AAYSIS O EQUECY ISIUIOS

. Grdnr & G. Grdnr

IS 0 02246 8 4

. Grdnr nd G. Grdnr CAMOG COCES A ECIQUES I MOE GEOGAY O 19 (Coces a eciques i Moe Geogay AAYSIS O EQUECY ISIUIOS CAMOG as ee ceae o i a eacig ee i e ie o quaiaie meos i uegauae geogay couses ese es ae amiae guies y o e eaces ye cea eoug o sue ucase as e asis o cass- wok Eac ook is wie y a auo cuey wokig wi e ecique Gaie a G Gaie o coce e escies (eicese Uiesiy a omey o Saosie oyecic 1 A ioucio o Mako cai aaysis - Cois isace ecay i saia ieacios - ayo COES 3 Uesaig caoica coeaio aaysis - Cak Some eoeica a aie asecs o saia ieacio age soig moes - S Oesaw 5 A ioucio o e suace aaysis - Uwi I IOUCIO Cassiicaio i geogay - oso (i e-equisies o eae 3 7 A ioucio o aco aayica eciques - Goa A Kiy (ii e coce o a equecy isiuio 3 icia comoes aaysis - S auey (iii yes o aa 5 9 Causa ieeces om icoomous aiaes - aiso II E EE O EAMIAIO O EQUECY ISIUIOS I GEOGAICA ESEAC 1 Ioucio o e use o ogi moes i geogay - Wigey (i esciie saisics 11 iea ogammig eemeay geogaica aicaios o e asoaio oem - A ay (ii Ieeia saisics 1 A ioucio o quaa aaysis - W omas (iii Causa eaaio o saia ocesses 13 A ioucio o ime-geogay - i (i Oe uses o equecy isiuios i geogay 9 1 A ioucio o ga eoeica meos i geogay - K ike III GAICA MEOS O EAMIAIO O EQUECY ISIUIOS 15 iea egessio i geogay - eguso (i Cosucio o isogams 11 1 1 oaiiy suace maig A ioucio wi eames a (ii Ieeaio o isogams oa ogams - Wigey (iii Cumuaie equecy gas a oaiiy os 17 Samig meos o geogaica eseac - C io eac (i Cocusios 1 Quesioaies a ieiews i geogaica eseac - I COMAISO EWEE OSEE EQUECY ISIUIOS A E OMA C io eac ISIUIO 19 Aaysis o equecy isiuios - Gaie G Gaie (i Aaysis ase uo mome measues Aaysis o coaiace a comaiso o egessio ies - Sik (ii Gaica esimaes o mome measues 1 A ioucio o e use o simuaeous-equaio egessio aaysis Oe meos o assessig skewess a kuosis 3 i geogay - o (iii e W es o omaiy 35 Oe ies i eaaio (i 37 ( o-aameic oea gooess-o-i ess h r, Cnpt nd hn n Mdrn Grph 3 (i e eecs o ouies a oe isuaces o oma prdd b th Std Grp n Qntttv Mthd, f isiuio o aa th Inttt f rth Grphr. 5 r dtl f brhp f th Std Grp, rt t th (ii Coice o meos o comaig equecy isiuios Inttt f rth Grphr, I Knntn Gr, ndn, S.W.. wi e oma isiuio h r pblhd b G Abtrt, Unvrt f Et Anl, rh, 4 , t h ll thr nr hld b ddrd. SAEGIES O USE WI O-OMA AA

(i Assumios o e iea moe 7 I IOUCIO (ii Saegies i e assumios o e iea moe ae o saisie (i e-equisies o eae (iii yes o asomaio 51 is mooga is iee o oie a coeie summay o meos I wic may e use o assess e isiuioa caaceisics o aa use COCUSIOS 55 i geogaica wok We assume o e eae oy a aiiy o eom sime aimeic oeaios aoug a asic uesaig o e use GOSSAY O EMS USE 5 a meaig o sime saisica ess wou aso e a aaage I e eciques ouie ae o e aie o age amous o aa e SYMOS USE 1 e aiiy o asae sime aimeic oeaios io comue coe EEECES is aso ecessay ose uy coesa wi e coce o a equecy 3 isiuio may wis o omi Secios I-III A gossay o ems use (iaics i e is icue o age ; is aso icues symos use i omuae

(ii e coce o a equecy isiuio

aa aayse i geogay oe cosis o a seies o igues wic measue some aiue o e geogaica individuals cocee Geo- gaica aa may o eame e e yie o wea e ui aea o a se o aises e aeage soe o a seies o aiage asis o e aua aia o a ume o meeooogica saios; e iiiuas i ese eames ae aises aiage asis a meeooogica saios eseciey Geogaica iiiuas may e eie egios aig a aea ee suc as amiisaie uis o aiage asis o may e egae as eig oi ocaios suc as iusia eeises o meeooogica saios Eac igue i suc a se o aa is eme a vrt wis eac measue caaceisic is eme a variable. o eame e weae a a seies o meeooogica saios may e escie y aiaes suc as emeaue aia a umiiy eac oseaio eig a aiae

Some imessio o e age a aeage aue o a aiae may e gaie y meey ookig oug a is o aiaes u o e sake o simiciy o caiy iomaio i goue o geeaise om is oe moe useu Eac seaae aue i a se o aa as geae imoace we i is cosiee i eaio o e oe aues as a o a umeica isiuio a as a sige isoae oseaio eseaio o goue aa is usuay eecue y eaaio o a equecy iagam usuay i e om o a . o eae a isogam e aiaes ae goue io casses sowig ow may iiiua aiaes a io eac cass is is eme e equecy o eac cass a a iagam sowig ese equecies i asceig magiue o cass aues i e om o a a ga is cae a isogam

e aoe ieas ae iusae y e oowig eame usig iy yeas o aia aa as e se o aiaes (ae I igue I ese ae a suse o seecio om a ossie yeay aia oas A suc ossie aia oas ae e population o iees a sma suses seece ae samples.

3 ae I Aua aia (mm 1931-19

Yea Weigo ak o ameo Yea Weigo ak o ameo

1931 131 91 19 1 39 1 193 1 9 197 9 79 1933 9 19 111 3 193 115 9 199 137 1935 1195 97 195 139 99 193 17 5 1951 1 1937 17 117 195 1157 795 193 1 1953 37 igue e asiio om 1939 13 91 195 1 17 isogam o equecy isiuio 19 11 15 1955 97 9 oe owee a e asiioa 191 13 737 195 93 77 se ia e equecy oygo is 19 113 77 1957 1111 95 oe omie 193 1 195 135 99 19 1157 71 1959 137 13 195 1159 7 19 15 13

e aiae cocee is aua aia a e iy aues o oa aia oe o eac yea om 1931 o 19 o e aia saios (Weigo ak a o ameo oie iy aiaes o eac saio e iy aiaes ae eac assige o a cass e cass imis eig a ieas o 15mm isogams o ese aa ae esee i igue I

e oea sea o equecies ewee e casses is cae e distribution o e aa a i is wi meos y wic e aue o suc isiuios may e aayse a is mooga is cocee oe a e age a isiuio o aiaes as a coce is ieee o e cass ieas use i e eaaio o e isogam a aoug a equecy isiuio is coeioay isuaise as a isogam i is moe coecy oug o as a geeaise coce eaig o e sea o coceaio o aues ieee o e casses use

We ae so a isuaise equecy isiuio i ems o isogams u i we imagie a ey age same aw om a ey age ouaio a e aiaes ae assige o casses wi ey sma cass ages o e aiae e e isogam ouce wou e i e om o a smoo cue isogams ae oe esee i is om o as equecy oygos (igue as ese moe eay acco wi ou coce o wa e ueyig equecy isiuio sou ook ike uiuece y e same ake o cass sie use i eaaio o e isogam

(iii yes o aa

I as so a ee assume a e aa o coce eese aues om a coiuous age owee i is ossie o ecogise o continuous a drt aa e ome may ake o ay aue wii a igue I equecy isogams o e aia aa o ae I e gie age a i geea ae measue aues suc as aia aea o eica scae is oe o equecy (; e ume o aiaes i eac cass eg o eie aios suc as ouaio esiy o eceage o a is gie aoe eac a e eigs o wic ae ooioa o e use iscee aa cosis o coue aues suc as umes o uiigs equecy associae wi eac cass o umis a ae geeay woe umes I acice e isicio may ecome somewa ue sice coiuous scae measuemes ae a 5 iie esouio a aimeic ouig o euces e ossie aues o a coiuous aiae so a coiuous aa may e cosiee iscee i oue o e eaes woe ume eeeess e isicio is a imoa oe as i eemies e way i wic aa sou e ae e eciques escie i is mooga ae iee imaiy o aaysis o coiuous aa aoug ey may aso e aie o iscee aa i some cases; a u iscussio o equecy isiuios o use wi iscee aa is gie y Gaes a Eige (197

ue seciic yes o coiuous aa wic ee o e ecogise icue cose aa wic may oy ake o aues ewee aicua imis as o eame a 1 i e case o may sae measues o a 1 as i e case o eceage aa a oe aios wic may age om o iiiy is isicio emas eaie cosieaio we asomaios ae eig cosiee (Secio

II E EE O EAMIAIO O EQUECY ISIUIOS I GEOGAICA ESEAC

e caace o equecy isiuios is imoa o a geogaes wo ae ikey o wis o use umeica meos i ay ac o e suec; some o e moe imoa aicaios ae escie eow e ways i wic equecy isiuios may e eamie o use ae is o a ouie a e some seciic aicaios ae escie

(i esciie saisics

A se o aiaes may e isuay summaise y a isogam u is is o aways coeie i wic case esciie saisics may e use e mos commo o ese ae e n a e tndrd dvtn, a ese escie eseciey e aeage aue o cea eecy o e aiaes a e isesio o aues o eie sie o e mea aue e cacuaio a use o ese a oe measues is escie ue eow (Secio I

Caaceisics o equecy isiuios may e succicy summaise y ese sime measues (igue 3A A ee sime om o saisica soa is e use o dl l, wic is e cass aig e iges equecy owee ese saisica soas may occasioay gie ise o eoeous o miseaig cocusios as i e case o equecy isiuios wic ae asymmeic o d, a equecy isiuios wic ae moe a oe moa cass (imoa imoa ec; igue 3 e mea oies a eiey ai iicaio o cea eecy oy o symmeic uimoa equecy isiuios aoug is use is oe oeae o sigy skewe u eeeess uimoa equecy isiuios I is eeoe imoa a e aue o equecy isiuios e eamie eoe suc esciie saisics igue 3 Mea a saa eiaio as esciios o equecy e emoye isiuios A Comiaios o ig a ow meas a saa eiaios esciie use o isogams is ey commo i geogaic eseac; equecy isiuios o wic e mea is o a ai some eames om uma geogay icue wok o oy (197 wo use iicaio o cea eecy o e isiuio isogams o comae uiig Socieies a oca Auoiies i ems o e ucase ices o oey ue a e age icome a socio-

