33:
1447:
from the arithmetic mean. Because the standard deviation squares its differences, it tends to give more weight to larger differences and less weight to smaller differences compared to the mean absolute difference. When the arithmetic mean is finite, the mean absolute difference will also be finite, even when the standard deviation is infinite. See the
1439:), while the standard deviation is the square root of the variance about the mean (the second conventional central moment). The differences between L-moments and conventional moments are first seen in comparing the mean absolute difference and the standard deviation (the first L-moment and first conventional moment are both the mean).
1446:
and the mean absolute difference measure dispersion—how spread out are the values of a population or the probabilities of a distribution. The mean absolute difference is not defined in terms of a specific measure of central tendency, whereas the standard deviation is defined in terms of the deviation
1422:
If a random variable has a positive mean, then its relative mean absolute difference will always be greater than or equal to zero. If, additionally, the random variable can only take on values that are greater than or equal to zero, then its relative mean absolute difference will be less than 2.
962:
1824:
835:
493:
1632:
1148:
679:
2781:
2099:
1312:
The relative mean absolute difference is invariant to positive scaling, commutes with negation, and varies under translation in proportion to the ratio of the original and translated arithmetic means. That is to say, if
1219:
1224:
The relative mean absolute difference quantifies the mean absolute difference in comparison to the size of the mean and is a dimensionless quantity. The relative mean absolute difference is equal to twice the
3041:
4085:
2486:
2280:
1864:), in part because of the difficulty of finding an unbiased estimation for multiplying by the inverse of the mean. For example, even where the sample is known to be taken from a random variable
3093:
850:
2348:
2314:
2847:
3964:
161:
3532:
3350:
1660:
1233:. This relationship gives complementary perspectives to both the relative mean absolute difference and the Gini coefficient, including alternative ways of calculating their values.
2988:
2887:
714:
4133:
228:
3875:
2710:
3774:
3814:
3739:
282:
3567:
2523:
3662:
3602:
3480:
3298:
3154:
2408:
3420:
2959:
4047:
3914:
3385:
2935:
2674:
2244:
2209:
4005:
2641:
2382:
1499:
3636:
3454:
3272:
3128:
1013:
4428:
4026:
3835:
3704:
3683:
3501:
3319:
3238:
3217:
3196:
3175:
2915:
2607:
2586:
2565:
2544:
2450:
2429:
540:
4438:
2716:
1988:
4400:
4189:
1170:
1241:
The mean absolute difference is invariant to translations and negation, and varies proportionally to positive scaling. That is to say, if
54:
4340:
2169:
504:
76:
975:
4153:
3977:
2994:
1458:
plays similar role to the mean absolute difference but the distance standard deviation works with centered distances. See also
4055:
2456:
2250:
3048:
957:{\displaystyle \mathrm {MD} =\int _{0}^{\infty }\int _{-\infty }^{\infty }2\,f(x)\,f(x+\delta )\,\delta \,dx\,d\delta .}
695:
4433:
4396:
4317:
Mills, Jeffrey A.; Zandvakili, Sourushe (1997). "Statistical
Inference via Bootstrapping for Measures of Inequality".
1455:
3970:
47:
41:
2320:
2286:
1819:{\displaystyle \mathrm {RMD} (S)={\frac {\sum _{i=1}^{n}\sum _{j=1}^{n}|y_{i}-y_{j}|}{(n-1)\sum _{i=1}^{n}y_{i}}}}
4443:
2787:
3920:
58:
4163:
3507:
3325:
2495:
251:
102:
830:{\displaystyle \mathrm {MD} =\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x)\,f(y)\,|x-y|\,dx\,dy.}
17:
4326:
3780:
2965:
1892:
128:
94:
2853:
4237:
4101:
174:
3841:
2680:
4263:
488:{\displaystyle \mathrm {MD} =E=E_{X}]={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}|x_{i}-y_{j}|.}
4331:
3745:
4148:
3787:
2613:
2354:
98:
21:
3710:
4296:
3608:
3426:
3244:
3100:
2893:
1443:
3538:
2502:
3641:
3573:
3459:
3277:
3133:
2387:
4448:
3391:
2941:
1627:{\displaystyle \mathrm {MD} (S)={\frac {\sum _{i=1}^{n}\sum _{j=1}^{n}|y_{i}-y_{j}|}{n(n-1)}}}
986:
4032:
3881:
3356:
2920:
2646:
2215:
2176:
4384:
4361:
4336:
4288:
3984:
2620:
2361:
1226:
1143:{\displaystyle \mathrm {MD} =\int _{0}^{1}\int _{0}^{1}|Q(F_{1})-Q(F_{2})|\,dF_{1}\,dF_{2}.}
151:
116:
4250:
1161:
674:{\displaystyle \mathrm {MD} =\sum _{i=1}^{n}\sum _{j=1}^{n}f(y_{i})f(y_{j})|y_{i}-y_{j}|.}
272:
112:
3615:
3433:
3251:
3107:
4401:"Gini's Mean difference: a superior measure of variability for non-normal distributions"
4267:
4190:"Gini's Mean Difference: A Superior Measure of Variability for Non-Normal Distributions"
1164:
AM, the relative mean absolute difference, sometimes denoted by Δ or RMD, is defined by
4219:
4011:
3820:
3689:
3668:
3486:
3304:
3223:
3202:
3181:
3160:
2900:
2592:
2571:
2550:
2529:
2435:
2414:
147:
124:
4422:
4136:
1230:
133:
146:
The mean absolute difference is defined as the "average" or "mean", formally the
4388:
1459:
4366:
4349:
2776:{\displaystyle {\frac {1}{k-1}}\,{\sqrt {\frac {k}{k-2}}}{\text{ for }}k>2}
2094:{\displaystyle \operatorname {E} (R(S))=\sum _{i=0}^{n}p^{i}(1-p)^{n-i}r_{i},}
4158:
1843:
1833:
1830:
1645:
1642:
1638:
2144:
Examples of mean absolute difference and relative mean absolute difference
4341:
10.1002/(SICI)1099-1255(199703)12:2<133::AID-JAE433>3.0.CO;2-H
254:
the (empirical) mean absolute difference of the sequence of sample values
4168:
1436:
138:(GMD). The mean absolute difference is sometimes denoted by Δ or as MD.
4300:
1432:
4220:"How Has the Literature on Gini's Index Evolved in the Past 80 Years?"
1214:{\displaystyle \mathrm {RMD} ={\frac {\mathrm {MD} }{\mathrm {AM} }}.}
4292:
4276:
4375:
Nair, U. S. (1936). "Standard Error of Gini's Mean
Difference".
1858:
There does not exist, in general, an unbiased estimator for RMD(
26:
1160:
When the probability distribution has a finite and nonzero
3036:{\displaystyle {\frac {2\theta }{\mathrm {B} (0.5,k)}}\,}
1855:) can be calculated using bootstrap sampling techniques.
