When calculating G, city areas are ranked from lowest to highest value in proportion of White area (p). Specifically, each can be defined in terms of the other two according to the following expressions. The curve is contrasted with the diagonal line that would result under conditions of exact equal distribution, and the value of G is given by the ratio of the area between the diagonal and.

The values of RRWi by RRBi fall on the midpoints of the line segments that form the segregation curve. The value of the term Σpw RRWi⋅ i is 0.5 because the mean of the relative rank position is necessarily 0 5. These results show that the value of the Gini Index (G) can be directly and precisely assigned to the terms of the group. difference in means (YW − YB) on residential outcomes (y) scored based on relative rank position on area group share (p).

Its lower bound (i.e., the minimum possible value) is given by Q/2 which occurs under complete segregation and varies in exact value with city's ethnic composition. The resulting curve is contrasted with the diagonal line between the start point (0,1) and end point (1,1) of the curve. G records the deviation quantitatively based on the ratio of the area between the curve and the diagonal to the total area under the diagonal.

The height of the triangle (h) is equal to the length of the line perpendicular to the diagonal and ending at the point (xD, yD) on the segregation curve. This line is one side of a right isosceles triangle whose base has a length equal to the value of D – the maximum vertical distance of the segregation curve from the diagonal. It also follows that the value of the Gini index (G) for the three-point segregation curve is given by.

One is that D is G≤ because the full segregation curve for G can never be "inside" the three-point segregation curve for D. Specifically, it first appears at the end of the line segment on the curve for the last area unit where p is P.<. This can be seen by a closer look at the construction of the segregation curve.

The graphs in the middle column are for substantial, but not complete, segregation where the value of G is 0.900.9 The two rows are for two conditions for the racial composition of cities. This can be seen in the two graphs in the middle column of the figure, which serve as an example in which the value of G is 0.900. The nature of the y-p relationship for the Gini index (G) is complex and difficult to summarize.

As the model for city C shows, this possibility arises because both groups can differ by relatively small amounts in p—the area score that defines S—and at the same time can differ by large amounts in y as noted for G and D.

The finding is that scores for G and D can be, and often are, much higher than scores for S when the two comparison groups are unbalanced in size. The scaling function y f p= ( ) that sets S to the desired group mean frame difference is developed below. Specifically, it is the exact one-to-one linear function f p( )i =pi which means that S can be placed in the mean-difference frame without reestimating p from the original or.

S is particularly attractive when put into the difference-of-means framework used here, because S can be expressed as a difference of means in pairwise scaled group contact, where group contact is based on the proportion of group area (p) to its "natural" metric. thus, no rescaling p as required for the other indices considered here. 1978: 353) were the first to show that in both groups, case S can be given as the simple difference between focal group contact with itself (i.e., generally, PXX, for white contact with whites, PWW) and of comparison group contact with the focal group (ie, overall, PYX, for black-white contact, PBW) based on. Here the ".XY" suffix on the signature contact indicates that the contact calculations are based only on counting the two sets in the partition comparison.

That is, their calculations only draw on counts for the two groups in the separation comparison. To conclude this discussion, the separation index (S) can be given as the group difference of mean values of mean pairwise contact with the reference group. Setting housing outcomes (yi) to the value of the area proportion White (pi) allows S to be placed in the notation of the difference between mean frames and restated as S = YW–YB.

The overall mean of X is the proportion of White in the city's population (P), so TSS= ∑(Xk−P)² with k used here to index individuals. Regardless, it is clear that the value of S is about the effect of race on contact with whites at the individual level, as reflected in the white-black difference in means of contact with whites (pi). Under equal distribution, S will be 0 because all pi=P, so the white and black means of contact with whites (pi) are the same, and knowledge of race will not improve the prediction of contact with whites (p) above the baseline to assume the overall city average (P).

S can also be obtained from the simple difference between pairwise white contact with whites (PWW) and pairwise black contact with whites (PBW); that is, S = PWW–PBW. Under the traditional eta squared or variance ratio interpretation, S indicates the strength of the association between race (ie, group membership) and contact with whites (p). Under the new interpretation of S as a difference between group means of contact with whites, S indicates the “influence” or “effect” of race (i.e., group membership) on contact with whites.

As a final comment, I note that the discussion here shows that S simultaneously registers two separate and distinct aspects of the relationship between race and contact with whites (p). Appendix E: Establishment of the scaling function y =f p( ) required to transform the Theil entropy index (H) as a group difference. Continuing with the familiar example of white-black segregation, the basis for scoring housing scores (y) such that the y scores fall in the same range as p (ie 0–1) gives the Theil index (H) as the difference in means (YW−YB ) can be determined as follows.

The expression on the far right is an adaptation of the formula for H given in James and Taeuber (1985). Similarly, replace wi and bi with alternative expressions based on ti, pi and qi. For actual calculations, E and ei will be expanded to their full expressions using the following substitutions.

The results for yi will give H as a group difference mean achievement of an important goal of the exercise. Instead, they will fall in the range –Q in P as pi changes from its minimum value of 0 to its maximum value of 1. Therefore, the range for y can be set to 0–1 by including the constant Q in function as below.

As a practical matter, exact equality of pi and P is very rare in conventional empirical analyzes of residential segregation in urban areas. Nevertheless, it is a logical possibility that it could occur in empirical studies of segregation and it certainly does. The procedure I follow is this: when pi is exactly P, assign a value for y based on the limiting values of y obtained by taking values of pi that are arbitrarily close to P but just not exactly P does not reach.

For example, the value of y can be determined this way by averaging two values of y obtained using pi = −P 0 0000001. The two values of y will be very close; so close that the graph of the y-p relationship will appear as a smooth, continuous function in which y increases monotonically as p goes from 0 to 1 with only an arbitrarily small "break" in the line exactly at the point where pi =P . Appendix F: Establishing the Scaling Function y f p= ( ) Required to convert the Hutchens index to the square root (R) as a difference.

Instead, the values of yi will range from –Q to P, as pi varies from the minimum value of 0 to the maximum value of 1. Accordingly, the range for y can be set to 0–1 by adding the constant Q to the function take as follows. In practice, exact equality of pi and P is very rare in conventional empirical analyzes of residential segregation in urban areas.

Nevertheless, it is a logical possibility in empirical studies, and it is especially likely to occur in methodological analyzes and simulation studies. The option I use is as follows: when pi is exactly P, assign a value for y based on the bounds for y obtained by taking values of pi that are arbitrarily close to P but not exactly P. The two values of y will be exceedingly close; so close, in fact, that a graph of the y-p relationship will be a smooth, continuous function, where y increases monotonically as p goes from 0 to 1 by only an arbitrary small.

When this approach is taken, an interesting regularity is observed; the value of y always converges to 0.50 as pi. However, the above procedure of replacing y with 0.5 when pi=P also produces a result consistent with the regularity that y Q= when pi=Q.