7 ecoomic gou o ei oowes; o owey (1975 wo esee isogams o e ouaio sies o Egis cosiuecies i 197 a 197 i oe o emosae e eecs o eecoa e-aoiome; a o Moi a Symos (1977 wo use isogams o escie asecs o ime a isace o ae o a se o aciiies i a suy o oimum ocaio Some eames om ysica geogay icue e wok o Waig a We (1975 wo esee isogams o souio oa i sames o seam wae om aious ock a a use yes; o Gaie (197 wo use isogams o aious a om aamees o eamie ieeces ewee a Sysems; a o i (1973 wo use isogams o aious caaceisics o umis o sugges ossie moes o umi omaio igue e eceage o aiaes wii 1 a 3 saa eiaios o e mea o e oma isiuio e mea is (ii Ieeia saisics iicae y a oe ie e ages ae i saa eiaios a e sae aeas eese e aoiae aea ue e cue a Saisica ess ae oe emoye eie o esais wee ece e eceage o aiaes wii e gie imis wic ae e ieeces ewee wo sames o aiaes ae sigiica o wee symmeic aou e mea ey cou ae aise y cace o o es wee ee is a sigiica co-aiaio ewee wo aiaes o a gie se o iiiuas o o eame e saia ae o seemes may e suie as a susiue eame oe may wis o es e yoesis a e mea aues o o a sueme o e moe uamea ecoomic socia oiica a aia a eigo ak a o ameo (ae 1 igue 1 ie ysica ocesses goeig ei isiuio is may e eome y sigiicay o aeaiey a ee is some egee o coeaio eamiig caaceisics o e equecy isiuios o ceai asecs ewee em e ess emoye ae o wo yes aameic a o- o e saia ae o seemes o eame i as ee ou a aameic Parametric ess ae usuay e moe oweu u ei i egua quaas ae oeai o a aea a e ume o seemes ueyig eoy is ase o e assumio a e aiaes cosiee occuig i eac ce is coue e y egaig ese cous as ae a equecy isiuio o a aicua ye cae a normal o aiaes i a equecy isiuio e om o e esuig equecy Gaussian isiuio A oma equecy isiuio as e oeies isiuio may gie a iicaio o aicua saia ocesses suc as o eig symmeic a uimoa a a ceai ooio o aiaes wi iusio o aom aceme I a aom ae wi a eaiey sma occu wii seciie imis aoe a eow e mea as seciie i ume o seemes e quaas e equecy isiuio o ois igue I is oe escie as eig e-sae; a moe igoous e quaa aoimaes o a aicua ye o isiuio kow as a eiiio is gie i e Gossay oisso isiuio a e cose e osee isiuio aoaces is e moe easoae i is o osuae a saiay aom ocess Sicy aameic ess sou oy e use o aa wic ae ueyig e geeaio o seemes A eame o e use o ese omay isiue o aa se ikey o e ecouee wi e eciques is wok y aey (19 i wic e equecy isiuios isiue eacy as e oma isiuio u i is imoa a o cous o ois i quaas ae use o es moes o saia iusio aameic ess sou o e use o aiaes wic ae maiesy o- e eciques emoye ae iscusse ue i aey (19 a a u omay isiue sice e saisica oceue is ase uo a accou o em is gie y oges (197 assumio o omaiy a ioaio o is assumio iaiaes ay cocusios wic may e aw om e aaysis owee o-aameic Saia ocesses may aso e iee om equecy isiuios saisica ess may e use waee e om o e equecy isiu- i oe ways o eame Aoai Oo (1973 as suggese a isace io u esus may e ess ecise a i e case o aameic ess gow ecay ocess o ay o e ouey o wok i Youaa om o-aameic ess ae owee equay aece y ioaio o e a cosieaio o e equecy isiuio o isace aee a assumio o ieeece (see Secio e Smi (1977 as cosiee a aiey o uses o wic oeies o e equecy isiuios o isaces ewee a ois i a egio may Agai ee ae may isaces o e use o ieeia saisics e u eaig o equecy isiuios i geogay o eame Gaie (1971 esaise a e equecy isiuios o esimae aiage (i Oe uses o equecy isiuios i geogay esiy o ee ock yes wee iee a oos (1977 geeae ceua ewoks y ee ocesses a comae ei equecy equecy isiuios may e emoye i may oe aeas o isiuios o coac umes geogay (igue 5 o eame ayo (1971 as suggese a meo o e measueme o e sae o geogaica egios is equies (iii Causa eaaio o saia ocesses e sae o e oeai y a egua gi o ois eie ysicay o oioay y meas o a comue ogam isaces ewee a Geogaica ocesses esuig i saia isiuios ae iicu ossie ais o ois wii e sae ae cacuae o measue a o osee a eamie a ecouse is oe mae o eamiaio o saia egae as aiaes i a equecy isiuio igue 5A iusaes aes as a suogae o eamiaio o ei causaie ocesses 9 e way i wic is may e use o assess ow Ageia comaes i sae wi a se o geomeic igues (ae ayo 1971 A simia meo as ee oose (Ogey 197 o measueme o aiage asi sae a oos (1977 uses e equecy isiuios o coac umes o ceua ewoks as a iicaio o ei oea sae a omaie ocesses I geomooogy ae (1959; 19 as suggese a e equecy isiuios o sames o aiue may e iagosic o ceai sages o eeome o uia aoms (igue 5 a Eas (197 as ue iesigae is ie equecy isiuios o caaceisics o seimes suc as gai sie o aice sae ae ee wiey aie i suies o eosiioa eiomes (igue 5C wee geomooogiss ae emoye may geoogica eciques Samig esigs o saia aa ae ee comae y eamiaio o e equecy isiuio o e esuig aiaes i simuaio suies (eg Keyes e a 197 a iay suies o aiage ewok ooogy (eg Kumei a ames 199 ae oe ee ae o use equecy isiuios o ewok caaceisics suc as ik egs a asi aeas o iesigae ewok eeome

III GAICA MEOS O EAMIAIO O EQUECY ISIUIOS

e uose o summaisig e icia caaceisics o a equecy isiuio is o make i easie o iee a o ai comaiso wi oe isiuios I is way e comeiy o e iomaio may e cosieay euce as iiiua aues ae geeaise wis is useuess is eace as e geea ae o e aa is e- aie

equecy isiuios may e egae as sime summaies o geeaisaios o e oigia aa I e eame o aia (igue 1 e isogams ao a sime isua imessio o aua aia a e wo saios a aues o e wo saios may useuy e comae o eac oe us Weigo ak aeas o ae a ige mea aue o aia a a geae isesio o aues a o ameo

(i Cosucio o isogams

e oceue o cosucig isogams is ioes gouig e aa io a ume o casses a couig e ume o aiaes wic occu i eac cass e equecy o occuece is eesee i e isogam y e eig o e aoiae a us iiiua aa ae goue a e iusae gaicay

e ume o casses use ca ay accoig o e caaceisics o e aa oie e woe age o aues is coee Oiousy isiuios wic coai age umes o aiaes omay usiy use o moe casses a ose wi ew u i is imoa o eaise a igue 5 Geogaica aicaios o equecy isiuios A Sae e coice o cass sie may ae a gea eec o e moe a age measueme emoyig e equecy isiuios o isaces ewee a oaie Oe awack o a gouig may e a oo muc iomaio ais o ois wii saes (ae ayo 1971 aom aaysis is os; a caeu aace as o eis ewee geeaisaio a amou Eac oi eeses a same o so eigs (ae ae 19 C o iomaio eaie I igue e Weigo ak oa aua Seime aaysis Eac oi eeses a same o ees e eosiioa aia aa ae use o iusae a ocess o cooe geeaisaio eiomes o wic ae sow y yes o symo (ae oo am a Kig 1971 11 1 igue Gouig o aiaes io equecy casses (Weigo ak aa

I e cee o igue e 3 aiaes ae oe iiiuay aog a ie eeseig e age o aa e 3 aiaes ae e aocae o casses o equa wi coeig e woe age o e aa i e o o e igue o ay oe o ese casses ee is ow a equecy cou o e ume o aiaes i a cass; ese cous may e e eesee gaicay y a equecy isogam (eg igues 1 a 7 ee ses o ieey-sie casses ae ee use o eie equecy cous o ese aa a ese esu i ogessiey moe geeaise isogams as e cass wi is icease I is ossie o assess e oss o iomaio esuig om suc cass gouig y comaig e sum o goue aues a eac sage emoyig e gou mea aues as a iicaio o e aiaes wii eac cass o e ugoue aa

I e geeaise equecy isogams is sum ca oy e aoimae sice e magiues o e oigia aues ae coceae igue 7 isogams o Weigo ak aa emoyig ieey sie a i igue 7 isea o eaig wi e oigia 3 aiaes i is cass ieas ecessay o ea wi oy 1 (Sage 1 (Sage o 3 (Sage 3 eeseaie aues o e casses Assumig a e cass mioi aues ae eeseaie o e aiaes coaie i eac cass e e sum o goue aiaes may e aoimae as-

Sage 1 1 casess (11 i eo cass iscoue

1 13 ae Iomaio oss o iee cass sies usig Weigo ak aa

Ugoue aa

Goue aa geeaisaio is aimeic oceue is iusae i ae e ieece ewee ese aues a e coesoig sum o e ugoue 1 gous aa is oe iusaio o e oea oss o iomaio associae wi gouig e aa I may e oe a ee is uouaey o sime (1 ( eaiosi ewee oss o iomaio a ume o casses aoe a Cass mea aue equecy i cass ouc o (1 ( a i is ossie o e cacuae sum o e e same as a o e 15375 aw aa esie e ac a some iomaio oss mus ieiay occu 15375 1 15 i ay suc gouig o aa 15 1 13775 755 55 e ume o casses seece is geeay a suecie ugeme 1315 5 175 o e a o e eseace u i e aa ae o e use o ue 1375 5 1375 aaysis eyo e sime esciie sage i is esseia a i e 115 7 155 cosucio o e isogam e cass iea emais cosa a aso 1775 5 a e woe age o oigia oseaios is coee Oe way o 115 aiig a a sou gouig is o eeime wi iee cass ieas 9375 1 9375 5 a o seec a wic aeas o gie e mos saisacoy esu 5 1 7775 owee ee ae moe oecie guieies aaiae o eame 7775 1 usege (191 gies e oowig omua o esimaig e ume (K o casses o e use 355

% wee = e oa ume o aiaes ieece om ugoue aa as % o ae ae 3 iusaes e use o is omua i cosucig a equecy isogam o e aia aa eee o aoe Si casses ae seece as oimum y is omua o age aa ses e suggesio o gous usege oiousy eas o a ueesimae o eame a isogam 1575 1 1575 o 1 aiaes wou equie aou 1 casses accoig o e omua 15 3 75 ye iuiie ugeme wou sugges is o e a oe-geeaise 175 1 175 eeseaio o e aa A aeaie aoac is oee y ooks 115 9 115 a Caues (1953 wo sugges- 975 5 75 5 175

Aicaio o is eaiosi is aso iusae i ae 3 esuig 33775 i a aue o 7 ie 7 casses A i oceue as ee suggese y ocie (1977 wo aocae e use o ieece om ugoue aa as % o ae 5% wic eaes a geae ume o casses o e emoye o age aases a i e cases o e eious suggesios iay Coo 3 gous a Cowe (19 sugges a mos equecy isogams sou ae 15 ewee si a siee casses Ceay ee ca e o coec ume 1 19 o casses sice oss o eai wi ay wi iee aases u cae 9 7 3 sou aways e ake o coose e ume o casses so a a comomise is esaise ewee ey ew casses sowig eaiey ie a oo 351 may casses giig oo eaie a cassiicaio I e case o e aia aa wi 3 aiaes o 7 casses wou e aoiae as use ieece om ugoue aa as % o ae 13% i igue 1 1 15 ae 3 eemiaio o ume o casses i isogam

ntbrr thd. o o casses (K igue 9 isogam o Weigo ak aia aa emoyig cass imis sow o e oom o igue

r nd Crrthr thd.

owe a o igue aiaes ae ee aocae o si casses e owes o wic commeces 1mm ige a i e equiae gouig sage o e ue a o e igue; is esus i a isogam (igue 9 wic ies i eai i o oea om om a o igue 1 o igi ues ca e gie o seecig e osiios o e Crxtn nd Cdn o 1 casses cass ouaies u is sou eeay e oe wi eeece o e scaegam o aues suc as a i igue so a cass ouaies o o ass oug ese gous o ois

Eas (1977 as oee some suggese guieies o seecio o cass ieas o mas a is may e eee o e case o isogam eaaio e cocue a cass ieas sou e goee y e oea sae o e isiuio ecagua isiuios i wic a casses ae aoimaey equa umes o aiaes equie igue Ga o equa iisios o e oea age omiay uimoa isiuios aciiae coice o equie cass ieas eae o e saa eiaio o e aa a ume o isogam skewe (i Eas emioogy -sae isiuios equie cass ieas casses wic a io a geomeic ogessio e ase o wic iceases as skewess iceases is ugeme oes o couse equie some a ioi aeciaio o e oea sae o e isiuio eoe e cass ieas ae seece a e isogam aw; is wou omay e gaie om isua isecio o e scaegam o aiaes (eg igue