4080:{\displaystyle {\frac {\pi }{\sqrt {2}}}\approx 2.2214}
2481:{\displaystyle {\frac {2}{\sqrt {\pi }}}\approx 1.1284}
164:
with the same (unknown) distribution henceforth called
111:, which is the mean absolute difference divided by the
107:
4104:
4058:
4035:
4014:
3987:
3923:
3884:
3844:
3823:
3790:
3748:
3713:
3692:
3671:
3644:
3618:
3576:
3541:
3510:
3489:
3462:
3436:
3394:
3359:
3328:
3307:
3280:
3254:
3226:
3205:
3184:
3163:
3136:
3110:
3051:
2997:
2968:
2944:
2923:
2903:
2856:
2790:
2719:
2683:
2649:
2623:
2595:
2574:
2553:
2532:
2505:
2459:
2438:
2417:
2390:
2364:
2323:
2289:
2275:{\displaystyle {\frac {1}{\sqrt {12}}}\approx 0.2887}
2253:
2218:
2179:
1991:
1663:
1502:
1173:
1016:
853:
717:
543:
285:
177:
16:
For differences with respect to a central point, see
119:. The mean absolute difference is also known as the
3088:{\displaystyle {\frac {2}{k\mathrm {B} (0.5,k)}}\,}
246:of a population distributed uniformly according to
4127:
4079:
4041:
4020:
3999:
3958:
3908:
3869:
3829:
3808:
3768:
3733:
3698:
3677:
3656:
3630:
3596:
3561:
3526:
3495:
3474:
3448:
3414:
3379:
3344:
3313:
3292:
3266:
3232:
3211:
3190:
3169:
3148:
3122:
3087:
3035:
2982:
2953:
2929:
2909:
2881:
2841:
2775:
2704:
2668:
2635:
2601:
2580:
2559:
2538:
2517:
2480:
2444:
2423:
2402:
2376:
2342:
2308:
2274:
2238:
2203:
2093:
1818:
1626:
1213:
1142:
956:
829:
673:
487:
275:of the absolute value of all possible differences:
222:
842:An alternative form of the equation is given by:
4225:. Department of Economics, Dalhousie University
4350:"The Standard Error of Gini's Mean Difference"
8:
4408:Metron – International Journal of Statistics
2343:{\displaystyle {\frac {2}{3}}\approx 0.6667}
2309:{\displaystyle {\frac {1}{3}}\approx 0.3333}
531:, are the values with nonzero probabilities:
2842:{\displaystyle {\frac {2k}{(k-1)(2k-1)}}\,}
4197:Metron International Journal of Statistics
3959:{\displaystyle 2(1-p){\text{ for }}p>0}
1431:The mean absolute difference is twice the
1319:is a random variable and c is a constant:
4365:
4330:
4105:
4103:
4059:
4057:
4034:
4013:
3986:
3942:
3922:
3883:
3845:
3843:
3822:
3789:
3752:
3747:
3717:
3712:
3691:
3670:
3643:
3617:
3580:
3575:
3545:
3540:
3527:{\displaystyle {\sqrt {3}}\approx 1.7321}
3511:
3509:
3488:
3461:
3435:
3398:
3393:
3363:
3358:
3345:{\displaystyle {\sqrt {2}}\approx 1.4142}
3329:
3327:
3306:
3279:
3253:
3225:
3204:
3183:
3162:
3135:
3109:
3084:
3061:
3052:
3050:
3032:
3009:
2998:
2996:
2976:
2969:
2967:
2943:
2922:
2902:
2878:
2857:
2855:
2838:
2791:
2789:
2759:
2739:
2738:
2720:
2718:
2684:
2682:
2654:
2648:
2622:
2594:
2573:
2552:
2531:
2504:
2460:
2458:
2437:
2416:
2389:
2363:
2324:
2322:
2290:
2288:
2254:
2252:
2222:
2217:
2178:
2082:
2066:
2044:
2034:
2023:
1990:
1807:
1797:
1786:
1760:
1754:
1741:
1732:
1726:
1715:
1705:
1694:
1687:
1664:
1662:
1596:
1590:
1577:
1568:
1562:
1551:
1541:
1530:
1523:
1503:
1501:
1198:
1190:
1188:
1174:
1172:
1131:
1123:
1117:
1109:
1104:
1095:
1073:
1058:
1052:
1047:
1037:
1032:
1017:
1015:
944:
937:
933:
914:
901:
892:
884:
874:
869:
854:
852:
817:
810:
805:
791:
790:
777:
759:
751:
741:
733:
718:
716:
663:
657:
644:
635:
626:
607:
591:
580:
570:
559:
544:
542:
477:
471:
458:
449:
443:
432:
422:
411:
399:
390:
376:
362:
349:
345:
332:
317:
303:
286:
284:
209:
195:
178:
176:
162:independently and identically distributed
77:Learn how and when to remove this message
4262:. Bologna: Tipografia di Paolo Cuppini.
2142:
1962:But the expected value of any estimator
40:This article includes a list of general
4277:"Measurement of Inequality and Incomes"
4180:
101:of two independent values drawn from a
4246:
4235:
7:
4429:Statistical deviation and dispersion
2983:{\displaystyle {\sqrt {k}}\,\theta }
238:Specifically, in the discrete case,
150:, of the absolute difference of two
4439:Theory of probability distributions
2882:{\displaystyle {\frac {2}{2k-1}}\,}
4106:
4036:
3062:
3010:
2164:Relative mean absolute difference
1992:
1671:
1668:
1665:
1507:
1504:
1202:
1199:
1194:
1191:
1181:
1178:
1175:
1021:
1018:
893:
888:
875:
858:
855:
760:
755:
742:
737:
722:
719:
548:
545:
290:
287:
182:
179:
46:it lacks sufficient corresponding
14:
4354:Annals of Mathematical Statistics
4128:{\displaystyle \mathrm {B} (x,y)}
1229:which is defined in terms of the
1156:Relative mean absolute difference
223:{\displaystyle \mathrm {MD} :=E.}
108:relative mean absolute difference
95:measure of statistical dispersion
3870:{\displaystyle {\sqrt {p(1-p)}}}
2705:{\displaystyle {\frac {k}{k-1}}}
976:cumulative distribution function
31:
4319:Journal of Applied Econometrics
1451:for some specific comparisons.
4203:(2). Springer Verlag: 285–316.
