:
{ block 1 , block 2 , … , block N } {\displaystyle \{{\text{block 1}},{\text{block 2}},\ldots ,{\text{block N}}\}}
Let's create a vector r {\displaystyle \mathbf {r} } which shows the number of rich people in each block of our city:
r = [ r 1 , r 2 , ⋯ , r N ] {\displaystyle \mathbf {r} =[r_{1},r_{2},\cdots ,r_{N}]}
Similarly, let's create a vector p {\displaystyle \mathbf {p} } which shows the number of poor people in each block of our city:
p = [ p 1 , p 2 , ⋯ , p N ] {\displaystyle \mathbf {p} =[p_{1},p_{2},\cdots ,p_{N}]}
Now, the L 1 {\displaystyle L^{1}} -norm of a vector is simply the sum of (the magnitude of) each entry in that vector. That is, for a vector v = [ v 1 , v 2 , ⋯ , v N ] {\displaystyle \mathbf {v} =[v_{1},v_{2},\cdots ,v_{N}]} , we have the L 1 {\displaystyle L^{1}} -norm:
| v | 1 = ∑ i = 1 N | v i | {\displaystyle |\mathbf {v} |_{1}=\sum _{i=1}^{N}|v_{i}|}
If we denote R {\displaystyle R} as the total number of rich people in our city, than a compact way to calculate R {\displaystyle R} would be to use the L 1 {\displaystyle L^{1}} -norm:
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