In the logical extreme where displacement is concentrated to the maximum possible extent, the value of S will be equal to the value of D. 4 The value of G can therefore be given as twice the value of the group difference in average percentile scores on area group proportion (p). This provides the initial basis for understanding how the value of S can move independently of the value of D.

There is a second basis why the value of S can move independently of the value of D. Under minimal segregation, all possible integration-promoting exchanges that do not change the value of D are implemented.

## Overview of D-S Differences in Responding to Integration-Promoting Exchanges

An exchange that promotes integration is “depolarizing” because it moves the two areas involved in the exchange closer together in terms of the White area ratio. D records integration-promoting exchanges only in circumstances where the racial makeup of the two areas involved in the exchange, on opposite sides of P, is the white ratio for the entire city. Otherwise, integration-promoting exchange results in a larger decrease in S when the two domains involved in the exchange are more polarized.

## Examples of D-S Differences in Responding to Integration-Promoting Exchanges

This gives a value of λ – the white-black mean difference for the population in both areas – of 100.0 points. After the exchange, the average contact with whites drops to 98.02 points for whites in both areas and rises to 1.98 points for blacks, giving a λ value of 96.04 points. For the subset of households in affected households, average contact with whites at baseline is 50.02 points for whites and 49.98 points for blacks, resulting in a white-black difference (λ) of 0.04 points.

From a substantive point of view, the first exchange has a greater impact on reducing group segregation because it does more to “deconcentrate” the distribution of groups across the two areas by bringing together white and black households from areas that were initially at opposite extremes of area racial affiliation. composition. The initial mean of contact with whites, as measured by D, is 100.0 for whites and 0.0 for blacks, resulting in a mean difference between whites and blacks of 100.0 for the population in affected neighborhoods. In this example, the average contact with whites, as measured by D, is 51.0 for whites and 49.0 for blacks at baseline, resulting in a mean difference between whites and blacks of 2.0 for the population in affected neighborhoods.

The impact on S can be assessed as before by examining the value of λ – the difference between white and black on segregation-relevant contact with whites for the population in the affected areas. The fourth exchange involves two areas that together have the same number of white and black households as in the two areas of the third exchange; 50 white and 150 black households respectively. Another important finding is that the impact on reducing S is much greater in the third exchange than in the fourth exchange.

As seen earlier with the first exchange, the third exchange does more to “deconcentrate” the group distributions.

## Implications of Analysis of Example Exchanges

It is that the value of D can remain fixed, while the value of S can become higher or lower, depending on whether integrating movements involving areas not above parity reduces polarization or whether separate movements affect the increase polarization. Stated differently, S will take on higher values if the population living in non-parity areas is concentrated to form racially polarized areas, and S will take on lower values if the population living in non-parity areas is widely dispersed to form areas that are at resemble each other. racial composition rather than being polarized. The first concentrates both groups in homogeneous areas and maximizes contact between the same groups and group separation.

At a given degree of displacement (D), the value of the segregation index (S) can vary significantly depending on whether group distributions within "odd" areas tend toward concentration or dispersion. When the concentration within the non-parity areas is the greatest, the value of S will be equal to the value of D. But when the concentration is the smallest – that is, when the groups are maximally dispersed throughout the non-parity areas, the value of S will be lower, sometimes much lower than the value of D.

At a given level of displacement, implement as many segregation-promoting exchanges as possible within non-parity areas. If such exchanges are possible, residential group distributions will shift toward the pattern of “prototypical segregation” and the value of S will increase. Ultimately, the value of S will increase until it reaches the value of D, and the D-S inequality will be zero.

If such exchanges can be made, group housing distributions will shift toward the pattern of "diffuse displacement" and the value of S will decrease.

## Clarifying the Potential for D-S Concordance and Discordance – Analytic Models

### Examples of Calculating Values of S Min Given Values of D and P

In this section I review examples to illustrate how SMin and SMax values can be calculated for a given combination of displacement (D) and racial composition of the city (P). The value of S under the "Maximum S" algorithm can be obtained using the formulas in Fig. Examination of the two general calculation formulas for S given below (as well as earlier) reveals that the "equalization zone" in the three areas of the analytical model under consideration can be ignored because the calculations for this zone give values of zero (0) and have no impact on the value of S.