4122:
4110:
3939:
3927:
3903:
3891:
3862:
3850:
3769:{\displaystyle 35/64=0.546875}
3078:
3066:
3026:
3014:
2832:
2817:
2814:
2802:
2063:
2050:
2013:
2010:
2004:
1998:
1779:
1767:
1761:
1733:
1681:
1675:
1618:
1606:
1597:
1569:
1517:
1511:
1427:Compared to standard deviation
1105:
1101:
1088:
1079:
1066:
1059:
930:
918:
911:
905:
806:
792:
787:
781:
774:
768:
664:
636:
632:
619:
613:
600:
478:
450:
384:
381:
377:
363:
359:
350:
338:
322:
318:
304:
300:
214:
210:
196:
192:
105:. A related statistic is the
20:. For paired differences, see
1:
3809:{\displaystyle 0\leq p\leq 1}
1849:Confidence intervals for RMD(
1448:
505:discrete probability function
123:(not to be confused with the
4312:. New York: Springer-Verlag.
4310:Ethical Social Index Numbers
3734:{\displaystyle 35/16=2.1875}
696:probability density function
242:For a random sample of size
4308:Chakravarty, S. R. (1990).
1842:), but is not, in general,
1456:distance standard deviation
4465:
3562:{\displaystyle 15/8=1.875}
2518:{\displaystyle \lambda =1}
15:
4389:10.1093/biomet/28.3-4.428
4218:Xu, Kuan (January 2004).
4188:Yitzhaki, Shlomo (2003).
3657:{\displaystyle \theta =1}
3597:{\displaystyle 5/8=0.625}
3475:{\displaystyle \theta =1}
3293:{\displaystyle \theta =1}
3149:{\displaystyle \theta =1}
2403:{\displaystyle \sigma =1}
1247:is a random variable and
271:can be calculated as the
115:, and equal to twice the
4348:Lomnicki, Z. A. (1952).
4260:Variabilità e Mutabilità
4164:Coefficient of variation
3415:{\displaystyle 3/4=0.75}
2954:{\displaystyle k\theta }
2161:Mean absolute difference
1982:)) will be of the form:
1454:The recently introduced
686:In the continuous case,
252:law of total expectation
121:absolute mean difference
103:probability distribution
91:mean absolute difference
4367:10.1214/aoms/1177729346
4042:{\displaystyle \infty }
3909:{\displaystyle 2p(1-p)}
3380:{\displaystyle 3/2=1.5}
2930:{\displaystyle \theta }
2669:{\displaystyle x_{m}=1}
2239:{\displaystyle 1/2=0.5}
2204:{\displaystyle a=0;b=1}
2121:)) can never equal RMD(
1907:) = 1) = 1 −
1474:from a random variable
61:more precise citations.
18:Mean absolute deviation
4275:Gini, Corrado (1921).
4258:Gini, Corrado (1912).
4245:Cite journal requires
4129:
4081:
4043:
4022:
4001:
4000:{\displaystyle \nu =2}
3960:
3910:
3871:
3831:
3810:
3770:
3735:
3700:
3679:
3658:
3632:
3598:
3563:
3528:
3497:
3476:
3450:
3416:
3381:
3346:
3315:
3294:
3268:
3234:
3213:
3192:
3171:
3150:
3124:
3089:
3037:
2984:
2955:
2931:
2911:
2883:
2843:
2777:
2706:
2670:
2637:
2636:{\displaystyle k>1}
2603:
2582:
2561:
2540:
2519:
2482:
2446:
2425:
2404:
2378:
2377:{\displaystyle \mu =0}
2344:
2310:
2276:
2240:
2205:
2095:
2039:
1893:Bernoulli distribution
1820:
1802:
1731:
1710:
1628:
1567:
1546:
1215:
1144:
958:
831:
675:
596:
575:
489:
448:
427:
224:
129:mean signed difference
4130:
4082:
4044:
4023:
4002:
3961:
3911:
3872:
3832:
3811:
3771:
3736:
3701:
3680:
3659:
3633:
3599:
3564:
3529:
3498:
3477:
3451:
3417:
3382:
3347:
3316:
3295:
3269:
3235:
3214:
3193:
3172:
3151:
3125:
3090:
3038:
2985:
2956:
2932:
2912:
2884:
2844:
2778:
2707:
2671:
2638:
2604:
2583:
2562:
2541:
2520:
2483:
2447:
2426:
2405:
2379:
2345:
2311:
2277:
2241:
2206:
2111:are constants. So E(
2096:
2019:
1821:
1782:
1711:
1690:
1629:
1547:
1526:
1216:
1145:
959:
832:
676:
576:
555:
490:
428:
407:
225:
97:equal to the average
4281:The Economic Journal
4102:
4056:
4033:
4012:
3985:
3921:
3882:
3842:
3821:
3788:
3746:
3711:
3690:
3669:
3642:
3616:
3574:
3539:
3508:
3487:
3460:
3434:
3392:
3357:
3326:
3305:
3278:
3252:
3224:
3203:
3182:
3161:
3134:
3108:
3049:
2995:
2966:
2942:
2921:
2901:
2854:
2788:
2717:
2681:
2647:
2621:
2593:
2572:
2551:
2530:
2503:
2457:
2436:
2415:
2388:
2362:
2321:
2287:
2251:
2216:
2177:
1989:
1661:
1500:
1470:For a random sample
1171:
1014:
851:
715:
541:
283:
175:
4268:1912vamu.book.....G
4149:Mean absolute error
3631:{\displaystyle k=4}
3449:{\displaystyle k=3}
3267:{\displaystyle k=2}
3123:{\displaystyle k=1}
2145:
1057:
1042:
897:
879:
764:
746:
99:absolute difference
22:Mean absolute error
4434:Summary statistics
4125:
4077:
4039:
4018:
3997:
3956:
3906:
3867:
3827:
3806:
3766:
3731:
3696:
3675:
3654:
3628:
3594:
3559:
3524:
3493:
3472:
3446:
3412:
3377:
3342:
3311:
3290:
3264:
3230:
3209:
3188:
3167:
3146:
3120:
3085:
3033:
2980:
2951:
2927:
2907:
2879:
2839:
2773:
2702:
2666:
2633:
2599:
2578:
2557:
2536:
2515:
2478:
2442:
2421:
2400:
2374:
2340:
2306:
2272:
2236:
2201:
2170:Continuous uniform
2158:Standard deviation
2143:
2091:
1816:
1654:). The statistic:
1624:
1444:standard deviation
1211:
1140:
1043:
1028:
1004:, it follows that:
954:
880:
865:
827:
747:
729:
671:
485:
220:
93:(univariate) is a
4095:
4094:
4069:
4068:
4021:{\displaystyle 0}
3945:
3865:
3830:{\displaystyle p}
3699:{\displaystyle 2}
3678:{\displaystyle 4}
3516:
3496:{\displaystyle 3}
3334:
3314:{\displaystyle 2}
3233:{\displaystyle 1}
3212:{\displaystyle 1}
3191:{\displaystyle 1}
3170:{\displaystyle 1}
3082:
3030:
2974:
2910:{\displaystyle k}
2876:
2836:
2762:
2757:
2756:
2736:
2700:
2602:{\displaystyle 1}
2581:{\displaystyle 1}
2560:{\displaystyle 1}
2539:{\displaystyle 1}
2470:
2469:
2445:{\displaystyle 1}
2424:{\displaystyle 0}
2332:
2298:
2264:
2263:
2135:between 0 and 1.
