¹ 369 U. S. 186 (March 26, 1962). For a summary of measures developed prior to this decision see extension of remarks of Senator Paul Douglas of Illinois, Congressional Record, Vol. 102, Part 4, 84th Cong., 2dsess., March 26, 1956, pp. 5536–48Google Scholar. For a similar discussion subsequent to the decision, see Goldberg, Arthur L., “The Statistics of Malapportionment,” pp. 90–106Google Scholar in “The Problem of Malapportionment: A Symposium on Baker v. Carr,” Yale Law Journal, Vol. 72 (November, 1962)Google Scholar. For an excellent article which focuses upon the normative question how simultaneously to maximize the three criteria (population equality, compactness, and contiguity) in order to produce optimally fair reapportionment, see Weaver, James B. and Hess, Sidney W., “A Procedure for Nonpartisan Districting: Development of Computer Techniques,” Yale Law Journal, Vol. 73 (December, 1963), pp. 288–308CrossRefGoogle Scholar.

² Conference of Research Scholars and Political Scientists, One Man—One Vote (New York, Twentieth Century Fund, 1962)Google Scholar.

³ de Grazia, Alfred, Apportionment and Representative Government (Washington, American Enterprise Institute for Public Policy Research, 1963)Google Scholar.

⁴ Klain, Maurice, “A New Look at the Constituencies: The Need for a Recount and a Reappraisal,” this Review, Vol. 49 (1955), pp. 1105–1119Google Scholar. Klain's findings were confirmed, and his data brought up to date, by Table 2 of David, and Eisenberg, , State Legislative Redistricting: Major Issues in the Wake of Judicial Decision (Chicago, Public Administration Service, 1962)Google Scholar.

⁵ Dauer, Manning J. and Kelsay, Robert G., “Unrepresentative States,” National Municipal Review, Vol. 44 (1955), pp. 515–575, 587CrossRefGoogle Scholar.

⁶ Derge, David A., “Metropolitan and Outstate Alignments in Illinois and Missouri Legislative Delegations,” this Review, Vol. 52 (1958), pp. 1051–1065Google Scholar; -Wahlke, John C., Eulau, Heinz, Buchanan, William and Ferguson, LeRoy C., The Legislative System (New York, 1962)Google Scholar; Wahlke, John C. and Eulau, Heinz (ed.), Legislative Behavior (New York, 1959)Google Scholar; Steiner, Gilbert and Gove, Samuel, Legislative Politics in Illinois (Urbana, University of Illinois Press, 1960)Google Scholar; Havens, Murray C., City vs. Farm (University, Alabama, Bureau of Public Administration, University of Alabama, 1957), 57 pp.Google Scholar; and Flinn, Thomas A., “The Outline of Ohio Politics,” Western Political Quarterly, Vol. 13 (September, 1960), pp. 702–21CrossRefGoogle Scholar.

⁷ Baker, Gordon E., Rural versus Urban Political Power (New York, 1955), pp. 16–17Google Scholar.

⁸ Tyler, Gus, “Court versus Legislature; The Socio-Politics of Malapportionment,” Law and Contemporary Problems, Vol. 27 (1962), p. 402CrossRefGoogle Scholar.

⁹ David, Paul T. and Eisenberg, Ralph, Devaluation of the Urban and Suburban Vote (Charlottesville, Virginia, Bureau of Public Administration, University of Virginia, Vol. 1: 1961 and Vol. 2: 1962)Google Scholar.

¹⁰ Ibid., Vol. 1, pp. 12–13.

¹¹ Ibid., Vol. 1, p. 15. This table incidentally, does not consistently compare the same class intervals.

¹² Alan L. Clem, “Legislative Malapportionment and the Mathematical Quagmire,” a paper presented at the annual meeting of the Midwest Conference of Political Scientists, held at the University of Notre Dame, South Bend, Indiana, April 27, 1962; mimeo., pp. 1–33. See also by the same author, “Measuring Legislative Malapportionment: In Search of a Better Yardstick,” Midwest Journal of Political Science, Vol. 7 (May, 1963), pp. 125–44CrossRefGoogle Scholar.

¹³ Ibid., Appendix Table G.

¹⁴ 369 U. S. 186, 255n. 7, 262–264, 342–343.

¹⁵ Clark's procedure for attributing representation in multi-county districts seems to be similar to that followed by David and Eisenberg, who apparently computed their index on the basis of an assignment of the total district population to each of the included counties; Clark, however, assigned equal shares of the total district population to each included county. Evidently, David and Eisenberg's procedure involves even more distortion than Clark's, as we shall exemplify with Michigan data. The 14th Michigan Senatorial District consists of two counties, Ingham and Livingston, whose populations are 211,296 and 38,233. David and Eisenberg assigned a vote value of 92 to each of these counties for the upper house of the Michigan legislature (op. cit. ftn. 9, supra, Vol. 2, pp. 76–77); using his formula for Tennessee, whose chambers of 33 and 99 are about the same as Michigan's 34 and 110, Clark would assign an index value of 1.5 to each county, while Harlan would assign 2.54 to Ingham and only 0.46 to Livingston. Moreover, and because of the not inconsiderable population differences, David and Eisenberg assign these two counties—with identical vote values for the senate—to different class intervals for purpose of their grouped county data analyses: Ingham is in the “100,000–499,999” category, while Livingston is in the “25,000–99,999” category.