The value of S is thus derived from calculations for the "above parity" and "below parity" areas and may be specified as either. Taking the example of a combination of displacement as measured with the dissimilarity index (D) set to 60 and the even city proportion of white (P) set to 0.90, the resulting S value under the "maximum S" model can be obtained by first determining values of the corresponding components of the components and then performs the calculations. The value of S in model A1 can be obtained by first determining the values of the corresponding component components and then performing further calculations.

The following calculations illustrate that any of the three expressions can be used to obtain SMin =0 1330. But it is occasionally relevant, perhaps most often for comparisons between whites and Latinos in the borderlands of the southwestern United States. As with Model A1, the value of S can be obtained from any of the following three equivalent expressions.

If D e.g. is 60 and pairwise city proportion White (P) is 0.30 (similar to the value of P for many White-Latino comparisons in Texas border region cities), the resulting value of S under Model A2 can be obtained by first establishing the values of relevant component terms and then perform calculations.

### Examining D, S Max , and S Min over Varying Combinations of D and P

The diagonal in the figure therefore serves as a reference line indicating the maximum extent to which groups can be residentially separated at a given level of displacement from even distribution. Note that it is not necessary to plot the same relationships for values of P above 50, they are identical to the relationships already shown for values of 1 − P already shown. It is important that all the curves fall below the diagonal and thus visually depict the fact that S can take a lower value than D at any combination of values for D and P when group displacement is distributed from uniform distribution in a way that promotes group residential mixing maximize instead of being concentrated in a way that maximizes group residential separation.

The maximum possible size of the D-S difference is the smallest when the two groups in the comparison are equal in size (i.e. P Q= =0 5. This curve documents that the absolute and relative size of the possible D-S difference can be substantial even when it The D-S difference when groups are of equal size reaches a maximum of 25 points when D is 50 and it is 20 points or more when D is in the range 28–72.

The D-SMin curves, plotted at selected values of P, deviate further from the diagonal as the racial composition of the city becomes increasingly unbalanced. The curves in the figure show that the level of group separation and racial polarization in the area, as measured by S, can vary widely between cities that are identical in terms of group movement relative to an even distribution (D) and relative group size (P ). Here the diagonal represents the values of D plotted by S when the displacement from a uniform distribution is maximally concentrated (SMax).

The curves in the figure show the values of D drawn by S when the shift from a uniform distribution is maximally dispersed.

Implications of Findings from Analytic Models for S Max

## Is Separation a Distinct Dimension of Segregation?

A basis for grouping the indices together is the similarity of the calculation formulas - the operational implementations of the division concepts embodied in the indices. The value of each of these indices records the weighted population mean of the quantitative score of the deviations of the pairwise racial composition (pi) from the pairwise racial composition of the city (P), normalized to the interval 0-1 where 0 indicates no. This principle has the practical consequence of requiring indices to rank separation comparisons in accordance with the principle of "separation curve dominance."15 Two widely used indices—the separation index (S) and the Theil entropy index (H)— do not meet this. criterion.

The shift would be compatible with the geometric interpretation of the gini index (G) in conjunction with the segregation curve and closely related interpretations of the vertical distance and extent of movement of the dissimilarity index (D). Two other measures—a symmetric version of the Atkinson index (A) and the Hutchens square root index (R)—could also be categorized as displacement measures. More precisely, the principle would require that the impact of the exchange on the index result increases with the value of |pi−pj| increases.

G may treat these exchanges differently, but not based on the quantitative magnitude of the level polarization; that is, not in proportion to the value of |pi−pj|. Instead, because G assesses group differences in rank, G will treat these exchanges differently when they differ in terms of the proportion of the combined group populations living in areas with values on racial composition (p) that fall between the values on racial composition. composition for the two areas involved in the exchange. Segregation indices that rank segregation equations in terms of the segregation curve are poor choices for measuring separation between groups and racial polarization.

18 As previously mentioned, G registers group differences in rank position on area group share (p) regardless of the quantitative size of the differences involved.