1952:)/(1 +
1874:) for an unknown
1814:
1622:
1466:Sample estimators
1206:
987:quantile function
405:
87:
86:
79:
4456:
4444:Scale statistics
4415:
4405:
4397:Yitzhaki, Shlomo
4392:
4383:(3–4): 428–436.
4371:
4369:
4344:
4334:
4313:
4304:
4287:(121): 124–126.
4271:
4254:
4248:
4243:
4241:
4233:
4231:
4230:
4224:
4205:
4204:
4194:
4185:
4134:
4132:
4131:
4126:
4109:
4088:
4086:
4084:
4083:
4078:
4070:
4064:
4060:
4048:
4046:
4045:
4040:
4027:
4025:
4024:
4019:
4006:
4004:
4003:
3998:
3965:
3963:
3962:
3957:
3946:
3943:
3915:
3913:
3912:
3907:
3876:
3874:
3873:
3868:
3866:
3846:
3836:
3834:
3833:
3828:
3815:
3813:
3812:
3807:
3775:
3773:
3772:
3767:
3756:
3740:
3738:
3737:
3732:
3721:
3705:
3703:
3702:
3697:
3684:
3682:
3681:
3676:
3663:
3661:
3660:
3655:
3637:
3635:
3634:
3629:
3603:
3601:
3600:
3595:
3584:
3568:
3566:
3565:
3560:
3549:
3533:
3531:
3530:
3525:
3517:
3512:
3502:
3500:
3499:
3494:
3481:
3479:
3478:
3473:
3455:
3453:
3452:
3447:
3421:
3419:
3418:
3413:
3402:
3386:
3384:
3383:
3378:
3367:
3351:
3349:
3348:
3343:
3335:
3330:
3320:
3318:
3317:
3312:
3299:
3297:
3296:
3291:
3273:
3271:
3270:
3265:
3239:
3237:
3236:
3231:
3218:
3216:
3215:
3210:
3197:
3195:
3194:
3189:
3176:
3174:
3173:
3168:
3155:
3153:
3152:
3147:
3129:
3127:
3126:
3121:
3094:
3092:
3091:
3086:
3083:
3081:
3065:
3053:
3042:
3040:
3039:
3034:
3031:
3029:
3013:
3007:
2999:
2989:
2987:
2986:
2981:
2975:
2970:
2960:
2958:
2957:
2952:
2936:
2934:
2933:
2928:
2916:
2914:
2913:
2908:
2888:
2886:
2885:
2880:
2877:
2875:
2858:
2848:
2846:
2845:
2840:
2837:
2835:
2800:
2792:
2782:
2780:
2779:
2774:
2763:
2760:
2758:
2755:
2741:
2740:
2737:
2735:
2721:
2711:
2709:
2708:
2703:
2701:
2699:
2685:
2675:
2673:
2672:
2667:
2659:
2658:
2642:
2640:
2639:
2634:
2608:
2606:
2605:
2600:
2587:
2585:
2584:
2579:
2566:
2564:
2563:
2558:
2545:
2543:
2542:
2537:
2524:
2522:
2521:
2516:
2487:
2485:
2484:
2479:
2471:
2465:
2461:
2451:
2449:
2448:
2443:
2430:
2428:
2427:
2422:
2409:
2407:
2406:
2401:
2383:
2381:
2380:
2375:
2349:
2347:
2346:
2341:
2333:
2325:
2315:
2313:
2312:
2307:
2299:
2291:
2281:
2279:
2278:
2273:
2265:
2259:
2255:
2245:
2243:
2242:
2237:
2226:
2210:
2208:
2207:
2202:
2146:
2100:
2098:
2097:
2092:
2087:
2086:
2077:
2076:
2049:
2048:
2038:
2033:
1957:
1928:
1911:
1890:
1825:
1823:
1822:
1817:
1815:
1813:
1812:
1811:
1801:
1796:
1765:
1764:
1759:
1758:
1746:
1745:
1736:
1730:
1725:
1709:
1704:
1688:
1674:
1633:
1631:
1630:
1625:
1623:
1621:
1601:
1600:
1595:
1594:
1582:
1581:
1572:
1566:
1561:
1545:
1540:
1524:
1510:
1493:, the statistic
1480:, consisting of
1227:Gini coefficient
1220:
1218:
1217:
1212:
1207:
1205:
1197:
1189:
1184:
1149:
1147:
1146:
1141:
1136:
1135:
1122:
1121:
1108:
1100:
1099:
1078:
1077:
1062:
1056:
1051:
1041:
1036:
1024:
963:
961:
960:
955:
896:
891:
878:
873:
861:
836:
834:
833:
828:
809:
795:
763:
758:
745:
740:
725:
680:
678:
677:
672:
667:
662:
661:
649:
648:
639:
631:
630:
612:
611:
595:
590:
574:
569:
551:
494:
492:
491:
486:
481:
476:
475:
463:
462:
453:
447:
442:
426:
421:
406:
404:
403:
391:
380:
366:
358:
357:
353:
337:
336:
321:
307:
293:
229:
227:
226:
221:
213:
199:
185:
152:random variables
117:Gini coefficient
82:
75:
71:
68:
62:
57:this article by
48:inline citations
35:
34:
27:
4464:
4463:
4459:
4458:
4457:
4455:
4454:
4453:
4419:
4418:
4403:
4395:
4374:
4347:
4332:10.1.1.172.5003
4316:
4307:
4293:10.2307/2223319
4274:
4257:
4244:
4234:
4228:
4226:
4222:
4217:
4214:
4209:
4208:
4192:
4187:
4186:
4182:
4177:
4145:
4100:
4099:
4054:
4053:
4051:
4031:
4030:
4010:
4009:
3983:
3982:
3944: for
3919:
3918:
3880:
3879:
3840:
3839:
3819:
3818:
3786:
3785:
3744:
3743:
3709:
3708:
3688:
3687:
3667:
3666:
3640:
3639:
3614:
3613:
3572:
3571:
3537:
3536:
3506:
3505:
3485:
3484:
3458:
3457:
3432:
3431:
3390:
3389:
3355:
3354:
3324:
3323:
3303:
3302:
3276:
3275:
3250:
3249:
3222:
3221:
3201:
3200:
3180:
3179:
3159:
3158:
3132:
3131:
3106:
3105:
3057:
3047:
3046:
3008:
3000:
2993:
2992:
2964:
2963:
2940:
2939:
2919:
2918:
2899:
2898:
2862:
2852:
2851:
2801:
2793:
2786:
2785:
2761: for
2745:
2725:
2715:
2714:
2689:
2679:
2678:
2650:
2645:
2644:
2619:
2618:
2591:
2590:
2570:
2569:
2549:
2548:
2528:
2527:
2501:
2500:
2455:
2454:
2434:
2433:
2413:
2412:
2386:
2385:
2360:
2359:
2319:
2318:
2285:
2284:
2249:
2248:
2214:
2213:
2175:
2174:
2141:
2110:
2078:
2062:
2040:
1987:
1986:
1948:(1 −
1933:
1913:
1896:
1889:) − 1
1879:
1803:
1766:
1750:
1737:
1689:
1659:
1658:
1602:
1586:
1573:
1525:
1498:
1497:
1492:
1468:
1429:
1251:is a constant:
1239:
1169:
1168:
1162:arithmetic mean
1158:
1127:
1113:
1091:
1069:
1012:
1011:
996:), then, since
849:
848:
713:
712:
653:
640:
622:
603:
539:
538:
522:
467:
454:
395:
341:
328:
281:
280:
273:arithmetic mean
262:
236:
173:
172:
144:
136:mean difference
113:arithmetic mean
83:
72:
66:
63:
53:Please help to
52:
36:
32:
25:
12:
11:
5:
4462:
4460:
4452:
4451:
4446:
4441:
4436:
4431:
4421:
4420:
4417:
4416:
4393:
4372:
4360:(4): 635–637.