Seciicaio o cass iea imis aso ees o e eome wi esec o e isicio emasise i Secio I ewee coiuous a iscee aa e esseia equieme is a a aues sou e wo o e meos o cacuaio o ume o casses ae mae assigae o casses wiou uceaiy o eame i ae e sime o ay y use o igue is os e eaiosis o casses ae gie as icusie imis a sice aua aia is eoe usege a ooks a Caues ewee K ume o casses a o e eaes miimee is suices o oie a uequioca cassiica- ume o aiaes o aues o om 1 o 1 a a aoiae io o e aa Sice mm is eoeicay ossie e casses ae ee ume o casses ca e ea o om is ga y ookig a e oioay commece wi - 19mm; us e is cass o o ameo eea aue o I sou e emasise a e ue ie is -79mm ae a 1-75mm eeses a suggese ue imi o K a ay coeie aue o K yig ewee e wo ies wou omay e suiae o smae aases; ue iicuies aise i i is wise o comae wo sames K may ee o e icease o age aases we ocies (1977 wi iee umes o aiaes i eac o wi makey iee omua is moe aoiae ages I ese cases i is ecessay o saaise e equecy o aiae scae as aoiae I e is case e equecies may e e sae o e esuig isogam is aso iuece y e osiio coee io eceages o e oa same o eac cass i eac o e cass ouaies as we as y e cass age o eame i e 17 1 o e isiuio is o e ig as i is case e isiuio is ae Cumuaie equecy aa-gous sai o e postively skewed, wi e moa cass (a cass wic coais e ages ume o aiaes o-cee o e e (eg igue 3 e Oe isiuios may e negatively skewed, wi e moa cass Weigo ak eig owas e ue e o e age (eg igue 3 ig I geea equecy isiuios ecouee i geogaic eseac sow Cass equecy % equecy Cumuaie % equecy ayig eecies owa osiie skewess aoug eceios o occu Symmeic isiuios (skewess = may owee si e o-oma 75-99 7 7 owig o e eisece o . Kuosis is oe escie as e eakeess o a equecy isiuio aoug iews ae ee eesse 9-19 5 17 33 a i may aeaiey e egae as a measue o e ee o uimoaiy esus muimoaiy o e isiuio (aigo 197 15-1199 9 3 533 a e eg o is ais (iuca 19 A platykurtic isiuio wi ow kuosis as ess aiaes i e casses ea e moa aue 1-139 1 3333 7 a/o moe susaia ais o e isiuio (igue 1 y coas a leptokurtic isiuio wi ig aues o kuosis as a age a 135-199 3 1 97 eece ume o aiaes i e ea-moa casses a ey og ais 15-19 1 333 9999 Skewess a kuosis ae usuay cosiee as eig seaae a ieee caaceisics o equecy isiuios owee some suies ae sow a e wo measues ae eae a a e eec o transformations (Secio o oe o e measues is usuay simia o o ameo e oe measue o eame ae (igue 5 as sow a seies o isiuios o wic kuosis as ee oe agais skewess eeaig Cass equecy % equecy Cumuaie % equecy a geeay U-sae eaiosi ewee e wo a Gaie (1973 emosae a simia U-sae eaiosi ewee e wo measues o -79 aious asomaios o e same aa wi miimum kuosis occuig we skewess was eaes o eo 75-99 13 333 333 e eisece o skewess a/o kuosis i geogaica aa is o 9-19 9 3 9333 sigiicace we usig ieeia saisics i eseac oceues as e moe oweu aameic ess o saisica ieece ema symmey 15-1199 1 333 9 a aicua assumios o sae suc a e aiaes cocee sou e aoimaey normally distributed. e mos imoa oeies o 1-139 9 e oma isiuio ae a i is symmeic a iecios occu i e equecy isiuio cue a oe saa eiaio aoe a 135-199 1 333 9999 eow e mea a a e aea ue e cue (ie ume o aiaes wii seciie imis om e mea is as sow i igue isogam isogam a eceage isogams emoye o e ae case e sae wee assesse isuay o y meas o iices suc as ose aa aues ca e coee io saaise uis suc as saa escie i Secio I is aso o iees i a i may ise sugges eiaios aoe a eow e mea; is is escie uy eow (Secio useu iomaio aou e ouaio cocee (ige 197 o ( Wis i is eeae o use e same cass ieas i eac o eame Mie (1955 cosiee e osiiey skewe isiuio o wo isogams o e comae (Mice 1975 i may e ecessay o igoe esoa icome i e Uie Saes a cocue a is skewess was is icie a i mus e ackowege a e maoiy o isogams ue agey o megig seea symmeica isiuios amey ose o eese a comomise ewee ose eaues o e isiuio i is me a wome seaaey a o iiiua occuaioa gous wii e wise o sow e esie o geeaise e aa e ee o eai as muc sees Skewe isiuios o sie-eae aiaes aso aise as a esu iomaio as ossie a e coice o easy o use a ossiy meaig- o ocesses cocee wi e ackig ogee o iiiuas i sace u cass ages o eame e equecy isiuios o aiage asi aea a eae measues ae iaiay ogaimic o osiiey skewe (Gaie 1973 (ii Ieeaio o isogams ecause e saia ackig o asis a e equiemes o e aom moe o aiage comosiio (Sma 197 equie may ey sma asis A isogam oies a sime isua imessio o e isiuio a eciig umes o age oes egaiey skewe isiuios ae o a se o aiaes o isace i e case o e equecy isogam muc ae a osiiey skewe oes aoug e sie isiuio o o o ameo aia (igue 1 ee is a geae ume o aiaes maeia o a oue eac may oow is ae a ige (197 owas e owe e o e age a comaaiey ew a e oe eeme oe a ee o siy-ou cesus aiaes wee aso eemey egaiey Asymmeica isiuios suc as is ae sai o e skewe I e ai 19 1 skewe Eemey osiiey skewe o -sae isiuios suc as a o oa ouaio e gi squae (ige 197 ae commoy ouce o cou a esiy measuemes o gi squae aa Quoie measues o gi squaes suc as eceage o oa ouaio i a aicua age caegoy ae ess eicay isiue owee a may e eie aoimaey oma o skewe ogaimicay isiue aa ae aso ey commo o eame esaas (197 eoe e equecy o semaic esose i a suy o e eceie eiome o oow a ogaimic isiuio a Gaie (1973 cocue a may iices o aiage asi om wee ogaimicay isiue

imoa isiuios ae aicuay commo i geomooogy; o eame e isiuio o seime aice sie may e imoa as i e case o eaces wic ae muc ie sa a age oues u ie maeia o iemeiae gae imoaiy oe imies a miue o wo ouaios wi simia equecy isiuio sae caaceisics u ieig mea aues o eame e equecy isiuio o ee sae i a ie may e imoa wee a mao iuay iouces maeia o a ouess ieig om a o e uk seam

Kuosis ca aso aise y e amiue o wo ouaios we e meas ae oy sigy iee u e saa eiaios ae e same us a aykuic uimoa equecy isiuio is ouce y miig ogee wo oma isiuios wi e same saa eiaio u wi meas ieig y wice e saa eiaio (iss 197 11 Coesey a eokuic isiuio is ouce y miig wo ouaios wi e same mea a ieig saa eiaios (iss 11- us kuosis i a isiuio may aso sugges a e same is eie om wo seaae ouaios Meos ae ee escie o esoig susece comosie isiuios io comoe as; o eame Muy (197 igue 1 Cumuaie equecy os o aia aa emoyig escies a meo comiig isua ugeme a egessio aaysis a gies eeeces o oe oceues ase uo o gaica a umeica goue aa meos A O oiay scaes O oaiiy scae isua eamiaio o aa i isogam om may oie a aequae cues; owee eaues om e oma ae o easy o uge a a eimiay iicaio o wee a isiuio is aoimaey oma ue moiicaio o e ecique as ee eise o aciiae o eame i igue 1 e isiuio o Weigo ak is oay comaiso wi e oma isiuio I is e cumuaie eceages moe oma a a o o ameo u i mus e emasise a ae oe agais cass imis i e same way u o ga ae kow is is oy a quick sime a suecie isua assessme a moe igoous ess o omaiy sou usuay e aie as prbblt ppr. is as eceage oaiiies se ou aog e eica ais a e oioa ais is eguay iie a use o o (iii Cumuaie equecy gas a oaiiy os cass ue ouay aues e ae is so esige a cumuaie e- ceages eeseig a oma isiuio we oe agais ue cass ouay aues wi a aog a saig ie I igue 1 e A aeaie gaica aoac o e esciio a eamiaio cumuaie eceage equecies o e Weigo ak aa a amos o geogaica aa is aoe y e ltv frn rph. eacy o a saig ie iicaig a isiuio ey cose o oma Cumuaie equecies may e oe eie o ugoue aa wic ae e oe isiuio ieges makey om a saig ie a eeoe ee ake y aue o o goue aa o eame i ae e goue om a oma isiuio us coimig e isua imessio gaie equecy aa use i cosucig e isogams o igue 1 ae ee om e isogams Agai as wi e cosucio o equecy isogams coee io eceages o e oa ume o aiaes i e isiuio a ocess o cooe geeaisaio aies accoig o e sie o e eceage equecies ae e cumuae uwas giig eceage equecy gouig seece Oiousy a moe accuae iicaio o e igues o e ooio o occuece i a aicua cass us a aue o e isiuio may e oaie y oig may sma gous; owe aue casses ese cumuaie eceages ae e oe agais is eas eeuay o e use o e oigia aiaes (ae 5 ese e ue cass imi oucig e cue iusae i igue 1A Suc may e ake om wic cumuaie eceages ae cacuae a oe cues ae oe kow as v, aoug a ee em is isiuio agais e aoiae aues Agai a saig ie eeses omaiy cue Uimoa isiuios esu i S-sae (sigmoia isiuio igue 11 sows e isiuio o aia o o ameo oe i

1 ae 5 Cumuaie equecy aa - iiiua aiaes

o ameo

Yea ak aue % equecy Cumuaie % equecy

1953 3 37 333 333 1933 9 333 199 333 999 19 7 71 333 1333 195 7 333 1 191 5 737 333 1999 195 77 333 333 19 3 77 333 197 79 333 999 195 1 795 333 3333 19 15 333 3 193 19 333 3999 19 1 3 333 333 193 17 333 193 1 9 333 999 193 15 5 333 5333 1951 1 333 5 193 13 9 333 5999 1957 1 95 333 333 1931 11 91 333 1955 1 9 333 999 1935 9 97 333 7333 1939 91 333 7 195 7 99 333 7999 195 99 333 333 195 5 17 333 19 1 333 999 1937 3 117 333 9333 1959 13 333 9 19 1 13 333 9999

is way a aoug cosieae o-omaiy is si eie i is see a e makey o-oma aeaace o igue 1 is ue ay o e gouig io aiicia casses

