ocr IB 1967 '70 OAK RIDGE NATIONAL LABORATORY operated by UNION CARBIDE CORPORATION NUCLEAR DIVISION for the • U.S. ATOMIC ENERGY COMMISSION ORNL - TM- 1919 INDICES OF QUALITATIVE VARIATION Allen R. Wilcox NOTICE This document contains information of a preliminary natur<> and was prepared primarily far internal use at the Oak Ridge National Laboratory. It is subject to revision or correction and therefore doe> not represent o finol report. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. .-------------------LEGAL NOTICE--------------~ This report was prepared as on account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, expressed or implied, with respect to the accuracy, compl""t"'n~<~~:oo;, nr ulltf!fulnes!l: of the information conto1ned in this report, or that the use of any information, apparatus, method, or process disc lased in this report may not infringe privately ownsd rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report. As used in the above, "person acting on behalf of the Commission" includes ony employee or contractor of the Commission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employment or contract with the Commission, or his employment with such contractor. ;., r-· ~ -~~:;---- ----- ------~·- ----- -·-- --- -·- - ---- ~ ·-----. LEGAL NOTICE ~ ! This report was prepared as an account of Government sponsor-ed. work. Neither the United ~ States, nor th2 Commission, nor any pefson acting on behalf of the Commission: ~ A. Makes any warranty or representation, expressed or tmpUeC:, wtt.h respect to the accu­ 0 racy, completeness, or usefulness of the information con·Atned in tlis report, or that the use UJ 0\ of any information, apparatus, method, or process disc;:losed In tJ:js report may not infringe +> g privately owned rights; or () s:: r-l Q) B. Aesur.~.es any Uabilttles with respect to the use of. or for damages resulting from the •.-! !!! 0\ 'r-;) . r-l use of any- information, apparatus, method, or process disclosed in: this report. .O'd+> ~ As ueed in the above, uperson acting on behalf of the Comn:isslon" includes any em­ !-t Q) () :;k ployee or cor.. tractor of the Commission, or employee of such cor.trnctor, to the extent that Poi fl.l Q) () E-i such employee or contractor of the Commission, or employee of 3uch contractor prepares, UJ.-l ~· disseminates, or provides access to, any information pursuant to tis employment or contract ..C:WG-i 0 ' with the Commission, or his employment with such contractor. () !-t o]) !-t Pl !-t § ------- m >< . ... ~ Q) Q) :>, ~ fl.l r-l Q) fl.l •.-! p:; ;3: !-t a w m 'd .,; fl.l Q) :> fl.l :<:i () Q) E-i :>! :<:i s:: Q) () *0 ~ Q) \() m..c: (\J ~ :<:i ~ s:: r=3 E-i 0 C/) ~E-i I 0 ~ <!! H Cfl +> bO s::p:; p:jQ) E-i ~ 0 s:: OP-i ~· Q) Q) s:: ·w ·.-I > >< 0~ ~ ~ ..c: fl.l . I UJ::t: 0 e;lfl.l 0 0 II\ +> >::: 0 ·.-! 0 () ~ ,Q) Poi 0 :U'd 0 ~ r-l "~ s:: :>, p:; :>,<H ...::t .~ ~ H ·.-I ...:IS::.now:>~ .n ·:U 'd ['- l=lr£1 ;:.:: ~ ~QJ O..C:e:l l=l s:: 1 C/) ~ l2:E-i'd .pp:; +> (1j ;:.:: E-i w _oo ~ p:; co 0 Q) r£1 r£1 !-tG-i H H ... .po ~-t:2i ro o s:: !-t 0 E-iWmHOr£1 Pl ;3: Or£! i:: 1-- :»o :£ § Q) ;:;SF-t'dW bO!-t~G-i 0 S:: !-t tfg 0? r-l u ·.-! Pl t-i .,., m oo +> r£1p:jOO :2; () 0 +>- ~~ li< ~ 0 0 'd Q) !-t m ·.-! r£1 0 Q ~ :<:i IE-f QJ!-100 !-t l=ll=l H m o ·~ .PCJ..C:r£1 C/) 1-!W.PO::X:: ~ ...:I r£1 OW::SCfl 0 0 ;o ~ C/) Pl m :::> 0 t; H PlW H 0 ;:::::> ::s..c: Q) Q) 0 § UJ.P ..c:..c: H +>+> UJG-i m o w 1-1 ;:;;: !-t 0 w m ..C:CJ ...:I () •.-! Q) :;:::; 1-!G-ir-lp::j mG-i uo W0..-1 fl.l +>G-i W "' !-t 0 ~-t :» ro () Q) fl.l s:: fl.l fl.l •.-! (i) ·.-! 0 ..1'1 bO ..c: ..c: E-1<:::!+>+> * THIS PAGE WAS INTENTIONALLY · LEFT BLANK 3 CONTENTS Introduction . ....................... • ............ ~ . 5 Formal Properties of the Indices •..•.•.•.... .................... 5 Description of the Indices . ................. MODVR • ••••••••••••••• ·••.•••••••••••. · • . • • • • . • . • • • • . • • • • . • • • • • . • 7 'RA.NVR ••••••••••••••••••••••••••••••••••.••• ! • • • • • • • • • • • • • • • • • • • • • 8 A"VDEV • •••••••• •.••• 8 MriDI~ •••.••.••• ·..•. ............................................ 9 VARNC ••..•..•.••...•••..•...••.•.•. !' ••••••••••••••••••.•••••••••• • 11 STDEV . •••••.•••••••••••.•.•.• ••.•••••••••••.•••.