¹⁶ Op. cit. ftn. 8, supra, pp. 393, 402.

¹⁷ Ibid., p. 391; op. cit. ftn. 5, supra; and Boyd, William J. D. (ed.), Compendium on Legislative Apportionment (New York, National Municipal League, 1962 ed.), pp. iii–ivGoogle Scholar.

¹⁸ Tyler's extremity ratio is only one step of complexity—since division is a slightly more complex mathematical operation than subtraction—removed from the range, the measure of dispersion deemed most appropriate (to characterize population differences among congressional representation districts in the states) by Mr. Justice Harlan, in the “Appendix” to his dissenting opinion in Wesberry v. Sanders, 32 L. W. 4142 at 4156–4157 (February 17, 1964). A typical sample of statisticians' comments upon the adequacy of the range as an index of variance follows; and these comments appear equally applicable to Tyler's ratio. Chaddock, Robert E., Principles and Methods of Statistics (Boston, 1925), p. 153Google Scholar: “While this is the simplest measure of variability it is also the least informing. It gives no idea of the nature of the distribution within these extreme limits. It is very unstable since by cutting off a single … item at either end of the scale or adding one the range may be entirely changed. It fails to characterize in a useful manner the series as a whole if stated alone, and ignores the degree of concentration almost entirely. It offers no basis for judging the typical character of the average itself.” [Italics in the original.] McNemar, Quinn, Psychological Statistics (New York, 2d ed. 1955), p. 19Google Scholar: “One may doubt whether the range (highest to lowest score) is of sufficient value in psychological research to justify its use as a measure of variation. It is, obviously, determined by the location of just 2 individual measures or scores and consequently tells us nothing about the general clustering of the scores about a central value.”

¹⁹ Op. cit. ftn. 9, supra, Vol. 1, p. 6.

²⁰ We have considered the possibility that one might rank the states according to the lowest Dauer-Kelsay index score observed for either chamber in each state. It might be assumed that this would rank states according to the maximal blocking power of minorities. Such an assumption is questionable on the grounds of political realism, because it omits one house from consideration. A state in which D-K indices for both houses are 17.0 and 17.0 certainly is different politically from one in which the two houses are 17.0 and 47.0, respectively. In the second state one would expect the activities of the malapportioned house to be spotlighted. We are not aware that any comparative empirical studies have been made of such situations. It is reasonable to hypothesize, however, that the combined pressures of a governor and a well-apportioned house might force political concessions from the malapportioned chamber. Where both houses are badly malapportioned, we hypothesize that still a different political situation would exist.

Cf. the discussion of the blocking power of legislative chambers, in Shapley, L. S. and Shubik, Martin, “A Method for Evaluating the Distribution of Power in a Committee System,” this Review, Vol. 48 (1954), pp. 787–92Google Scholar.

²¹ See note 6 supra.

²² Op cit. ftn. 9, supra, vol. 1, p. 10n. 9.

²³ Guilford, Joy Paul, Fundamental Statistics in Psychology and Education (New York, 1956, 3d ed.), p. 85Google Scholar.

²⁴ Ibid., p. 101.

²⁵ Cf. McNemar, op. cit. ftn. 18, supra, p. 29: “the ratios [for g₁ and g₂ are pure numbers, i.e., are not inches or pounds or IQ's or minutes. If we have the distribution of the weights and of the heights for 1000 individuals, the measure of skewness for the height distribution may be compared directly with that for the weight distribution. This is true by virtue of the fact that for each we are expressing the third moment relative to the amount of variability, both in inches for one distribution, both in pounds for the other. Likewise, it can be reasoned that the measures of kurtosis for different distributions are comparable, although the distributions involve different measurement units.”

²⁶ For good introductory discussions of the normal curve, see Hagood, Margaret J. and Price, Daniel O., Statistics for Sociologists (New York, rev. ed., 1952)Google Scholar, ch. 14; or Walker, Helen M. and Lev, Joseph, Elementary Statistical Methods (New York, rev. ed., 1958)Google Scholar, ch. 12.

²⁷ In theory, the tails of a normal curve extend to infinity, but this assumes (in terms of our problem) a legislative chamber that consists of an infinitely large number of members. In practice, the range of empirical approximations of the normal curve is limited by the size of the chamber, and with an N of 400 to 500 usually will extend over ±3 standard deviations; with an N of 100, the mean range drops to about ±2.5 standarddeviations; for 67 it is about ±2.4 σ; and with an N of 35 the mean range is only about ±2.1 standard deviations, while with an N of 17 the range averages about ±1.8 σ. See Snedecor, George W., Statistical Methods: Applied to Experiments in Agriculture and Biology (Ames, Iowa, The Collegiate Press, 4th ed., 1946), p. 98Google Scholar. For our own empirical analysis, based upon the sizes in March 1962, none of the “lower” chambers of state legislatures was smaller than 35, although 17 of the state senates included between 17 and 33 members. Conversely, none of the “upper” chambers was larger than 67, although three-fourths (37) of the lower chambers were larger than 67.