A aaage o is meo o eeseaio o e aa is a i moes e iscee aa y a coiuous ucio e scae o ois eig geeaise y a ie om is iemeiae aues may e ea o o eae saaise comaisos o e mae ewee a ume o isiuios o eame om igue 1 i may e eemie a a oi o e ie eeseig e Weigo ak isiuio wi a cumuaie equecy o 5% aso eeses a aue o e aiae aia o aou 119mm us i is geeaise ie is eeseaie o e aiae e iy e ce o yeas wi ae geae a 119mm aia a iy e ce ess 119mm eeoe eeses a miway o dn aue wic ca e comae om oe isiuio o aoe igue 1 Kuosis i equecy isiuios y comaiso e meia o e o ameo aa is 5 mm 3 is ocess ee o e coie o eemiaio o e meia sice ay ae Meos o comaig equecy isiuios oe eceage equecy aues may e ea om e ga Commoy use aues ae e 5% a 75% ois (owe a ue quaies

e eceage cumuaie equecy ga as ee aicuay wiey Asec Gaica meos Saisica ess use i seimeoogy o e oaya o gai sie isiuios I cosiee a esciie saisics is ie a wie aiey o esciie measues o sie isiuios ae ee ase o aious comiaios o eceage ois ese ae escie Skewess isogam ises g1 ue eow (Secio I easos skewess (i Cocusios Gaica skewess om Gaica eeseaios o geogaica aiaes i isogam cumuaie oaiiy cumuaie equecy a oaiiy o om oie sime a eecie o meas o esciio a eemeay aaysis o e aa O ese meos e ae oes e mos igoous a useu aoac a eaes may Momea skewess isiuios o e oe o oe ga o comaaie uoses uike isogams wic ae o e comae seaaey y e use o ese gaica meos come aa may e summaise a comae o oe isiuios a e comeiy o e aa is cosieay euce wis Sma sames age sames e mos imoa eemes ae eaie Some suecie imessio o e saisica omaiy o e aa may aso e gaie eseciay i cumuaie Kuosis isogam Geay mea age/ses equecy gas ae emoye moe soisicae gaica meos aso eiaio ises g eis (eg ukey 1957 Gaica kuosis om aio cumuaie owee ese meos ae esseiay esciie oos aowig oaiiy o oy a suecie isua comaiso o isiuios o moe eac comaisos eie ewee isiuios o wi e oma isiuio Momea kuosis i as ee wiey accee a ieeia saisics mus e emoye ese ae eiewe i e succeeig secios owee eoe eaig ga- ica oceues i is wo oig a some saisicias (eg Gaaesika 1977 17-; Wik a Gaaesika 19 ae ow ess saguie aou e Oea isogam Saio Wiks Komogoo- ieiae ee o ieeia ess a sugges a i may ways gaica gooess o W Smio es oceues suc as oaiiy os oe e saes aoac o e i Cumuaie oaiiy Ci-squae oem o es Ci-squae usig og- ikeioo I COMAISO EWEE OSEE EQUECY ISIUIOS A E OMA aio ISIUIO oi Ci- squae o aig esaise i secios I a II a ee o e eamiaio g 1 a g o equecy isiuios a aig suggese i secio III a uey isua meos ae o uy aequae o is uose i is ow ecessay o eamie umeica oceues o comaig a osee equecy isiuio wi e oma isiuio owee i mus e emasise Ouies isogam ios ga age/s a e ieeia saisics o e escie mus o e use iy es es I a same o aiaes is suc a i is o ossie o eec e u yoesis o o ieece om oma a e cose sigiicace ee e woke mus o ecessaiy immeiaey acce e same as eig eecy oma - i is meey o uoe o-omaiy o oe o sicy gooess o i omaiy is cosieaio is aicuay eea o sma sames o wic i is eas moe aoiae o ik i ems o egee o o- omaiy ae a e sigiicace o o-omaiy May meos ae aaiae o comaig a osee equecy isiuio wi e oma

5 isiuio (eg Gaaesika 1977 1- u ose wic ae mos ikey o e me wii geogaica wok ae ouie i ae o aiace o mome wo e skewess o mome ee a e kuosis Saisica ess ase uo iewig e aa as coiuous aa ae is o mome ou ige owe momes aso eis u ae ey seom eamie Oea gooess-o-i ess ase uo o-aameic measues emoye Saa eiaio (s is gie y- ae e iscusse; is ae oceue moes e oma equecy isiuio y iscee aa i gous a is eeoe ess ikey o e as ecise as e eac measues Eac measues ae oy ae o cosie ceai asecs o e osee isiuio owee a o is oea skewess (Sk y- sae i eaio o e oma

(i Aaysis ase uo mome measues

Mome measues o equecy isiuios ae ee meioe i secio III i e coe o ei uses o e esciio o e sae o equecy isiuios ou mome measues ae omay ecogise a kuosis (K y- a may e eie as-

ese ae e usua esciie momea measues emoye a gie imesioess measues o isiuio sae o a oma isiuio e skewess is oiousy eo as e isiuio is symmeic a wee u sas o e eiaio o eac aue om e mea a [ ] e kuosis aue is ee Skewess may e osiie o egaie iica- e symo imies summaio oe a aiaes i e isiuio ig e iecio o asymmey o e isiuio (igue 3 ; kuosis ca cocee us i moe coeioa u uwiey emioogy- oy e osiie i aue

o es e sigiicace o eaues o a equecy isiuio om e oma i ems o asymmey a kuosis a seies o measues ase uo e momes escie aoe as ee oose Usig e oaio as aoe ese may e cacuae om a se o saisics kow as ises K saisics-

e sigiicace o eaues o ese g saisics om omaiy ae e

7 ese y wo ue aues cacuae as- ae 7 Cacuaio o mome measues

Weigo ak (Cacuaios ae sow i eai o oy e is a as aiaes i e isiuio; a summaio sigs ae uesoo o ay o summaio oe a iy aiaes

isiuio a ae oiousy eee oy uo e same sie om ese aiaces e aios-

ae cacuae a iay eee seaaey o a ae o e nrl prbblt ntrl (suc as ae A i iss (197 I is ae e cacuae aue is egae as e saaise oma eiae igoig e sig a e oaiiy ea om e ae is oue ecause is is a wo-aie es oaiiy aues smae a e seece sigiicace ee equie eecio o e u yoesis o o sigiica ieece om e oma isiuio

I oy oe o e wo es saisics iicaes a sigiica ieece om e oma isiuio e a comie Ci-squae saisic is cacuae om-

e cose sigiicace ee equies eecio o e u yoesis o o ieece ewee osee a oma isiuios is ae egy oceue is iusae i ae 7 o mos aicaios e ecessay aimeic is es eome y comue Meos ae aso aaiae (eg Coe a Kig 19 111-5; iss 197 1- y wic ese measues may e cacuae o goue aa

(ii Gaica esimaes o mome measues

Aoimaios o mome measues may e oaie y gaica meas usig iomaio eie om e cumuaie oaiiy os eiousy ouie I was suggese i Secio III a iemeiae aues o e aiae scae cou e ea o o ay eceage oaiiy eese- ig a aue o e aiae eow wic e sae eceage o e same ies us i igue 1 o Weigo ak 5% o e aiaes ie eow

1 mm aia eeoe 1mm aia may e sai o eese e 5 prnt oi o e isiuio i eig uesoo a cumuaio is caie ou i e mae escie e emioogy emoye ee o is wi e 5 eeoe o e isiuios i ae 1 5 o Weigo ak is 1mm a 5 o o ameo is 775 mm

y seecig ceai ciica eceage aues a seies o oceues as ee oose y wic esimaes o e ou mome measues may e eie om e cumuaie oaiiy o A simia oceue is wiey use i seimeoogy wee gai sie aa ae usuay measue o a ogaimic scae cae e i ( scae

h n. eaig o e 5 aue yies e meia as a iicaio o cea eecy owee is is seiousy i eo as a esimae o e mea o skewe isiuios (igue 3 a ok a Wa (1957 suggese a e ais o e isiuio cou e icooae io e cacuaio o e mea y-

Mea = 1 + 5 +

McCammo (19 eauae is as e mos eecie eisig gaica measue o e mea u oose a sigy ee measue as-

Mea = 5 + 15 + 5 + 35 + 5 + 55 + 5 + 75 + 5 + 95 ieece ewee e osee a e oma isiuio cao eeoe 1 e eece a say 1 as e auae aue is age a is Stndrd dvtn. is as ee e suec o muc iesigaio y o ameo seimeoogiss sice we aie o gai sie aa i gies a iicaio o soig wic is a key eeme i geeic ieeaio Equiae aues o o ameo ae- ok a Wa (1957 suggese e use o- Mea = 737mm Saa eiaio = 11mm Saa eiaio = - 1 + 95 - 5 Skewess = 111 Kuosis = 55 McCammo (19 agai cocue a is was e mos eecie eisig measue u suggese as a ossie imoeme- om e ae e associae oaiiies ae- Saa eiaio = 7 + + 9 + 97 - 3 - 1 - - 3 91

Sn. ok a Wa (1957 sugges a ie wic ey eme Icusie Gaic Skewess as- As o o ese ae ess a say 1 e u yoesis may e eece a is isiuio oes aea o e sigiicay o-oma Skewess = (1 + - 5 + (5 + 95 - 5 ( - 1 (95 - 5 I oy oe o ese saisics a iicae o-omaiy e e oi saisic cou ae ee cacuae as- is ages om -1 o +1 wi mos aues yig ewee - a +

isiuio a e o ameo aa may eeoe e eece a is ee 31 3 ecause e i scae is eese i a age i aues eese sma gai iamees a ice esa e skewess measue eie om seimeoogy ae Gaica esimaio o mome measues as escie aoe as a sig wic eoes e eese o a coeio- ay eece us osiie skewess om is gaica measue is Weigo ak equiae o coeioa egaie skewess a ice esa om igue 0:

Krt. Mea ok a Wa (1957 sugges- Mea = 1 + 5 + Kuosis = P95 - P5 3 (75-5 om igue 1 1 = 15mm

is ages om a maemaica miimum o 1 o a osee maimum o 5 = 119mm aou wi e oma isiuio aig a aue o 1 ese = 13mm gaic measues ae iusae i ae wic suggess e wo isiuios o e iee i om a o Weigo ak aig kuosis ° Mea = 15 + 119 + 13 = 1173mm a skewess cose o e oma isiuio a o ameos isiu- 3 io aig sig egaie (ie osiie i coeioa ems skewess a ow kuosis ese gaica measues ae o ee wiey use ousie Simiay McCammos mea = 7+1+1+1115+11+11+1+195+135+1 seimeoogy aoug ee wou seem o e scoe o ei aicaio i 1 a eas e eoaoy ases o iesigaios i oe ies ue = 11755 = 1175.5 aias o e iices eis (eg ae 19 eseciay i seimeoo- 1 gica ieaue o wic geogaes wi i a ioucio i CAM Kig (19 7-91 - 1 + 95 - 5 = 13 - 15 + 1 - 7 (iii Oe meos o assessig skewess a kuosis = 1739 Esimaes o skewess may e aiy oaie om e eaiosi ewee e mea a oe measues o cea eecy as saaise McCammos saa eiaio = 1+1315+1395+155-35-95-13-15 y e aiaiiy usig e omuae- 91 = 173

Skewess = 3(mea-meia Saa eiaio Skewess = (1 + - 5 + (5 + 95 - 5 o Skewess = - ( - 1 (95 - 5 Saa eiaio = (15 + 13 - 119 + (7 + 1 - 119 (13 - 15 (1 - 7 ese imesioess aios ae kow as e easoia coeicies o =-9 skewess a oiousy ao oy a ey geeaise iicaio o isiuioa asymmey e is oe wic is e mos wiey use Kurtosis ages om -3 o +3 wi eo eeseig a symmeic isiuio a Kuosis = 95 - 5 aues ewee -1 a +1 eeseig oy moeae skewess e seco (75 - 5 ie oes o ae eie imis a is eeoe o esse aue + 1 - 7 = 1.03 A sime oceue o esig kuosis as ee suggese y Geay (195 - 1 (193 is is ase uo e saisic- o ameo om igue 11- Mea = 5mm Saa eiaio = 15mm Skewess = - Kuosis = 9

3 33 e mome-ase measues eseciay kuosis sou o omay e use o ess a aou o 5 aiaes o aases o ess a aoimaey aiaes e oceue o Saio a Wik (195 gie eow sou eeay e emoye A eceio o is esicio is e Geay aio wic ca e aie o isiuios wi as ew as e aiaes

o a oma isiuio is aio as a aue o 7979 e sigii- (i e W es o omaiy (Saio a Wik 195 cace o eaues om is aue may e ese y eeece o aes suc as a coaie i Geay (193 usig (-1 egees o eeom is iS a eac es o omaiy o sma sames a is ase uo o aykuic isiuios e aio is age a 7979 a o e seco mome o e isiuio emoyig e oowig oceue- eokuic isiuios is ess A aaage o is saisic is a is samig isiuio as ee auae ow o ey sma aues o y Geay (193; a iusaio o is cacuaio a use is gie i ae 9

ae 9 Cacuaio o aoimae iicaios o mome measues Weigo ak easoia Skewess = 3(mea - meia = 3(111 - 119= --7 saa eiaio 11