4345:
4325:(2): 133–150.
4314:
4305:
4272:
4255:
4247:|journal=
4213:
4210:
4207:
4206:
4179:
4178:
4176:
4173:
4172:
4171:
4166:
4161:
4156:
4154:Mean deviation
4151:
4144:
4141:
4140:
4139:
4124:
4121:
4118:
4115:
4112:
4108:
4093:
4092:
4089:
4076:
4073:
4067:
4063:
4049:
4038:
4028:
4017:
4007:
3996:
3993:
3990:
3980:
3967:
3966:
3955:
3952:
3949:
3941:
3938:
3935:
3932:
3929:
3926:
3916:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3877:
3864:
3861:
3858:
3855:
3852:
3849:
3837:
3826:
3816:
3805:
3802:
3799:
3796:
3793:
3783:
3777:
3776:
3765:
3762:
3759:
3755:
3751:
3741:
3730:
3727:
3724:
3720:
3716:
3706:
3695:
3685:
3674:
3664:
3653:
3650:
3647:
3627:
3624:
3621:
3611:
3605:
3604:
3593:
3590:
3587:
3583:
3579:
3569:
3558:
3555:
3552:
3548:
3544:
3534:
3523:
3520:
3515:
3503:
3492:
3482:
3471:
3468:
3465:
3445:
3442:
3439:
3429:
3423:
3422:
3411:
3408:
3405:
3401:
3397:
3387:
3376:
3373:
3370:
3366:
3362:
3352:
3341:
3338:
3333:
3321:
3310:
3300:
3289:
3286:
3283:
3263:
3260:
3257:
3247:
3241:
3240:
3229:
3219:
3208:
3198:
3187:
3177:
3166:
3156:
3145:
3142:
3139:
3119:
3116:
3113:
3103:
3097:
3096:
3080:
3077:
3074:
3071:
3068:
3064:
3060:
3056:
3044:
3028:
3025:
3022:
3019:
3016:
3012:
3006:
3003:
2990:
2979:
2973:
2961:
2950:
2947:
2937:
2926:
2906:
2896:
2890:
2889:
2874:
2871:
2868:
2865:
2861:
2849:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2799:
2796:
2783:
2772:
2769:
2766:
2754:
2751:
2748:
2744:
2734:
2731:
2728:
2724:
2712:
2698:
2695:
2692:
2688:
2676:
2665:
2662:
2657:
2653:
2632:
2629:
2626:
2616:
2610:
2609:
2598:
2588:
2577:
2567:
2556:
2546:
2535:
2525:
2514:
2511:
2508:
2498:
2492:
2491:
2488:
2477:
2474:
2468:
2464:
2452:
2441:
2431:
2420:
2410:
2399:
2396:
2393:
2373:
2370:
2367:
2357:
2351:
2350:
2339:
2336:
2331:
2328:
2316:
2305:
2302:
2297:
2294:
2282:
2271:
2268:
2262:
2258:
2246:
2235:
2232:
2229:
2225:
2221:
2211:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2172:
2166:
2165:
2162:
2159:
2156:
2153:
2150:
2140:
2137:
2108:
2102:
2101:
2090:
2085:
2081:
2075:
2072:
2069:
2065:
2061:
2058:
2055:
2052:
2047:
2043:
2037:
2032:
2029:
2026:
2022:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1960:
1959:
1827:
1826:
1810:
1806:
1800:
1795:
1792:
1789:
1785:
1781:
1778:
1775:
1772:
1769:
1763:
1757:
1753:
1749:
1744:
1740:
1735:
1729:
1724:
1721:
1718:
1714:
1708:
1703:
1700:
1697:
1693:
1686:
1683:
1680:
1677:
1673:
1670:
1667:
1635:
1634:
1620:
1617:
1614:
1611:
1608:
1605:
1599:
1593:
1589:
1585:
1580:
1576:
1571:
1565:
1560:
1557:
1554:
1550:
1544:
1539:
1536:
1533:
1529:
1522:
1519:
1516:
1513:
1509:
1506:
1488:
1467:
1464:
1428:
1425:
1420:
1419:
1397:
1382:
1310:
1309:
1287:
1272:
1238:
1235:
1222:
1221:
1210:
1204:
1201:
1196:
1193:
1187:
1183:
1180:
1177:
1157:
1154:
1153:
1152:
1151:
1150:
1139:
1134:
1130:
1126:
1120:
1116:
1112:
1107:
1103:
1098:
1094:
1090:
1087:
1084:
1081:
1076:
1072:
1068:
1065:
1061:
1055:
1050:
1046:
1040:
1035:
1031:
1027:
1023:
1020:
1006:
1005:
967:
966:
965:
964:
953:
950:
947:
943:
940:
936:
932:
929:
926:
923:
920:
917:
913:
910:
907:
904:
900:
895:
890:
887:
883:
877:
872:
868:
864:
860:
857:
840:
839:
838:
837:
826:
823:
820:
816:
813:
808:
804:
801:
798:
794:
789:
786:
783:
780:
776:
773:
770:
767:
762:
757:
754:
750:
744:
739:
736:
732:
728:
724:
721:
707:
706:
684:
683:
682:
681:
670:
666:
660:
656:
652:
647:
643:
638:
634:
629:
625:
621:
618:
615:
610:
606:
602:
599:
594:
589:
586:
583:
579:
573:
568:
565:
562:
558:
554:
550:
547:
533:
532:
518:
496:
495:
484:
480:
474:
470:
466:
461:
457:
452:
446:
441:
438:
435:
431:
425:
420:
417:
414:
410:
402:
398:
394:
389:
386:
383:
379:
375:
372:
369:
365:
361:
356:
352:
348:
344:
340:
335:
331:
327:
324:
320:
316:
313:
310:
306:
302:
299:
296:
292:
289:
277:
276:
258:
235:
232:
231:
230:
219:
216:
212:
208:
205:
202:
198:
194:
191:
188:
184:
181:
148:expected value
143:
140:
125:absolute value
85:
84:
39:
37:
30:
13:
10:
9:
6:
4:
3:
2:
4461:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4426:
4424:
4413:
4409:
4402:
4398:
4394:
4390:
4386:
4382:
4378:
4373:
4368:
4363:
4359:
4355:
4351:
4346:
4342:
4338:
4333:
4328:
4324:
4320:
4315:
4311:
4306:
4302:
4298:
4294:
4290:
4286:
4282:
4278:
4273:
4269:
4265:
4261:
4256:
4252:
4239:
4221:
4216:
4215:
4211:
4202:
4198:
4191:
4184:
4181:
4174:
4170:
4167:
4165:
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4146:
4142:
4138:
4137:Beta function
4119:
4116:
4113:
4097:
4096:
4090:
4074:
4071:
4065:
4061:
4050:
4029:
4015:
4008:
3994:
3991:
3988:
3981:
3979:
3975:
3974:
3969:
3968:
3953:
3950:
3947:
3936:
3933:
3930:
3924:
3917:
3900:
3897:
3894:
3888:
3885:
3878:
3859:
3856:
3853:
3847:
3838:
3824:
3817:
3803:
3800:
3797:
3794:
3791:
3784:
3782:
3779:
3778:
3763:
3760:
3757:
3753:
3749:
3742:
3728:
3725:
3722:
3718:
3714:
3707:
3693:
3686:
3672:
3665:
3651:
3648:
3645:
3625:
3622:
3619:
3612:
3610:
3607:
3606:
3591:
3588:
3585:
3581:
3577:
3570:
3556:
3553:
3550:
3546:
3542:
3535:
3521:
3518:
3513:
3504:
3490:
3483:
3469:
3466:
3463:
3443:
3440:
3437:
3430:
3428:
3425:
3424:
3409:
3406:
3403:
3399:
3395:
3388:
3374:
3371:
3368:
3364:
3360:
3353:
3339:
3336:
3331:
3322:
3308:
3301:
3287:
3284:
3281:
3261:
3258:
3255:
3248:
3246:
3243:
3242:
3227:
3220:
3206:
3199:
3185:
3178:
3164:
3157:
3143:
3140:
3137:
3117:
3114:
3111:
3104:
3102:
3099:
3098:
3075:
3072:
3069:
3058:
3054:
3045:
3023:
3020:
3017:
3004:
3001:
2991:
2977:
2971:
2962:
2948:
2945:
2938:
2924:
2904:
2897:
2895:
2892:
2891:
2872:
2869:
2866:
2863:
2859:
2850:
2829:
2826:
2823:
2820:
2811:
2808:
2805:
2797:
2794:
2784:
2770:
2767:
2764:
2752:
2749:
2746:
2742:
2732:
2729:
2726:
2722:
2713:
2696:
2693:
2690:
2686:
2677:
2663:
2660:
2655:
2651:
2630:
2627:
2624:
2617:
2615:
2612:
2611:
2596:
2589:
2575:
2568:
2554:
2547:
2533:
2526:
2512:
2509:
2506:
2499:
2497:
2494:
2493:
2489:
2475:
2472:
2466:
2462:
2453:
2439:
2432:
2418:
2411:
2397:
2394:
2391:
2371:
2368:
2365:
2358:
2356:
2353:
2352:
2337:
2334:
2329:
2326:
2317:
2303:
2300:
2295:
2292:
2283:
2269:
2266:
2260:
2256:
2247:
2233:
2230:
2227:
2223:
2219:
2212:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2173:
2171:
2168:
2167:
2163:
2160:
2157:
2154:
2151:
2148:
2147:
2138:
2136:
2134:
2130:
2126:
2125:
2120:
2119:
2114:
2107:
2088:
2083:
2079:
2073:
2070:
2067:
2059:
2056:
2053:
2045:
2041:
2035:
2030:
2027:
2024:
2020:
2016:
2007:
2001:
1995:
1985:
1984:
1983:
1981:
1977:
1976:
1971:
1970:
1965:
1955:
1951:
1947:
1943:
1939:
1938:
1932:
1931:
1930:
1927:
1923:
1919:
1918:
1910:
1906:
1902:
1901:
1894:
1888:
1884:
1883:
1877:
1873:
1869:
1868:
1863:
1862:
1856:
1854:
1853:
1847:
1845:
1841:
1840:
1835:
1832:
1808:
1804:
1798:
1793:
1790:
1787:
1783:
1776:
1773:
1770:
1755:
1751:
1747:
1742:
1738:
1727:
1722:
1719:
1716:
1712:
1706:
1701:
1698:
1695:
1691:
1684:
1678:
1657:
1656:
1655:
1653:
1652:
1647:
1644:
1640:
1615:
1612:
1609:
1603:
1591:
1587:
1583:
1578:
1574:
1563:
1558:
1555:
1552:
1548:
1542:
1537:
1534:
1531:
1527:
1520:
1514:
1496:
1495:
1494:
1491:
1487:
1483:
1479:
1478:
1473:
1465:
1463:
1461:
1457:
1452:
1450:
1445:
1440:
1438:
1434:
1426:
1424:
1417:
1413:
1412:
1407:
1406:
1402:
1398:
1395:
1394:
1389:
1388:
1383:
1380:
1379:
1374:
1370:
1369:
1364:
1360:
1359:
1354:
1350:
1349:
1344:
1343:
1338:
1337:
1332:
1328:
1327:
1322:
1321:
1320:
1318:
1317:
1307:
1306:
1301:
1297:
1296:
1292:
1288:
1285:
1284:
1279:
1278:
1273:
1270:
1269:
1264:
1260:
1259:
1254:
1253:
1252:
1250:
1246:
1245:
1236:
1234:
1232:
1228:
1208:
1185:
1167:
1166:
1165:
1163:
1155:
1137:
1132:
1128:
1124:
1118:
1114:
1110:
1096:
1092:
1085:
1082:
1074:
1070:
1063:
1053:
1048:
1044:
1038:
1033:
1029:
1025:
1010:
1009:
1008:
1007:
1003:
999:
998:f(x)=dF(x)/dx
995:
991:
988:
984:
980:
977:
973:
969:
968:
951:
948:
945:
941:
938:
934:
927:
924:
921:
915:
908:
902:
898:
885:
881:
870:
866:
862:
847:
846:
845:
844:
843:
824:
821:
818:
814:
811:
802:
799:
796:
784:
778:
771:
765:
752:
748:
734:
730:
726:
