# Notes on Segregation Indexes

PPA
786 Urban Policy

Professor Yinger

In order to study racial and ethnic segregation, social scientists have developed several indexes to measure segregation. These notes provide an introduction to two of the most widely used indexes.

I. The Dissimilarity IndexThe index of segregation that appears most frequently in the literature on segregation in housing is called the dissimilarity index. This index, which has also been used to study segregation in education and employment, is designed to show how evenly two groups, such as blacks and whites, are spread throughout an area. It is based on the absolute deviations of racial composition in small areas (usually census blocks or tracts) from the racial composition of an area (usually a metropolitan area) as a whole.

To be specific, the numerator of the index, say

D, is the sum of the absolute deviations from the area-wide racial composition, or_{n }

where

T= the total population on block_{i}I,pis the proportion of group_{i}B, henceforth blacks, on blockI, andpis the proportion black in the entire area. The dissimilarity index is defined by dividingDby its maximum possible value. This value, say_{n}D, occurs when there is complete segregation, that is, when_{m }pequals either zero or one for all_{i}I. In symbols,

where

T= the total population in the area,B= the total black population, andW= the total white population.Thus, the dissimilarity index is

The value of this index clearly ranges from zero to one. Some scholars multiply the index by 100 so that it will range from zero to 100.

Some algebraic manipulation transforms

Dinto a form that is more easily calculated and interpreted.

where

T= the total population on block_{i}I,B= the black population on block_{i}I, andW= the white population on block_{i}I. Hence, one simple way to calculateDis to calculate the sum of the absolute differences between the shares of the black and white populations that live on each block.Now if integration is completely even throughout the urban area, then each block must contain the same share of the area's blacks and of its whites. Let

B* be the level of_{i}Bthat is associated with this completely even complete integration on block_{i}I. Then it must be true that

Actual levels of

Bdeviate from_{i}B* by some amount, say_{i}B, or_{i}^{d}

It follows that

This expression has a clear interpretation:

Dis the minimum percentage of the black population that would have to move in order to achieve completely even integration, that is, the same racial composition on every block. Note that the ½ in the formula indicates that in order to achieve this racial balance, only one-half of blacks would have to move; some of the blacks living on blocks whereBis greater than_{i}B* would move to blocks where blacks are relatively under represented. Note also that an analogous formula can be derived for whites; in fact,_{i}Dalso is the minimum percentage of the white population that would have to move in order to achieve racial balance.The data in Table 1 are provided to illustrate the calculation of

D. Using any of the above versions of the formula, these data result in a value ofDequal to 2/3 or, if multiplied by 100, 66.67. Verify this result to check your understanding of the formulas. Also note that to have the same racial composition, namely 50 percent black on every block, all 30 black residents of block 4 would have to move to block 1 (which would then have 30 blacks and 30 whites) and 10 residents of block 3 would have to move to block 2 (which would leave 10 blacks and 10 whites on block 3 and 20 blacks and 20 whites on block 2). Thus 30 + 10 = 40 blacks, all originally located on blocks with a racial composition above 50 percent black, would have to move, which is 2/3 of the total--the same asD.Note further that

Dis unaffected by a movement of blacks and whites between blocks that are all above (or all below) the racial composition of the entire urban area (which is 50 percent in Table 1). For example,Dis still 2/3 if blocks 3 and 4 both have 25 black inhabitants. However,Dgoes down if some blacks in blocks with an above-average percentage black either move to blocks with a below-average percentage black or simply leave the urban area. (As a further test of your understanding, you might want to verify these results.)

The Isolation IndexAnother widely used index, called the isolation index,

I, indicates the extent to which blacks are isolated from whites in their residential (or school or work) locations.^{(1)}As we will see, this index differs from the dissimilarity index in that it is higher if those blacks living on blocks with an above-average percentage black are concentrated in the blocks where the percentage black is highest.

In symbols, the isolation index is

For each block in the summation, the first term measures blacks on a block as a share of total blacks in the area and the second term measures the racial composition of the block, which is the measure of exposure to other blacks. In effect, therefore,

Iis nothing more than a weighted average of black exposure to blacks, where the weights are population shares.Using the data in Table 1, the value of

Iturns out to be .7778 or, multiplied by 100, 77.78. (Verify this.) If the population is redistributed so that 25 blacks live in blocks 3 and 4,Idrops to 75.0. This proves thatI, unlikeD, depends on the extent to which blacks are concentrated on the blocks with the highest percentage black. Note also thatIshares two features withD, namely that it decreases if blacks are shifted from blocks with an above-average percentage black to blocks with a below-average percentage black or if blacks in tracts with an above-average percentage black simply leave the area.

Further References

For further discussion of segregation indexes and the many dimensions of segregation, see D.S. Massey and N.A. Denton, "Hypersegregation in U.S. Metropolitan Areas: Black and Hispanic Segregation Along Five Dimensions,"

Social Forces 25(August 1989): 373-3921; D. Massey and N. Denton,American Apartheid: Segregation and the Making of the Underclass(Cambridge, MA: Harvard University Press, 1993); and N.A. Denton, "Are African Americans Still Hypersegregated?" InResidential Apartheid: The American Legacy, edited by R. Bullard, J.E. Grigsby, III, and C. Lee (Los Angeles: CAAS Publications, 1994), pp. 49-81. Recent values of several segregation indexes in U.S. cities, along with a detailed discussion of the causes of segregation, can be found in J. Yinger,Closed Doors, Opportunities Lost(Russell Sage Foundation, 1995), chapter 7.

Table 1

Illustrative Data for Calculating Segregation Indexes

I

T

_{i}B

_{i}W

_{i}1

30

0

30

2

30

10

20

3

30

20

10

4

30

30

0

Total

120

60

60

^{ 1. }An alternative approach is to measure black exposure to whites using the so-called exposure index. To maximize comparability with the dissimilarity index, these notes focus on black isolation from whites (or exposure to blacks), but it is worth noting that with just two groups, the index of exposure to whites is just one minus the index of exposure to blacks.