Geary kurtosis

( Cacuae a saisic as-

o ameo easoia Skewess = 3(737 - 5 = 19 11

Geary kurtosis (5 Cacuae e aio W as-

( ee is W aue o e ae i Saio a Wik (195i wic smae aues o W a ose auae iicae sigiica o- om e ae i Geay (193 o iss (197 ae A e ue a omaiy a e seece sigiicace ee e ecessay coeic- owe ciica ees o e Q saisic ae a e 5 sigiicace ee ies (a a eceage ois o W ae ee auae y Saio 5 a 7 wi 3 egees o eeom Sice o cacuae aues a Wik o aues o u o 5 e oceue is iusae i ae wii e age gie y ese aues i is o ossie o eec e ae 1 om wic i is cea a e es is oe wic as e u yoesis o o ieece ewee e osee isiuio a e I aaage o cosieig e asoue aues o a o e aiaes oma isiuio o eie se o aa i aie comiaios eecig ei ak oe wii e oea isiuio A ue sime esimae o kuosis ai e as (195 age/ saa eiaio es (easo a Sees 19 is cosiee eow (e eec o ouies Secio I (i

3 35 ae 1 Cacuaio o Saio a Wiks W saisic ( o-aameic oea gooess-o-i ess o ameo e oceues eamie aoe ae egae e aiaes as sames aa ae ake i ae 5 Mea = 737mm om a ueyig coiuous isiuio I e emaiig se o oceues a o-aameic es is use o comae e osee equecy isiuio i ems o equecies i casses wi a iscee eese- aio o e oma isiuio usig e same casses a aig e same mea a saa eiaio as e osee isiuio e ess ae eeoe esimaig e ieece ewee wo isiuios wi e same mea a isesio u ossiy iee saes i oe wos e osee isiuio a a eece oma isiuio

1 5 13 37 75 3 wo o-aameic ess ae commoy emoye i is aoac e Ci-squae a Komogoo-Smio ess u e oea oceue 9 13 35 133 may e summaise y e oowig ses-

3 7 117 39 1 (1 Cacuae mea a saa eiaio o e aa

1 1 71 7 15 ( Gou e aa io casses as i eaig o aw a isogam o Ci-squae ceai esicios o e ume o casses ay; 5 17 17 7 1 555 ese ae eaie eow

13 99 737 1 5 (3 Oai e eece equecies i eac o e cose casses ue e assumio o a oma isiuio wi e same mea a 7 115 99 77 3113 saa eiaio as e osee isiuio is oceue wic yies e saa agais wic e osee isiuio 119 91 77 3 75 may e comae y oe o e o-aameic ess is aso eaie ue eow 9 13 97 79 135 1399 ( Cacuae e aoiae es saisics accoig o e es 1 9 795 131 119 cose

11 97 91 15 99 9 Fitting the

1 537 95 53 5 is is oe y usig a ae o e oma oaiiy iega (eg ae A i iss (197 Sages i e oceue wic is iusae 13 31 9 3 175 y ae 11 ae- (1 1 7 5 3 Coe e cass ouaies io uis o saa eiaio aoe a eow e mea 15 7 5 9 5 71 =

is aue is less a e auae aue o 97 a a 5 sigiicace ee a e u yoesis o o ieece ewee e osee a oma isiuios may e eece ( Cosu a ae o e oma oaiiy iega o i e oma oaiiy a a aue wi a ewee e wo cass ouaies o e Weigo ak aa a W saisic o 91 esus; sice o coce is is equiae o e aea ue e oma isiuio is is geae a e auae aue a e equie sigiicace ee cue (igue a is gie i e oma oaiiy iega ae e u yoesis may o e eece i uis aage so a e aea ue e oa cue is 1 37 3 Eac ume i e ae gies e aea ue e cue ewee e mea a e aue o saaise eiae o a aicua aue o eiae a e aea ewee wo aues o saaise eiae (ie cass ouaies is ou y suacio o e wo aea aues (A oious eceio o is is e cass coaiig e mea o wic e auae aues o e cass imis eie sie o e cass coaiig e mea sou e ae ogee a suace om 1 o gie e aea aoiae o e cea cass coaiig e mea

(3 Ceck a ese aeas sum o 1 We cumuae o a casses e saa eiae aues sou oa 1 u may oe e sigy ess as e eoeica oma isiuio as ais wic ee o iiiy i eie iecio a ese cao e ie o e osee isiuio

( Muiy e cass oaiiies (ie aeas ue e cue y e oa equecy o gie e eece ume o aiaes i eac cass assumig a a iscee eeseaio o e oma isiuio is ie o e aa I ae 11 e ouc o cass oaiiy eie as escie aoe a same sie yies e aues sow i e coum eae eece equecy

e osee a eece isiuios may e comae y e Ci- Squae es o e Komogoo-Smio es I e ae is o e use o comaiso is as se se is omie a e eece ooios ouce i se ( ae use isea

ChSrd tt.

o aow e use o a Ci-Squae es o comaiso ewee a osee a oma isiuios i is ecessay o e ume o gous emoye i e equecy isiuio o e eaiey age as may as 1 o gous eig eeae Waso (1957 oie ou a aues i is aicaio e o e age a ei coec aue aoug e ieece is isigiica o 1 o moe gous; Cayes (195 sugges- e a eac cass sou e o age a a e saa eiaio a sou eeay e ess a a quae I is aso esseia a e eece aues cacuae o e e casses i e isiuio ae o ess a oe As is is omay e case is equieme may e cicumee y comiig e eeme casses i e isiuio wi ei aoiig casses ui e equieme is osee O couse e osee equecies mus aso e comie o e same casses Sages i e es as iusae i ae 11 ae e-

(1 o eac oe o e k casses cacuae-

3 39 ae 1 Ci-squae es usig og-ikeioo aio

o ameo

- 75 1 1 - 1 7- 75 5 3 9 751- 5 5 533 - 13 - 9 5 577 3 57 - 9 I some cases a mao o-omaiy i e osee aa may e co- 91- 975 ceae y comiig e e caegoies i oe o aow use o e es 97-15 7 39 177 151-115 11-1 1 11-175 5 17-135 1351-15 1 1 317 1-15 1 317 wee a E ae eie as eoe e susci i ees o eac o e k casses a e summaio ees oe e k casses comisig e

e osee eo equecies ae i e ais o e isiuio a eces e = k - I a ume o osee eo equecies occu i e ei oa eece equecy is geae a 7 eeoe K-1 egees ai o e isiuio a ei oa eece equecy is geae a o eeom ae aoiae ie 11 o equa o 7 e = k - 1 I oe o moe casses ae a osee eo equecy wii e isiuio ae a i e ais a i e auae aue o 11 egees o eeom a e 5 ee is 1975 a e u yoesis o o ieece ewee osee a eec- e isiuios may eeoe e eece a is ee

The Kolmogorov-Smirnov test. Weigo ak_

is es oceue as o equiemes coceig e ume o casses emoye aoug a easoae ume is equie o aow a meaigu es a e geae e ume emoye e moe eecie is e o- ceue Sages i e oceue as iusae i ae 13 ae-

(1 Cumuae e aa io a cumuaie equecy isiuio as i eaig a cumuaie equecy cue o oaiiy o (Secio III

( Coe e cumuaie equecy o eac cass i e osee isiuio io a cumuaie ooio y iiig eac y e same sie o e same o e eece isiuio ie use e iomaio gie y se o iig e oma isiuio

1 ae 13 Komogoo-Smio gooess-o-i es (3 Cacuae e ieece ewee e osee cumuaie ooio a e eece cumuaie ooio as cacuae o e oma o ameo aa isiuio o eac cass i e isogam

Cass Osee Cumuaie Eece Cumuaie ( i e ages o e ieeces cacuae i se (3 (mm equecy osee equecy oaiiy eece igoig e sig as ume as ooio (om ae 11 oaiiy o (5 e meo o eemiig e sigiicace o ees uo e same sie Wee is ess a 3 eeece is mae o a ae o -75 1 1 3 557 557 ciica aues (eg Siege 195 ae E o oe 35 a saisic K s is cacuae as- 75-75 5 173 13 75-5 5 11 37 1775 37 5-9 19 3 591 9-975 3 73 19 735 975-15 7 9 97 1311 1 15-115 9 97 11 937 (i e eec o ouies a oe isuaces o oma isiuio o aa 115-1 9 97 79 933 1-175 9 97 13 971551 e gaica meos iscusse i secio III may someimes iicae a a aicua same is susaiay oma ece o oe o a ew 175-135 9 97 1799 9733 ey eeme oseaios o tlr i e isiuio Suc mig 135-15 1 3 1 377 97711 o eame e e case i iiiua oica som aia oas ae eamie wee e uk o aa eese oma cycoic o coecie 15-15 3 1 3 9775 soms u a ew ey ig aues ae cause y e occasioa assage o uicaes oe e saio ese ouies eeoe eog o a iee ouaio a eeig o e uose o e iesigaio i may e Maimum ieece ewee e eece (ie oma a osee cumuaie oug ecessay o omi em om ay ue aaysis wo saisica ooios is 11 o cass 975-15mm oceues may e suggese o eae e ecisio o e mae wee a eeme aue eeses a ue ouie ios ga es is aicae om aes suc as a i Siege (195 e ciica ee is o sames wic ae om 3 o 3 aiaes ese ae ake i oe o a 5 sigiicace ee Sice e cacuae aue oes o ecee is sie wi e susece aomaous aue 1 aig e ak o 1 e e u yoesis o o ieece ewee e osee a oma ieece ewee e aue o is aiae a is eaes eigous is isiuios may o e eece e uge i eaio o e oea age o e isiuio accoig o oe o e oowig omuae; cose accoig o same sie- Weigo ak aa

Maimum cass ieece 5173 Agai eeoe e u yoesis o ewee 3 a icusie may o e eece

o ewee a 13 icusie

o ewee 13 a 3 icusie

o a 13 aiaes eie aoiae omua may e use e sigii- cace o e aoiae aue is e uge y eeece o e ae esee i io (1951 a eouce y iss (197; is oceue is iusae y ae 15

3 ae 15 Cacuaio o saisics coceig ouies a gie y easo a Sees (19 o iss (197 (197 ae A1 as iusae y ae 15 Use o is ae oceue sou e eome o ameo aa wi cauio as i may o oy iicae eaue om omaiy ue o e esece o ouies u aso ue o e eisece o kuosis as oo sma Dixon's gap test. ese ae ee ake accoig o sie i ae 5 e a aio wi e ouce y a aykuic isiuio susece aomaous aiae is a o 19 aig e aue o 13 mm Oe oms o isuace may aso aec e omaiy o aicua e es saisic use is- sames is may e ue o e miig o wo ouaios as o eame i a makey imoa equecy isiuio o oa aia amou o sige aia ees wic cou esu om sma amous o aia eig ouce y coecioa mecaisms a age amous y cycoic mecaisms A comou isiuio eeoe esus may aso aec e aue o a equecy isiuio is occus we a e sigiicace o is is uge y eeece o e ae i io (1951 same as ee cu o a a gie oi suc as may occu we oo eaks o iss (197 As i ecees e auae aue o 57 a a sigiicace aoe a gie ee ae o aaiae i a seies o eak iscage aues ee o 1 i is uge o eese a sigiica eaue om oma ecause o isumea imiaios o aiues a ig iscages a a ouie eeoe eiss I is ouie is eece e saisic A simia om o isiuioa isuace may occu i cesus aa we may e e ecacuae o e emaiig 9 aiaes o esais wee i oe o eai coieiaiy aa ae suesse o ae aomy a ue ouie eiss I is case- aie o sma aeas o ey ow ouaio esiy I e ume o aues omie om a isiuio is o ee kow e e isiuio is sai o e truncated ae a cesoe e oceue o aaysis o suc isiuios is eyo e scoe o is ioucoy mooga u e ieese eae wi i ue eai i iss (197 15- 3 is is ess a e auae aue o 31 a e 5 sigiicace ee a e u yoesis may eeoe o e eece eeoe oy oe (ii Coice o meos o comaig equecy isiuios wi e aiae as ee ieiie as a ouie oma isiuio

