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A Measure of the Population Quality of Legislative Apportionment*

Published online by Cambridge University Press:  01 August 2014

Henry F. Kaiser
Henry F. Kaiser*
Affiliation:
University of Wisconsin
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Extract

This paper attempts to examine carefully the problem of determining a numerical measure which reflects the quality of the population apportionment in a legislative body. We desire a single index, an index which takes advantage of the available population data, and an index which is readily understood. It should be said that we are not concerned with the much more difficult problem of legislative districting—actually drawing district boundaries on a map—but concentrate our attention only on the quality of apportionment in the particular, but important, sense of population equality.

In order to accomplish this purpose I review a number of increasingly complex possibilities and eventually arrive at a measure which superficially seems somewhat esoteric, but which turns out ultimately to be easily interpretable.

It will be convenient to proceed heuristically with the development by carrying along a small concrete example. For this purpose, consider the six congressional districts of Connecticut.

Type
Articles
Information
American Political Science Review , Volume 62 , Issue 1 , March 1968 , pp. 208 - 215
Copyright
Copyright © American Political Science Association 1968

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References

1 These data were obtained from: U.S. Bureau of the Census. Supplement to Congressional District Data Book. Redistricted States. Connecticut (Districts of the 89th Congress), U.S. Government Printing Office. Washington, D.C., 1965.

2 Let nj be the population of the jth of k districts and let n = Σn j be the total population. Now the average district population is n/k so that a population ratio rj is knj/n. Then Σrj = Σknj / n = kn/n = k. Since the sum of the k population ratios is k, their average value must be one.

3 Schubert, Glendon and Press, Charles, “Measuring Malapportionment,” this Review, 58 (June, 1964), 302–327.Google Scholar

4 It is readily seen that the worst possible apportionment occurs when the entire population is in one of the k districts. When this occurs we have one population ratio deviation of k — 1 and k — 1 deviations of — 1. The sum of the squared deviations is then (k — 1)2 + (k — 1)(—1)2 = (k — 1)(k — 1 + 1) = (k — 1)k. The mean squared deviation, or variance, is (k — 1)k/k = k — 1, and the root mean square deviation, or standard deviation, is √k — 1.

5 Kendall, Maurice G., The Advanced Theory of Statistics, Volume I (London, 1948, 4th ed.) pp. 81–82.Google Scholar

6 Schubert, Glendon and Press, Charles, “Malapportionment Remeasured,” this Review, 58 (December, 1964), 966–970.Google Scholar

7 Consider a k district legislature with half the districts having population ratios r 1 and the other half having population ratios r 2, r2 ≠ r 1. Clearly, Πrj goes to zero as k becomes large. However, in this example, g = √r1r2, regardless of k.

8 Kendall, op. cit., pp. 33–34.

9 A Fortran version of a program to compute b, given the district populations, is available from the writer. More generally, please write regarding any computational problem involving the measure b for a particular set of data.

10 Dauer, Manning J. and Kelsay, Robert G., “Unrepresentative States,” National Municipal Review, 44 (1955), 515–575, 587.CrossRefGoogle Scholar

11 The discussion here has been rendered somewhat cumbersome by having defined 6 to range from zero to one. Had we originally defined it as half as large it would numerically have been more directly comparable with the Dauer-Kelsay index. However, most indices in science vary from zero to one and consequently we have followed this convention in our definition of b.

12 Regardless of k, the number of districts, it is always possible to construct an example where the Dauer-Kelsay index is at least 0.50, and still have one of the districts with no population.

13 The maximum possible value for the Dauer-Kelsay index is (k + 2)/2k for k even, and (k + 1)/2k for k odd. The minimum possible value is 2/k for k even, and 1/k for k odd. From these formulas it is seen that when k gets small the Dauer-Kelsay index becomes spuriously inflated.

14 These data are for districts of the 89th Congress, using population figures from the 1960 census. The basic reference is: U.S. Bureau of the Census. Congressional District Data Book. (Districts of the 88th Congress), U.S. Government Printing Office. Washington, D.C., 1963. Eight states redistricted between the 88th and 89th Congress; revised figures for these states were obtained from pamphlets, for which a typical reference is given in footnote 1.

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