New criteria for indexes of residential segregation are developed. It is argued that a pattern of random segregation rather than complete desegregation should be used as a baseline for measuring segregation. It is shown that any index whose relationship to one baseline is independent of the proportion black in a city, necessarily has a dependent relationship with respect to the other baseline. The index of dissimilarity is adjusted to serve as a measure of deviation from random segregation. Eta-square, which was shown by Duncan and Duncan to depend on the proportion black, is shown to be independent of the proportion black when random segregation is used as a baseline. It is argued that segregation should be measured from a situation of complete desegregation when its effects are of concern, but that it should be measured from random segregation when its causes are being analyzed.