David et.al's . A wie age o meos o aaysig a esig e aue o equecy isiuios as ee esee; ese ae ee summaise i ae

e gaica meos a e associae gaica iices o symmey e sigiicace o is may e uge y eeece o e ae gie y a kuosis sou e egae oy as quick iicaios o e aue o easo a Sees (19 i wic e auae aue a e 5 ee equecy isiuios Moe igoous iicaos o symmey a kuosis is 9 Sice e cacuae aue ecees e auae aue e u ae aoe y e moe aoious cacuaio o momea skewess a yoesis o o ieece om e oma isiuio may e eece a kuosis A eac saisica es o skewess is a eeoe om e sae sigiicace ee momea skewess i e om o ises g1 saisic A ume o meos eiss o esig kuosis a coice ewee em is coiioe Weigo ak aa Corresponding values are:- y e same sie o age sames ises g saisic ase o momea kuosis is mos oweu u ai e as aio is easie Dixon's gap test. eeoe o ouie eiss o cacuae o sma sames e Geay mea eiaio aio sou e emoye o eiie igue ca e gie o isiguisig age a David e al's statistic. eeoe o sigiicay iee sma sames u oies a useu guieie om oma I e isiuio is o e ese agais e oma y a oea es ae a y seaae eamiaio o skewess a kuosis e o age sames ai e a (195 sugges e use o e Saio a Wiks W es sou e use o sma sames a oe o

e o-aameic gooess-o-i ess o age sames I g 1 a g saisics ae ee cacuae a oy oe aows eecio o e u yoesis o o ieece e a oi saisic may e cacuae om ese is is owee a oi skewess a kuosis saisic ae a a ue gooess-o-i es usig e ake oaio as aoe is is eee o a ae suc as

5 I e isiuio o coce seems o e o-oma oy ecause o suggese a e Geay mea eiaio aio is moe eecie i eecig e eisece o oe o moe ouies e a es may e use o assess og-aieess e mome-ase measues ae geeay oweu aoug wee suc ouies o eis ios ga es may e use o sames i mus e sesse a ese ike e measues o Geay a ai e a o 3 o ess aiaes a ai e as es o age sames Use ay oy o wo seciic aiues o o-omaiy amey asymmey a o e ae mus e oe wi cauio owee as i comous ay- kuosis ae a oea omaiy e gooess-o-i ess ae moe kuosis a e eec o ouies geea i a ey ae cocee wi e oea aue o e isiu- io e W es is oay e mos eecie oceue o sma sames ae 1 Summay o saisica measues emoye u is somewa aoious i cacuaio a is eiciecy ecies o age sames y comaiso wi e mome measues o ameo Weigo ak Use o e meos ouie i is cae eeoe oes a Pearson's skewness Sn 19 -7 eecie meas y wic equecy isiuios may e escie eamie Gaica skewess -5 -9 a ese agais e oma isiuio e gooess-o-i meos may aso e eee o aow comaiso ewee a osee isiuio a Momea skewess 111 -1 ay eoeica isiuio suc as e oisso iomia o gamma isi- S S uios I essece e oceue is e same as a escie aoe ece a eece equecies ae geeae accoig o e isiu- io cocee ae a y cosieaio o e oma isiuio Graphical kurtosis Krt 9 13 eais o is ae eyo e scoe o is mooga u may e ou Momea kuosis 55 9 i es suc as Keeig (19 -9 a Coo a Cowe (19 5-3 Geays aio 755 777 S S e eciques escie aoe eeoe oe a iicaio as o e aue o a equecy isiuio e couses oe o e woke we e isiuio is o-oma o oes o saisy oe isiuioa S S equiemes ae eamie i Secio S S S S S S SAEGIES O USE WI O-OMA AA

(i Assumios o e iea moe Otlr Dixon's gap test S S ai eas aio S S e eceig secios ae eamie a ouie meos y wic e isiuioa caaceisics o aa may e assesse Aeio is ow ue o e saegies wic may e emoye i e aicaio o S = o sigiicay iee om oma a a eas 5 ee ese meos suggess a e equecy isiuio o coce is o a om a oes o saisy e equie isiuio o a aicua use S = Sigiicay iee om oma a a eas 5 ee suc as e use o aameic saisica oceues oowig Mice (197 a oye (1973 i may e suggese a e saisica eciques oe a e sig o gaica skewess is e eese o a associae wi e iea moe ioe e oowig asic assumios coeioay use (see e i oe o imoace (Kuska 19-

Comaiso ewee e ess aaiae may aso e mae i ems o Indpndn f brvtn. is imies a oseaios ae mae i a ei eiciecy i eecig e u yoesis o o ieece ewee aom mae so a eac aiae is ieee o eey oe aiae e osee isiuio a e oma we a ea ieece oes eis is assumio is ey oe coaee i geogaica wok a (ae 1 Saio a Wik (195 comae a age o ess o eecig trrltn oe eiss o eame e aue o a aiae a a o-omaiy i a aiey o iee isiuios ey cocue a mome i ime suc as a aiy aia igue is saisicay eee e Komogoo-Smio es is geeay ieio o e es aoug uo e aue o e eceig ay ecause we a y eios e o wi a ew eceios; ese eceios ey aiue o e aes sa moe a oe aiy oseaio eio e aiaes ae us sai ae aiay gou amagamaio oceue Mice (1971 simiay o e auocoeae i e ime omai Auocoeaio may aso occu ois ou a e Komogoo-Smio es aciiaes esig we e i e sace omai wee o eame aia igues o saios i eece equecy equiemes ae o saisie ai eas aio eigouig ocaios wi e o e coeae wi oe aoe Saia is oy eecie o o-skewe isiuios a Geay (1935 as auocoeaio i geogaica eseac as ee e suec o some ece wok (eg Ci a O 1973; ee 197

7 as gie seea oe eames o comueise simuaio wok casig Addtvt o lnrt. is imies a e mai eecs eamie i a ou o e ousess o aameic ess aaysis comie i a iea asio wiou comicaig ieacios o eai e oseaios o a aicua aiae o eame e coce o ousess is si oe wic is eaiey ueie; Gegoy a Gaie (1975 aeme o eamie e coos eee uo ue (197 a aey (19 ae emasise a a coiuum om ous aiage esiy y o aia a geoogy i Souwes Ega u o o-ous mus eis a e ecise egee o ousess o a seciic wee uae o eom a u wo-way aaysis o o acos ecause es i a aicua aicaio mus ee o oy uo e egee o aia is cosey eae o aiue wic is i u eae o ock assumio ioaio i e aa u aso uo e same sies e aue ye e ige aeas eig ueai y moe esisa ocks us ock o e ioaio e samig a es coiios a e aue o e ye a aia wee ou o e eae o oe aoe a ei ieeces wic e eseace wises o make ousess oes o ee- seaae eecs cou o e eaiy ieiie oe oe a uiesa aacea a i is eisece is o e eie uo i aicua aicaios cae sou e ake o seciy uy e coiios is imies a e aiace o e nt f, r ntnt, vrn. ue wic i is use (Mice 197 oseaios is cosa a is eeoe ieee o o e eece aue o e oseaios a o e same sie is assumio is ae 17 Some commo o-aameic saisica ess a ei eaie usuay mae as i geay simiies e esimaio o e saisica owe-eiciecies (ae Siege 195 aamees o coce a is o aicua imoace i egessio a i e Aaysis o aiace we e es o saisica ieece aameic es o-aameic equiae eaie owe ewee gou meas assumes hdtt o equaiy ewee e gou eiciecy (% aiaces es Was es 75 rlt. is assumio is ciicay imoa i e esig o yoeses as i aows e use o a age oy o sime saa esig Meia es 3 - 95 oceues o wic isiuio ucios ae ee eesiey auae Ma-Wiey U 95 (ii Saegies i e assumios o e iea moe ae o saisie Komogoo-Smio 9

e i a ou o ese ou assumios eeoe coce is- es iema wo-way aaysis o 1 iuioa caaceisics o e aa Mice (197 as eiewe aiace e saegies wic may e use i e aa o o saisy a o some o Kuska Wais oe-way aaysis 955 e assumios a suggess a e oios aaiae icue- o aiace

ai n th rbtn f prtr tttl tt. A saisica easo Seama ak coeaio 91 es is sai o e ous i ioaio o assumios coceig e coeicie equecy isiuio o e aa oes o aec e oaiiy isiu- io o e es saisic Cosieae eos ae ee mae o eamie Kea ak coeaio 91 e imac o ioaio o isiuioa assumios o esig oceues coeicie a muc eay wok gae a eas some suo o e oio o some wokes wo ee o use ese oweu saisica ess esie some ioaio o owe-eiciecy is cacuae as escie y Siege (195 --1 e assumios o wic ey es I 1953 o iouce e em usig e assumio a a equiemes o e aoiae aameic ousess o escie e aiiy o a saisica es o aoi a ye es ae me rrr esie assumio ioaio a i 1955 oie ou a may o e ess use o eamie assumios suc as a o omogeeiy o h f nnprtr tttl tt. isiuio-ee o o- aiace ae emsees ikey o ea o eoeous cocusios i ie aameic ess oe a seco aeaie we e assumios iee i is kow aou e ae isiuio e ague sogy agais e use aameic saisica ess ae ioae ese ess ae geeay co- o o-aameic ess eeig o ey uo e ousess o aameic siee o e ess oweu (ae 17 i a ey ae ess ikey o eec measues aicuay we gou o same sies wee equa a sigiica ieece we oe eiss u i muc geogaica wok is oem ca e eaiy cicumee y akig a age same o iiiuas owee sice aou 19 i as ee ossie o iesigae e eec uemoe e o-quoe ieeces i owe-eiciecy ewee e wo o o-omaiy y meas o comue simuaio eciques wi e esu yes o es ae i mos cases ee eauae o siuaios wee a a ess coiece is ow ace i e ousess o aameic measues e assumios o e aameic es ae uie u o o siuaios (ogg 197 o eame Kowaski (197 cocue om a eesie wee some assumio ioaio occus (aey 19 a aock (19 simuaio suy a e ouc mome coeaio coeicie is quie as oe a rltv ffn o a es is a ucio o same sie sesiie o e eecs o o-omaiy a ecommee a is use e sigiicace ee a ye o ieeia yoesis o-aameic ess coie o eay oma aa ois a em (191 use simuae aso ae a weakess a i ei ese sae o eeome ey cao sames o 1 aiaes o eac simia cocusios a ogg (197 9 accommoae come eseac esigs ey o owee oe a aeaie ae 1 Cass ouaies emoye i igues 13 a 1 we aameic assumios ae ioae a ue eais o ei use may e ou i es suc as Siege (195 o Was (19 ogaimic asomaio Cue oo asomaio

Transformation of th . I aa ca e ecas io a om saisyig e oma isiuio e geae a moe oweu esouces eis o Logarithm Equivalent original Cube root Equivalent original esig saisica yoeses (Muee 199 data value data value