²⁸ McNemar, op. cit. ftn. 18, supra, p. 33: “In order to write the equation of a particular normal curve, i.e., one which corresponds to a particular distribution, we need to know N, M, and σ. This is the basis for the fact that, when we have the usual bell-shaped distribution, we need only the mean and standard deviation to describe it adequately. But in order to say that a given distribution is really normal, it is necessary to show that the g's … are zero or approximately zero.”

²⁹ With a normal curve for which the abscissal value of sigma is 5% of the mean, ICV would be .95; but with a sigma equal to two-thirds of the mean, ICV would drop to .60. Obviously, an ICV of much less than .60 (which is equivalent to a CV of .67) would not be associated with a normal curve, because an ICV of .50 implies a CV of 1 (with sigma equal to the mean); and with a symmetrical distribution and an absolute zero point for abscissal values, this would signify a range of ±1 standard deviation; but the expected minimal range for any of our empirical distributions—if they are normal—is ±1.8 sigma (when N = 17).

³⁰ We are not unaware of McNemar's admonition that “The nature of the research, the type of variable being studied, and also the size of the sample are factors which need to be considered in making a decision as to the necessity for computing measures of skewness and kurtosis. It is sel-dom advisable to compute these measures when N is less than 100.” Op. cit. ftn. 18, supra, p. 30. Cf. Hagood and Price, op. cit. ftn. 26, supra, p. 214: “The reason that higher moments [than the second] are not frequently computed for distributions observed among only a small number of cases is that they are summarizing measures less stable than the mean and standard deviation.” As our footnote 27, supra, suggests, over a third of the lower and all of the upper chambers of the fifty state legislatures each had fewer than 100 members at the time for which our empirical data were collected. But all except 3 of the lower chambers each had more than 50 members, and 90% of the senates had at least 25. Consequently, we think that more confidence ought to be reposed in the g scores for the lower chambers than in those for the senates.

³¹ For an excellent elementary discussion of the g indices and their formulas, see Hagood and Price, op. cit. ftn. 26, supra, pp. 210–17; and cf. McNemar, op. cit. ftn. 18, supra, pp. 27–31.

³² As Hagood and Price point out, “For a distribution to approach normality closely the range of its possible values must extend several standard deviation units on either side of the mean. For many characteristics, the measures of which can take only positive values, the range is cut off on the left side within two or three standard deviation units of the mean, causing a skew to the right.” Op. cit. ftn. 26, supra, p. 270. For some of our empirical distributions for representation in state legislative chambers, the range is cut off at less than two standard deviations on the left or negative side of the mean.

³³ We do not insist upon the validity, but only upon the plausibility of our political assumptions; any reader who disputes them is free, of course, to take our ICV and G score data and assign any weights he chooses. It was quite laborious to produce the data, but it is only a matter of simple arithmetic and an hour or less at a calculating machine to compute another set of apportionment scores, based on different weights.

³⁴ McNemar states that “a typical frequency curve (or polygon) or a frequency distribution can be roughly characterized as one which shows 4 chief features: a clustering of individuals toward some central value, dispersion about this value, symmetry or lack of symmetry, and flatness or steepness. Many variables or traits yield distributions which are said to be approximately bell-shaped, but such a description is not adequate for scientific purposes. One wishes to know about what particular value and with how much scatter the individual scores are distributed, to what extent the distribution is symmetrical, and to what extent it is peaked or flat. That is, we need measures of central value or tendency, measures of scatter or dispersion or variability, and measures of skewness (lack of symmetry) and of kurtosis (peakedness or flatness.) With such measures, one can describe the distribution mathematically, and in such a way that a statistically trained contemporary, say in Melbourne, can picture to himself the frequency distribution.” Op. cit. ftn. 18, supra, p. 13. Hagood and Price are in accord: “In a thorough analysis and description of a quantitative distribution, the aspect of form must be treated just as the aspects of central tendency and dispersion are treated, and if there is to be a generalization of the results, tests of hypothesis about form must be made.” Op. cit. ftn. 26, supra, p. 270.

³⁵ Gray v. Sanders, 372 U. S. 368 (March 18, 1963), declaring illegal Georgia's “county unit” system for statewide primary elections; and Wesberry v. Sanders, 32 L.W. 4142–4157 (February 17, 1964), invalidating Georgia's districting for the national House of Representatives.

³⁶ The Supreme Court ruled, in Wesberry v. Sanders, ibid., that Georgia's districting violated the principle of “one person, one vote” and therefore the popular election clause (Art. I, sec. 2) of the Constitution. Mr. Justice Harlan explicitly argued, in dissent, that if the conclusion of the Court's seven-man majority were correct, then the congressional districting in most of the other states also would be unconstitutional if appraised under the same standard that was applied to Georgia. The considerable skewness and kurtosis, for the distribution of representational units of the national House of Representatives, tend to support Justice Harlan's inference.