••••••••· ••••••••• 14 Irn.EL. • • • • ·• • • • • • • • • • • • • • • • • • • • • • •. •. • • • • • • • • • • • • • • • • • • • • • • • • • • • .• • • • 15 . Summary and Conclusion ....••.•. .. .. 16 Appendix A: Input Preparation, Sample Problem, and Output for the NOMSTAT Computer Program .•.••.••.••••...••••.•.•.•..•••••.•• . l '7 Appendix B: Fortran List of the NOMSTAT Computer Program •.• ·•.••.. 23 4 Prefatory Note The statistical exploration +eported in this memorandum was under­ taken to help us to better implement our responsibilities for the analysis of American public. opinion on defense policy in general, and continental defense in particular. As familiarity with surveys will certify, many of the most interesting questions and sets of responses only meet nominal criteria, i.e., cannot be placed in relative positions on a scale. Accord­ ingly, the analyst who wishes to treat these questions with a desirable de­ gree of r~gO! and sophistication needs measures appropriate to the~, in the author's phrase, measures of qualitative variation. This memorandum is a modest effort to assist the analyst who desires to improve his ability to explore essentially qualitative data. As such, we hope it will be help­ ful to a variety of social scientists who confront such data in the course of em:Pirical ;r·e;:;earch. Davis B. Bobrow 5 t.:':l INDICES OF QUALITATIVE VARIATION Allen R. Wilcox I. INTRODUCTION In textbook treatments of measures of variation, statistics are always mentioned which measure ·the variation·of a univariate distribu­ tion when the variable under consideration satisfies the requirements of an ordinal, interval, or ratio scale. Most commonly, the range, the semi-interquartile range, the average deviation, the standard de­ viation, and the variance are presented and discussed. However, the presentation and discussion of measures of variation suitable for use with variables that satisfy only the requirements of a nominal scale is often completely ·absent. In addition, there appears to be no uni­ fied discussion of those·,..few., appropriate measures scattered through the literature. Based on the_ assumption that such measures may have considerable utility for the statistical handling of qualitative data, this paper-represents a first attempt to gather together and to gener~ ate alternative indices of qualitative variation. The treatment is introductory throughout. A·more intensive mathematical and empirical treatment of these measures will hopefully be provided by others who .wish to probe more deeply into the characteristics and utility of the statistics. II. FOP~ PROPERTIES OF THE INDICES The discussion is limited to measures that satisfy certain formal conditions. The f~rst three of these conditions represent what are considered to be desirable formal properties of such indices. First and second, the maximum and/or minimum values that such an index may obtain should not depend on the magnitude of either of the two basic parameters of a qu.a.l:i.ta.t.:i.ve d.ist.:r:i.bu.t:i.on--the number. of cases and. the number of categories. These conditions facilitate comparison of the values of a particular index, _even when they are derived from radically different distributions. Third, they must all have a standard range of values: in th~ case, from 0 to 1. This condition facilitates com­ parison among indices for the same distribution. 6 The final condition specifies the forms that a distribution must have when an index based on it obtains the maximum and minimum values. With the number of cases placed at 100 and the number of categories at 4, Fig. 1 illustrates with histograms the minimum-related and maximum­ related forms, respectively. Thus, the minimum value of 0 occurs if, and only if, all cases fall within ~ cat.egory. Conversely, the max­ imum value of l occurs if, and only if, an identical number of cases l fall within each category. This condition delineates the general 11 type 11 of measure to which this paper is being addressed. It is assumed that a.n indc][ that dooo not oatiofy thlo oond.i tion oan be more \u:rafully re­ lated to some other concept or idea. Measures that satisfy this condi­ tion can be thought of as indices of 11 generalized qualitative variation. 112 F 100 F 100 r r e e q q t u u e e n n 25 c c -l y 0 I I y 0 I I I I A B c D A B c D Category Category Fig.