9 793 933 1 933 577 97 9 97 9 1 1 3 1 133 11 33 179 17 115 igue 13 Gouig o aiaes io casses o asome aa (Weigo ak aa is sou e comae wi igue o e 37 11 11 1331 uasome aa 31 159 1133 15 e icie ueyig e asomaio ocess is iusae y igue 3133 1359 117 159 13 wic emoys a simia aoac o igue i a e Weigo 317 1 ak aia aa ae oe aog e cea ie eeseig e aiae scae owee uike igue e isogam equecies aoe 3 151 e ie ae o eie o equa-sie cass ieas u ae o casses wic ae ouaies o uequa sies e sie o ese casses iceases ewee aiaes asome i a come mae owee may aso- uwas u i e ogaims (o cue oos o e ouay aues ae maios ae ee suggese a a ie eiew is ow esee; a useu eamie (ae 1 e i wi e see a ese o oow a egua ioucoy suey is Gie (1973 5- a ue eais may e sequece I eec eeoe e aiae scae is isoe y emoyig ou i oye (1973 say ogaimic cass ouaies I acica ems is is moe easiy oe y akig e ogaims o e aa a oig em o a iea scae (iii yes o asomaio iie io equa as (ae 19; igue 1 y wic e same esus ae accomise A gea aiey o yes o asomaio may e ecogise; oye (1973 eamies ieee asomaios i geea use some o wic ae secia cases o oes asomaios may e cassiie accoig o e assumio ioaio coece ie aiiiy omogeeiy o omaiy; aeaiey ey may e cassiie accoig o e maemaica oeaios ioe i e asomaio ocess e ae aoac is use ee Eac o e mao gous o asomaio accoig o ese cieia wi ow e iey eamie wi eeece o aicua asomaios a o igue 1 ogaimicay asome aiaes oe i e same mae ei eecs o e ioaios o assumios eie i e o-asome as igue (Weigo ak aa aa e oigia aiaes ae iicae y e asome aiaes y y e agmaic aioae o asomaio may owee ea o o i- eeaioa a oeaioa iicuies o eame asomaio Power Transformations. ese cosis o aisig a o e aiaes ( i a iaiae coeaio y a ogaimic asomaio o o ses o o a gie owe aa imies e eisece o a eoeia eaiosi ewee e aiaes Aoe oem is a i a asomaio is caie ou i oe o coec oe ioae assumio e aoe may e ioae o a geae ee a eiousy o eame i e Aaysis o aiace wee w is a cosa wic is oe eo a is a cosa o a ai- a se o aa wic is omosceasic ewee gous u wic is skewe cua asomaio aicua aues o wic ae ee commoy use wii gous may e mae eeosceasic y omaisaio o iiiua ae- gou aa A ia oem is a coceig ieeaio o e esuig eaiosi ewee ses o asome aa i a i may e ey iicu o ecogise causa mecaisms ueyig eaiosis 5 51 owe (p) asomaio

- 1 ecioca 33 Cue oo 5 Squae oo

e eec o ese asomaios is o euce osiie skew o icease egaie skew e ecioca is e mos oweu a e squae oo as e eas eec egaiey skewe aa may aso e asome i e same way i ey ae is o a suace om a cosa ( ie

mus e age a e ages oseaio A aeaie o is is o use a owe ( geae a oe Eemey osiiey skewe isi- uios may e mae symmeic y e ecioca asomaio; is is mos commoy emoye we suc a asomaio ossesses ysica meaig a ece ieeaiiy as i Eas e as (1975 use o i o asom e esos e ouseo cesus aiae o ouseos e eso e cue-oo asomaio as ee ou useu o emoa o moeae osiie skewess a as ee aiy wiey aie i aaysis o aia aa (eg Si 1953; owe 195 e squae oo asomaio emoes eaiey sig osiie skewess a may e use wee e aa ae a oisso isiuio i wic e aiace iceases ieay wi e gou meas is asomaio as aso ou aicaios i aia suies eae comou asomaios icue e coa asomaio (eema a ukey 195-

wic is useu o isiuios coaiig a moe a eo aues o owes oe a ose gie aoe may aso e emoye o eame Eas e a (1975 ese e eec o eig asomaios o is ye wi owes agig om + o -3 a Ascome (1953 ou e asomaio

isiuio I o-iege aues o ae acceae o ieeaie gous e a trtv oceue may e emoye o es e eec o successiey cagig e owe emoye ui a mos eay oma isiu- io is eace is aoac as ee use y o a Co (19 o eeo a asomaio

i wic a aue o may e ou ieaiey (uo a uy 197 emiig asomaio o a isiuio wi eo skewess (ikey 1975 1977 A a eame o is oceue e aues o ecessay o yie comeey symmeic isiuios o e o ameo a Weigo ak aa ae -3 a 13 eseciey

ukey (1957; see aso Mosee a ukey(1977 as aso aeme o esoe owe asomaios io amiies y eseig cas wic oay e eec o asomaios o e om-

5 53 moiicaios ae gie y oye (1973 a Keeig (19 wo aso iscuss e aious aues gie o w a i aicua cicumsaces o a aues o ess a 1 wee c is a cosa

Logarithmic transformations. e ogaimic asomaio (Aiciso a e agua asomaio aoe sou o e couse wi e ac ow 1957 may e eesee i e geea case y si asomaio y = og ( + w y = ac si ( + w wee w is a cosa wic wi oe e eo May geogaica aases wee w is a cosa wic may e eo; is asomaio as aso ee oow a og-oma isiuio a ei osiie skewess ca e emoe use o iomia aiaes a o eceage aa (i 197 as a ae- y akig ogaims o e oigia aiaes e eec o is is ess aie o e ogi asomaio I sou e oe a cosieae seee a e ecioca asomaio u moe so a e cue-oo aacy eiss i e saisica ieaue coceig e ems agua asomaio (Gaie 1973 e ogaimic asomaio as ee iese sie a ac si asomaios a ey ae oe icoecy wiey emoye aicuay i Aaysis o aiace we e aiaces use syoymousy e usage aoe ee oows a o oye (1973 a o gous ae ee ou o icease as e squae o e gou mea Kea a ucka (1971 iceases a aso i egessio wee ogaimic asomaio o e oigia aiaes suggess a owe ucio eaiosi ewee e A ue igeomeic asomaio as ee suggese y Co (i oigia aiaes o e om Eas e a 1975 amey w a eig cosas is may e aie o aiaes wic ecause o ei eiiio o eame as sae aio iices ae coie o e imie age - 1 Eeme aues o skewess may e coece y aicaio o e ey cao eeoe e cosiee omay isiue wiou asoma- ogaimic asomaio o e ogaimicay asome aa (Gaie io as e oma isiuio as a iiiey sma u iie equecy 1973; is is eme e og og asomaio a as ee use i o occuece a iiiy o e aiae scae e asomaio as o geogaica suies (eg a ae 197 i may e eesee y- eec o e symmey o e isiuio u gies i ous o us a mius iiiy Oe asomaio oceues ess commoy use i y = og (og geogaica wok ae eamie y oye (1973 e ogi asomaio e use o asomaios eeoe oes a i aoac o e oem o assumio ioaio i e aameic saisica moe; i ca- o soe ioaio o e ieeece equieme u ca aow e use o as aso ee use i geogaica wok (Wigey 1973 I equies e a oweu oy o saisica eoy i may cases a a kowege o as- aiaes o e wii e age om eo o oe a asoms em o e omaioa oceues is eeoe esseia i may geogaica suies ewee us a mius iiiy i oe wos om a cose o a o- cose ume sysem Aoug some cosue eec emais i is quie e use a cacuaio o e moe imoa asomaios ouie eecie i eucig asymmey; i is oe emoye o eceage aa is iusae y ae 19 wic sows oigia aa asome aiaes suiay ecas o ie ewee eo a oe a esuig aues o momea skewess a kuosis o e aia aa eiousy eamie e isiuios o e asome aa ae aso e wo gous o asomaio eamie aoe ae e mos commoy iusae y igue 15 o e o ameo aa use i geogaica suies owee oes ae occasioay o use a ese wi e iey ouie Some icue igoomeic ucios i ei cacuaio; e mos wiey use is e agua o iese sie asomaio- I COCUSIOS

om e ouie esee i e eceig secios i is cea a ee is a muiiciy o meos aaiae o e aaysis o equecy wee w a ae cosas wic may e eo is e oa ume a isiuios a aicuay o ei comaiso wi e oma isi- is e ume o successes i a binomial aiae Ac si mea si -1 uio owee ae sou sugges wic ecique is e mos ao- ie eese o e sie nt ecioca o sie; e asome aa ae iae accoig o e cieia o e aims ogee wi aaiae comua- eeoe ake as eig e age i egees wic ossesses e sie ioa owe o a aicua suy a e same sie aaiae I sou aue cacuae e asomaio is use o saiise e aiace o aso e oe i mi a as ae 1 emosaes e aaiae ess iomia aiaes wic ae aues om eo o oe A ae o aues ae ieig egees o eecieess i eecig e u yoesis o is asomaio is gie y ise a Yaes (19 a ue owee i is wo e-emasisig a e oma isiuio is oy

5 55 oe o a iiie ume o ossie moes a ee is oig sacosac aou omaiy a a asomaio eeses oy oe ossie souio o e oem o o-omaiy Geogaes ae iceas- igy ecomig amiia wi iee yes o equecy isiuio a some guiace o e mos accessie souces o a ioucio o ese is gie i e iiogay

A ia oi woy o emasis is a is mooga as ee co- cee soey wi uiaiae isiuioa caaceisics owee geo- gaica oems ae iceasigy muiaiae i aue wee oe is cocee wi eaiosis ewee wo o moe aiaes us i oe is emoyig aameic eciques oe sou cosie wee aa ae mui- aiae-oma i ei equecy isiuio (Kowaski 197; Gaeesika 1977 1-195 is ie as ee ie eoe i geogay aoug a eaiey accessie geoogica suy ase uo wok y Maia (197 is a o eyme (1971

easos a eciques o eamiaio o e equecy isiuios o geogaica aa ae ee eoe i is ioucoy mooga; e ecessay moiaio sou ie i e esie o e eseace wee a uegauae isseaio o moe aace ee o eac as muc iomaio as ossie om is aa wiou oucig miseaig co- cusios y coaeig e icies uo wic e asic saisica oos ae oue e oouiy is geeay ese wi e amos uiesa aaiaiiy o cacuaig o comuig aciiies a i is o e oe a uue geogaes wi caiaise uo ese oouiies