711:
710:
709:
708:
704:
700:
697:
693:
689:
688:
687:
668:
658:
654:
650:
645:
641:
627:
623:
616:
608:
604:
597:
592:
587:
584:
581:
577:
571:
566:
563:
560:
556:
552:
537:
536:
535:
534:
530:
526:
521:
517:
513:
509:
506:
502:
498:
497:
482:
472:
468:
464:
459:
455:
444:
439:
436:
433:
429:
423:
418:
415:
412:
408:
400:
396:
392:
387:
373:
370:
367:
354:
346:
342:
333:
329:
325:
314:
311:
308:
297:
294:
279:
278:
274:
270:
266:
261:
257:
253:
249:
245:
241:
240:
239:
233:
217:
206:
203:
200:
189:
186:
171:
170:
169:
167:
163:
160:
156:
153:
149:
141:
139:
137:
135:
130:
126:
122:
118:
114:
110:
109:
104:
100:
96:
92:
81:
78:
70:
67:November 2010
60:
56:
50:
49:
43:
38:
29:
28:
23:
19:
4411:
4407:
4380:
4376:
4357:
4353:
4322:
4318:
4309:
4284:
4280:
4259:
4238:cite journal
4227:. Retrieved
4200:
4196:
4183:
3972:
2149:Distribution
2132:
2128:
2123:
2122:
2117:
2116:
2112:
2105:
2103:
1979:
1974:
1973:
1968:
1967:
1963:
1961:
1953:
1949:
1945:
1941:
1936:
1935:
1925:
1921:
1916:
1915:
1908:
1904:
1899:
1898:
1886:
1881:
1880:
1875:
1871:
1866:
1865:
1860:
1859:
1857:
1851:
1850:
1848:
1838:
1837:
1828:
1650:
1649:
1636:
1489:
1485:
1481:
1476:
1475:
1471:
1469:
1460:E-statistics
1453:
1441:
1435:(the second
1430:
1421:
1415:
1410:
1409:
1404:
1403:
1400:
1392:
1391:
1386:
1385:
1377:
1376:
1372:
1367:
1366:
1362:
1357:
1356:
1352:
1347:
1346:
1341:
1340:
1335:
1334:
1330:
1325:
1324:
1315:
1314:
1311:
1304:
1303:
1299:
1294:
1293:
1290:
1282:
1281:
1276:
1275:
1267:
1266:
1262:
1257:
1256:
1248:
1243:
1242:
1240:
1231:Lorenz curve
1223:
1159:
1001:
997:
993:
989:
982:
978:
971:
841:
702:
698:
691:
685:
528:
524:
519:
515:
511:
507:
500:
268:
264:
259:
255:
247:
243:
237:
165:
158:
154:
145:
132:
120:
106:
90:
88:
73:
64:
45:
2496:Exponential
2131:)) for all
234:Calculation
59:introducing
4423:Categories
4414:: 285–316.
4377:Biometrika
4229:2006-06-01
4175:References
4091:undefined
3971:Student's
2490:undefined
2152:Parameters
2104:where the
1895:, so that
1831:consistent
1639:consistent
1237:Properties
142:Definition
131:) and the
42:references
4327:CiteSeerX
4159:Estimator
4072:≈
4062:π
4037:∞
3989:ν
3934:−
3898:−
3857:−
3801:≤
3795:≤
3781:Bernoulli
3646:θ
3519:≈
3464:θ
3337:≈
3282:θ
3138:θ
3005:θ
2978:θ
2949:θ
2925:θ
2870:−
2827:−
2809:−
2750:−
2730:−
2694:−
2507:λ
2473:≈
2467:π
2392:σ
2366:μ
2335:≈
2301:≈
2267:≈
2071:−
2057:−
2021:∑
1996:
1972:) of RMD(
1924:) = 2) =
1834:estimator
1784:∑
1774:−
1748:−
1713:∑
1692:∑
1646:estimator
1613:−
1584:−
1549:∑
1528:∑
1442:Both the
1390:) = −RMD(
1361:) / (1 +
1339:) · mean(
1083:−
1045:∫
1030:∫
1002:Q(F(x))=x
949:δ
935:δ
928:δ
894:∞
889:∞
886:−
882:∫
876:∞
867:∫
800:−
761:∞
756:∞
753:−
749:∫
743:∞
738:∞
735:−
731:∫
651:−
578:∑
557:∑
514:), where
465:−
430:∑
409:∑
371:−
312:−
250:, by the
204:−
4449:Distance
4399:(2003).
4169:L-moment
4143:See also
3764:0.546875
2139:Examples
1891:has the
1844:unbiased
1643:unbiased
1449:examples
1437:L-moment
1408:) = RMD(
1375:≠ −mean(
1355:) = RMD(
1345:)/(mean(
1333:) = RMD(
4301:2223319
4264:Bibcode
4212:Sources
4135:is the
4087:
4052:
1929:, then
1836:of RMD(
1484:values
1433:L-scale
1418:> 0.
1371:)) for
1365:/ mean(
1280:) = MD(
1265:) = MD(
985:) with
527:= 1 to
267:= 1 to
127:of the
55:improve
4329:
4299:
4075:2.2214
3729:2.1875
3522:1.7321
3340:1.4142
2614:Pareto
2476:1.1284
2355:Normal
2338:0.6667
2304:0.3333
2270:0.2887
1944:)) = 2
1878:, and
1648:of MD(
1414:) for
1396:), and
1286:), and
974:has a
694:has a
503:has a
44:, but
4404:(PDF)
4297:JSTOR
4223:(PDF)
4193:(PDF)
3609:Gamma
3592:0.625
3557:1.875
3427:Gamma
3245:Gamma
3101:Gamma
2894:Gamma
1829:is a
1637:is a
1384:RMD(−
1302:| MD(
1298:) = |
4251:help
3978:d.f.
3976:, 2
3951:>
3410:0.75
2768:>
2628:>
2155:Mean
1934:RMD(
1912:and
1641:and
1399:RMD(
1351:) +
1323:RMD(
1274:MD(−
1000:and
157:and
134:Gini
89:The
4385:doi
4362:doi
4337:doi
4289:doi
3375:1.5
3070:0.5
3018:0.5
2234:0.5
1914:Pr(
1897:Pr(
1289:MD(
1255:MD(
970:if
690:if
499:if
4425::
4412:61
4410:.
