60 2015 3 100 100 100 100 100 100 100 100 1 100 100 100 3 1 100
100 100 1982 : 237 100 1982 1969 : 17 100.0 1985 : 68 1 1 50 41.1100210/51141.09589 100.1 50 41.0 100.0 3 19821985 1 100 1 2 1 2 2 N1473 2 3 2 1980 50 m 30 40 50 60 70 80 28 104 210 117 49 3 5.5 20.4 41.0 22.9 9.6 0.6 511 100.0 63.3 25.5 1.8 7.3 1.4 0.8 100.1* 7 8 9 10 11 12 13 14 * 100 1549 100.0 11985 : 67 2Bohrnstedt & Knoke19881990 : 28 32012 : 90 2 99.9, 100.1 Bohrnstedt & Knoke19881990 2 3 2 100 3 2 306 772 384 72 8 3 2 0.1 19.8 49.8 24.8 4.6 0.5 0.2 0.1
60 2015 3 100 100100.0100.00 3 3 99.9 100.0 100 13 Mosteller, Youtz, & Zahn1967 Diaconis & Freedman1979 3 4 1 100 4 3 1 100 100 16π 1/2 47 79.8 100 3 1 100 4 100 5 100 100 100 100 4 3 19821985 100 6
100 100 Balinski & Young, 198219872006 4 3 Balinski & Young19821987 2006 2 i i$1"!!!" n i N i p i$100"(n i$n ) m d i m u i p i p 315.315 3 d i d 315.3p 3 2 u i u 315.4p 3 2 q i q 315.3p 3 2 r i r 315.2 100 q 3 p i!d i#u i!p i q i$d i p i!d i#u i!p i q i$u i m 1$10 m!1 p i m q i
60 2015 3 m 5 m#1 5 q i!i q i 100 100 i r i i r i$"q i r i$q i m 1!10 m!1 2 q i 53.1 0.1 531 3.7 0.1 37 1!10 m!1 1 1 2 23 3 3 0.01 2 0.023 0.03 3 100 3.1 100 2 1 1985 : 68 1987 : 148 1 3 2 1982 4 1982 : 237 1985 : 68 1982 19821985
100 4 100 1982 : 237 i n i p i q i r i 1 2 3 70 54 23 47.619 36.734 15.646 47.6 36.7 15.6 47.7 36.7 15.6 147 100 99.9 100.0 2 q i r i 7 1 1 19821985 6!max(r i!q i )"min(r i!q i )!max(r i!p i )"min(r i!p i ) max(x!y)x y min(x!y) 100.2 99.7 100 2 1982 1985 1 1 4 100 19821985
60 2015 3 2 "max(q i!p i )!min(q i!p i ) "max(q i!p i )"min(q i!p i ) 1.45 2 1 1.451 21.45 2 1 1 2 1.45 1 1.45 2 1.45 1.38 1.451.38 100 7 3. 2 100 1 1 Balinski & Young, 198219872006
100 m 1 m 100 2 m 1 1"10 m!1 32 100!i max(r i!p i )!min(r i!p i ) 5 1982 : 237 5 1 i n i p i q i d i r i 1 2 3 4 5 6 7 971 950 795 721 579 347 262 20.994 20.540 17.189 15.589 12.518 7.502 5.664 21 21 17 16 13 8 6 20 20 17 15 12 7 5 1 4 7 3 5 6 2 21 21 17 16 12* 7* 6 4625 100 102 96 100 * 4 100 4.1 100 1982 : 237 100
!i q i#100 r i#q i!i q i#100 r i#q i!(1"10 m!1 ) ri#q i!i q i!100 r i#q i"(1"10 m!1 ) ri#q i 60 2015 3 4 4 100 1 m 1 m 1 1 1 1 2 100 s s 102 1 1 3 1 1 100 6 1 2 6 2 1 1 2 1 2 a 29.8 25.1 b 52.3 43.5 c 17.9 31.4 N 1 100.0 413 100.0 577 2
100 6 2 100 1 2 100 24 234 23 6 1 4 2 2 3 i h (i"!h i h ) 4 j i k (j"!k ) j i k 4.2 3 33.33 2 33.3 99.9 3 33.4
60 2015 3 99.99 6 16.66 2 16.7 100.2 100 2 16.6 100 100 100 4.3 100 1 100 100 1 100 7 8 7 8 1 2 3 1 2 3 a b c.01 43.51 12.48 43.65 43.55 12.8 43.41 43.88 12.71 a b c 12 13 43 13 100 100 100 100 101 100
100 7 7 8 2 101 2 100 911 3 1982 1985 9 10 9 1 1 2 3 a b c 12 43 13 43 13 100 100 100 9 2 7 a b 9 b a 10 b 7 1 2 b 10 1 2 11 c 7 2 3 c 11 2 3 7 8 100 10 2 11 3 1 2 3 1 2 3 a b c 12 43 13 43 13 a b c 12 12 43 13 100 100 100 100 100 100 2 1 7 3 8
60 2015 3 7 2 7 3 100 8 7 100 100 100 3 100 5 100 100 3 3 1 100 2 100 100 100 100 100 100 100
100 100 2 100.1 99.8100.1 100100 100 100 8 100 3 100 100 100 100 100 100 100 m m 100 100 2 100 100 100
60 2015 3 100 100 6 100 100 100 100 4 4 5 100 100 1 2 100 100.0100.00 3 100 3 100 19821985 4 100 100 4
100 100 100 100 100 100 100 100.1 99.9 100.1100.099.9 4 9 100 100 100 0 1 1 1 1 2
60 2015 3 2 1 1 1 10 11 100 100 1987 : 148 99.9100.1 1 100.0 2010 : 148 99.9 2000 : 150 10 794 319 40.1 100319/790.1763 2 40.2 100.1 31910 0.1 1996 : 78 1987 : 169 Balinski & Young, 19821987 Balinski1996 m Diaconis & Freedman, 19791982 Excel5
100 i 1 (q i#1)#q i q i#(q i!1) q i 1 q i! 1 2 1 2 1.5! 1.5 (1"2) $1!4142 1.45 2 d u x u#x$x#d x 2$du x$(du) max(1"p#q)!min(1"p#q) d u 2du#(d#u) d$1"u$2 1.333 1.412 7 3 2 1 abc 33.4533.033.55 2 33.433.35 33.25 1 333334 100 2 33 99 2 a 34 a 1 2 2 b c 34 2 a b c b c 100 100 2 Balinski & Young, 19821987 1 1 1
60 2015 3 Balinski, M. L., 1996, How should data be rounded?, Institute of Mathematical Statistics Lecture NotesMonograph Series, 28 : 33. Balinski, M. L., and Young, H. P., 1982, Fair Representation, Yale University.M. L. H. P. 1987 Bohrnstedt, G. W., and Knoke D., 1988, Statistics for Social Data Analysis 2nd. Ed., Peacock. 1990 Diaconis, P., and Freedman, D., 1979, On rounding percentages, Journal of the American Statistical Association, 74 : 359 364. 200616 : 265 276. 1996 Mosteller, F., Youtz, C., and Zahn, D., 1967, The distribution of sums of rounded percentages, Demography, 4 : 850 858. 2012 4 1985 2000 2010 ABCD 1987 1969 1982 3 2014 10 31