4406:.
4381:28
4379:.
4358:23
4356:.
4352:.
4335:.
4323:12
4321:.
4295:.
4285:31
4283:.
4279:.
4242::
4240:}}
4236:{{
4201:61
4199:.
4195:.
4098:†
3758:64
3750:35
3723:16
3715:35
3638:;
3543:15
3456:;
3274:;
3130:;
3095:†
2917:;
2643:;
2384:;
2261:12
1846:.
1462:.
1381:),
1329:+
1308:).
1271:),
1261:+
705:):
523:,
263:,
187::=
168:.
4391:.
4387::
4370:.
4364::
4343:.
4339::
4303:.
4291::
4270:.
4266::
4253:)
4249:(
4232:.
4123:)
4120:y
4117:,
4114:x
4111:(
4107:B
4066:2
4016:0
3995:2
3992:=
3973:t
3954:0
3948:p
3940:)
3937:p
3931:1
3928:(
3925:2
3904:)
3901:p
3895:1
3892:(
3889:p
3886:2
3863:)
3860:p
3854:1
3851:(
3848:p
3825:p
3804:1
3798:p
3792:0
3761:=
3754:/
3726:=
3719:/
3694:2
3673:4
3652:1
3649:=
3626:4
3623:=
3620:k
3589:=
3586:8
3582:/
3578:5
3554:=
3551:8
3547:/
3514:3
3491:3
3470:1
3467:=
3444:3
3441:=
3438:k
3407:=
3404:4
3400:/
3396:3
3372:=
3369:2
3365:/
3361:3
3332:2
3309:2
3288:1
3285:=
3262:2
3259:=
3256:k
3228:1
3207:1
3186:1
3165:1
3144:1
3141:=
3118:1
3115:=
3112:k
3079:)
3076:k
3073:,
3067:(
3063:B
3059:k
3055:2
3043:†
3027:)
3024:k
3021:,
3015:(
3011:B
3002:2
2972:k
2946:k
2905:k
2873:1
2867:k
2864:2
2860:2
2833:)
2830:1
2824:k
2821:2
2818:(
2815:)
2812:1
2806:k
2803:(
2798:k
2795:2
2771:2
2765:k
2753:2
2747:k
2743:k
2733:1
2727:k
2723:1
2697:1
2691:k
2687:k
2664:1
2661:=
2656:m
2652:x
2631:1
2625:k
2597:1
2576:1
2555:1
2534:1
2513:1
2510:=
2463:2
2440:1
2419:0
2398:1
2395:=
2372:0
2369:=
2330:3
2327:2
2296:3
2293:1
2257:1
2231:=
2228:2
2224:/
2220:1
2199:1
2196:=
2193:b
2190:;
2187:0
2184:=
2181:a
2133:p
2129:p
2127:(
2124:X
2118:S
2115:(
2113:R
2109:i
2106:r
2089:,
2084:i
2080:r
2074:i
2068:n
2064:)
2060:p
2054:1
2051:(
2046:i
2042:p
2036:n
2031:0
2028:=
2025:i
2017:=
2014:)
2011:)
2008:S
2005:(
2002:R
1999:(
1993:E
1980:p
1978:(
1975:X
1969:S
1966:(
1964:R
1958:.
1956:)
1954:p
1950:p
1946:p
1942:p
1940:(
1937:X
1926:p
1922:p
1920:(
1917:X
1909:p
1905:p
1903:(
1900:X
1887:p
1885:(
1882:X
1876:p
1872:p
1870:(
1867:X
1861:X
1852:X
1839:X
1809:i
1805:y
1799:n
1794:1
1791:=
1788:i
1780:)
1777:1
1771:n
1768:(
1762:|
1756:j
1752:y
1743:i
1739:y
1734:|
1728:n
1723:1
1720:=
1717:j
1707:n
1702:1
1699:=
1696:i
1685:=
1682:)
1679:S
1676:(
1672:D
1669:M
1666:R
1651:X
1619:)
1616:1
1610:n
1607:(
1604:n
1598:|
1592:j
1588:y
1579:i
1575:y
1570:|
1564:n
1559:1
1556:=
1553:j
1543:n
1538:1
1535:=
1532:i
1521:=
1518:)
1515:S
1512:(
1508:D
1505:M
1490:i
1486:y
1482:n
1477:X
1472:S
1416:c
1411:X
1405:X
1401:c
1393:X
1387:X
1378:X
1373:c
1368:X
1363:c
1358:X
1353:c
1348:X
1342:X
1336:X
1331:c
1326:X
1316:X
1305:X
1300:c
1295:X
1291:c
1283:X
1277:X
1268:X
1263:c
1258:X
1249:c
1244:X
1209:.
1203:M
1200:A
1195:D
1192:M
1186:=
1182:D
1179:M
1176:R
1138:.
1133:2
1129:F
1125:d
1119:1
1115:F
1111:d
1106:|
1102:)
1097:2
1093:F
1089:(
1086:Q
1080:)
1075:1
1071:F
1067:(
1064:Q
1060:|
1054:1
1049:0
1039:1
1034:0
1026:=
1022:D
1019:M
994:F
992:(
990:Q
983:x
981:(
979:F
972:Q
952:.
946:d
942:x
939:d
931:)
925:+
922:x
919:(
916:f
912:)
909:x
906:(
903:f
899:2
871:0
863:=
859:D
856:M
825:.
822:y
819:d
815:x
812:d
807:|
803:y
797:x
793:|
788:)
785:y
782:(
779:f
775:)
772:x
769:(
766:f
727:=
723:D
720:M
703:x
701:(
699:f
692:Q
669:.
665:|
659:j
655:y
646:i
642:y
637:|
633:)
628:j
624:y
620:(
617:f
614:)
609:i
605:y
601:(
598:f
593:n
588:1
585:=
582:j
572:n
567:1
564:=
561:i
553:=
549:D
546:M
529:n
525:i
520:i
516:y
512:y
510:(
508:f
501:Q
483:.
479:|
473:j
469:y
460:i
456:x
451:|
445:n
440:1
437:=
434:j
424:n
419:1
416:=
413:i
401:2
397:n
393:1
388:=
385:]
382:]
378:|
374:Y
368:X
364:|
360:[
355:X
351:|
347:Y
343:E
339:[
334:X
330:E
326:=
323:]
319:|
315:Y
309:X
305:|
301:[
298:E
295:=
291:D
288:M
269:n
265:i
260:i
256:y
248:Q
244:n
218:.
215:]
211:|
207:Y
201:X
197:|
193:[
190:E
183:D
180:M
166:Q
159:Y
155:X
80:)
74:(
69:)
65:(
51:.
24:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.