5 57 GOSSAY O EMS USE (iaics i e Krt. A ie esciig e sae o a equecy isiuio eig eie om e ou mome o e isiuio I is usuay iee- Atrrltn. e iea coeaio ewee memes o seies o e as a iicaio o e eakeess o e isiuio a e ee oseaios oee i ime o sace Saia auocoeaio occus o is ais ecause oseaios o ocaios i cose oimiy o oe aoe wi e o e simia a ikewise emoa auocoeaio occus ecause ptrt. A isiuio wi a ig aue o kuosis is sai o e oseaios o momes i ime cose o oe aoe wi e o e simia eokuic Suc isiuios ae moe eake a e oma isiuio o ae oge ais (see igue 1 nl vrt. I a ia ca ae oe o wo muuay ecusie oucomes (eg yes/o o/o eas/ais a i e oaiiy o oe oucome Mn. e aimeic aeage o a se o aiaes; e usua iicao o ( is cosa oe a seies o ieee ias e e oaiiy cea eecy o oaiig suc oucomes is- Mdn. a aue o a aiae wic iies e oa equecy o aiaes io wo aes we ake i oe I may e eie as e aue o e mie akig oseaio we e same sie is o y aeagig e wo cea-akig oseaios we e same sie is ee o y eaig o e iie eceage oi om a cumuaie oaiiy ga e e isiuio o is eme e iomia isiuio a aiaes meia oes a aeaie o e mea as a iicaio o cea eecy aig is isiuio ae iomia aiaes Mdl l. See moe Cnrn. A cesoe isiuio is oe i wic a kow ume o iiiuas ae aues age o smae a e eeme iiiuas o Md. a cass i a equecy isiuio wic as e iges equecy wic aa ae aaiae (See aso ucaio o occuece is eme e moa cass o coiuous isiuios e cass mi-aue ca e eme e moe o moa aue o e isiuio Cntn vrt ae aiaes wic ca ake ay aue wii e oea Aeaiey o iscee aa e moe is a sige aue aig e age o eac aiae ey ae usuay eie y some om o measueme iges equecy o occuece o ome y makig a aio o oe aiaes Mnt r. Measues o e sae a ocaio o a equecy isiu- Cltv frn rph. A ga sowig a equecy isiuio i e io ase uo e cacuaio o momes aou e mea (See mea saa om o e oa equecy o occuece o ess a eac cass imi eiaio aiace skewess kuosis oe agais eac cass imi tvl d. egaiey skewe isiuios ae asymmeic isiu- rt o iscoiuous aiaes ae aiaes wic ca ae oy a ios i wic e ai o e isiuio is owas e owe e o e iscoiuous se o aues ose mos commoy me ae aiaes eie aiae scae couig nprtr. o-aameic saisica ess ae ose wic o o i- rn dtrbtn. A saeme o ow e equecies o aiaes i a cooae assumios coceig e equecy isiuios o e aa ouaio ae isiue accoig o e aues o e aiae wi i ei sigiicace esig oceue wic i is cocee rl dtrbtn. See Gaussia Gn. A Gaussia o oma isiuio is a uimoa a symmeic equecy isiuio aig eie ooios o oseaios wii rll dtrbtd. See Gaussia seciie aues o e aiae aoe a eow e mea (igue rl prbblt ntrl. A saaise ucio giig e aea ue I is oe escie as e-sae u is moe igoousy eie y- e oma isiuio cue o gie cass ieas om e mea is aea is ooioa o e equecy o occuece o aiaes i eac cass v. e cue sowig e eaiosi o cumuae equecy agais e wic is uy eaie i mos saa saisica es aiae aue is commoy kow as a ogie owee is em sou trdtt. e oey o uequa aiaces o saa eiaios sicy e coie o ose cases wee e cue is sigmoia a e em isiuio cue is o e eee i geea aicaios tr. A a ga i wic a a is aw o eac cass o a aiae wi e eig o eac a eig ooioa o e equecy o aiaes Otlr. aiaes wic ae eyo e imis o a equecy isiuio i eac cass accoig o some aicua eiiio ae eme ouies Sice may isiuios icuig e oma ae ais wic ee o iiiy dtt. e oey o equa aiaces o saa eiaios ieiicaio o ouies ca usuay oy e eome accoig o some Indvdl. A eiy o wic measuemes ae mae seece sigiicace ee Itrtv. A comuaioa oceue wic eies uo e eomace o rtr. aameic saisica ess ae ose wic icooae e same sequece o aimeic oeaios a ume o imes is sai o e assumios usuay o omaiy coceig e equecy isiuios o ieaie Suc a ocess usuay coeges o e esie souio ae a e aa i ei sigiicace esig oceue ew ieaios 5 59 rnt pnt. A aue o a aiae eow wic a gie eceage o oseaios ie is eme a aicua eceage oi o e isiu- Symos use io is is mos easiy eie om e cumuaie oaiiy ga o e isiuio Alphbt - coeicies i W es om ae i Saio a Wiks (195 ltrt. A isiuio wi a ow aue o kuosis is sai o e ay- kuic Suc isiuios ae ess eake a e oma isiuio o - cacuae aue i W es ae soe ais (see igue 1 - cass aues i cacuaio o iomaio oss o ieey- pltn. e eie gou o iiiuas o iees sie casses o isogams

C = ouc o aiae cass mi-oi a equecy tvl d. osiiey skewe isiuios ae asymmeic isiu- ios i wic e ai o e isiuio is owas e ue e o e - egees o eeom i saisica esig aiae scae - mea eiaio om mea o a saisica es is e oaiiy a i eecs e r - ages ieece ewee osee a eece cumuaie aeaie yoesis we a aeaie is ase equecy isiuios i Komogoo-Smio gooess-o-i rbblt ppr. Ga ae wi oe ais so esige a a cumuaie es eceage equecy ga o ay oma equecy isiuio wi o eece equecy i cass i Ci-squae gooess-o-i es as a saig ie - - iemeiae aue o Ci-squae cacuae o eac cass i A cumuaie equecy ga oe wi e equecies rbblt o Ci-squae gooess-o-i es o a oaiiy scae o wic a oma isiuio os as a saig ie - equecy i eac cass i igues ltv ffn (o owe-eiciecy o a es is e aio o same - saisics cacuae i e sigiicace esig o mome sies cocee wi wo ess o a saisica yoesis ecessay o measues yie e same pr agais e same yoesis - cosa i asomaio A suse o e ouaio Spl. - saisics cacuae i e sigiicace esig o mome Sd. A isiuio wic is o symmeic aou e mea is skewe measues Sn. A ie o e ee o wic a isiuio is o symmeic - ume o casses i a equecy gouig aou e mea Seea iices eis u e usua oe is a ase uo - es saisic cacuae i e Komogoo-Smio gooess- e i mome o e isiuio o-i es we 35 Stndrd dvtn. A ie o e aiaiiy o a equecy isiuio - momea kuosis I is ase uo e seco mome o e equecy isiuio a is equa o e squae oo o e aiace - ogaim o e ase e rnfrtn. e ocess o aa asomaio is a y wic some - miimees aimeic oeaio is eome o a aiaes i a equecy isiuio - is o ou momes o a equecy isiuio rntd. A ucae isiuio is oe i wic a ukow ume o iiiuas ae aues o e aiae o coce age o smae a e - ume o aiaes eeme iiiuas o wic aa ae aaiae (See aso cesoig - osee equecies i a isiuio i Ci-squae p I rrr. A ye I eo i saisica esig oceue is o eec gooess-o-i es e u yoesis (o o ieece we i ac i is ue a o owe o eoe i asomaio geuie ieece eiss - - eoes a aicua eceage oi o a isiuio rbl. A quaiy o ie o wic measuemes ae mae - es saisic cacuae i ai e as saisica oceue rn. I geea ems aiace is e egee o aiaiiy o a aa se I moe ecise ems e aiace o a equecy isiuio is equa - es saisics cacuae i ios ga es o e saa eiaio squae a is simiay a ie o e aia- iiy o e isiuio eig cacuae om is seco mome Aaysis - saa eiaio o aiace is a aameic saisica es wic eamies e eaie - momea skewess ee o aiaio o wii a ewee ses o aiaes - cass ouay aues i e Ci-squae gooess-o-i es vrt. Oe oseaio o a aiae o oe iiiua - e eiaio o a aiae om e mea aise o e owe iicae (see aso 1 1 - saisics cacuae i e sigiicace esig o mome measues IIOGAY (Mos useu ioucoy maeia is aseiske - cosa i asomaio A es a aes coaiig susaie iscussio o eea coces - aiaes oe o omig e aa o coce Aiciso a AC ow (1957 h nrl trbtn. Camige - e mea o e aa o coce aock M (19 Sl Sttt. ew Yok - es saisics cacuae i e sigiicace esig o iss CI (197 Sttt fr lt, o ew Yok mome measues ooks CE a Caues (1953 ndb f Stttl Mthd n - cass imis saaise accoig o ei eiaio om Mtrl. oo MSO e mea i eaio o saa eiaio Aso kow as oo saaise oma eiaes Coe a CAM Kig (19 Qntttv Grph. Comack M (1971 h Stttl Arnt. Eiug Coo E a Cowe (19 Appld Gnrl Sttt. ew Yok - sigma Iicaes summaio oe e age gie y e i oo aues sow - i is case om oe o aies G (1971 Cptr rrn n Qntttv l. io W (1951 aios ioig eeme aues Annl f Mthtl Sttt, -7 ookam C a CAM Kig (1971 rl Anl n Grphl. - Ci-squae es saisic wi a we-ocumee isiuio wic ca e use i a ume o sigiicace-ess oo Eo (1977 Sttt n Grph. Oo - Ci-squae aue eie y e og-ikeioo meo Eas IS (1977 e seecio o cass ieas rntn f th 9-1 - mouus o asoue aue o iesecie o Inttt f rth Grphr, Sr, ageaic sig Eas IS Caea a W i (1975 Spf trnfrtn r nr. Cesus eseac Ui Uiesiy o uam Wokig ae Gaes CG a G Eige (197 A geeaise se o iscee isiuios wi OA ogam Mthtl Gl, 1- Gie E (1973 trl Intrprttn. oo Gaeesika (1977 Mthd fr Stttl t Anl f Mltvrt Obrvtn. ew Yok Gugi G a oes (1975 oaiiy i geogaica eseac aicaios a oems h Stttn, 3 157-77 ammo a McCuag (197 Qntttv hn n Grph. • Oo oye M (1973 asomaios - a ioucio a a iiogay Intrntnl Stttl v, 1 3-3 Keeig ES (19 Intrdtn t Stttl Infrn. iceo Kig (199 Stttl Anl n Grph. Egewoo Cis Kumei WC a A Gayi (195 An Intrdtn t Stttl Mdl n Gl. ew Yok ewis (1977 Mp nd Sttt. oo Mice (197 ee aoaces o esoig oems aisig om assumio ioaio uig saisica aaysis i geogaica eseac Chr d Grph d Qb, 1 57-3 Mosee a W ukey (1977 t Anl nd rn. eaig Mass 3 Gaie (197 a eauaio a e umeica eimiaio o Muy E (197 O e esouio o mie equecy isiuios io aua egios Grph ln, 3 11-3 oma comoes Mthtl Gl, 55- aey W (19 Geogaica ocesses a oi aes esig ocie G (1977 Infrntl Sttt fr Grphr. oo moes o iusio y quaa samig rntn f th easo ES a MA Sees (19 e aio o age o saa Inttt f rth Grphr, 1-95 eiaio i e same oma same tr, 51 -7 aey W (19 Some meooogica oems i e use o e eyma eyme A (1971 Muiaiae omaiy i moomeic aaysis ye A a egaie iomia oaiiy isiuios i e rnl f th Intrntnl Atn f Mthtl Gl, aaysis o saia seies rntn f th Inttt f 3 357- rth Grphr, 5-95 Saio SS a M Wik (195 A aaysis o aiace es o i A (1973 e isiuio o umis i Couy ow Iea 3 - omaiy (comee sames tr, 5 591-11 Annl, Atn f Arn Grphr, Siege S (195 nprtr Sttt fr th hvrl Sn. Kasky K (193 Strtr f trnprttn ntr: rltnhp ew Yok btn ntr tr nd rnl hrtrt. eame o Geogay es ae o Uiesiy o Cicago Smi M (1975 ttrn n n Grph. ewo Ao Keyes U asogu E Kumey a M ye (197 Comaiso ayo (1977 Qntttv thd n Grph: n Intrdtn t o seea samig esigs o geogaica aa Grphl Sptl Anl. oso Anl, 95-33 Wik M a Gaaesika (19 oaiiy oig meos o Kie E (1973 Saisica eciques i geogay some aicaios e aaysis o aa tr, 55 1-17 a oems Grphl vpnt, 33-9 Eames o e use o equecy coces i Geogay Kumei WC a W ames (199 equecy isiuios o seam ik egs rnl f Gl, 77 5-5 Aoai Oo G (1973 ouey o agicuua wok i Youaa a ae (197 Some asecs o iea kas eessio eeome i Annl, Atn f Arn Grphr, 3 5-9 i sou cea Keucky Annl, Atn f Arn oy M (197 e sucue o mogage iace uiig socieies Grphr, 57 9-71 a e iis socia omaio rntn f th Inttt f Moga C (197 Some eames o aiage comoe aaysis rth Grphr, Sr, 1 5-71 Ar, 37-1 oos (1977 Coac ume oeies i e suy o ceua Moi a Symos (1977 Eiciecy a equiy asecs o oimum ewoks Grphl Anl, 9 379-7 ocaio Grphl Anl, 9 15-5 Cak (1973 omaiy asomaio a e icia comoes Ogey E (197 aiage asi aia a sae aamees om mome souio a emiica oe Ar, 5 11-3 measues Cndn Grphr, 1 3- esaas M (197 Semaic sucue a eceie eiome ige (197 omaiy asomaios a gi squae aa Grphl Anl, 53-7 Ar, -5 e Smi M (1977 isace isiuios a i eaiou i eie owey G (1975 e eisiuio o aiameay seas i e Uie egios Grphl Anl, 9 33-5 Kigom emes a oiios Ar, 7 1-1 ookam C a CAM Kig (1971 rl Anl n Grphl, Si CK (1953 Cue-oo oma eciiaio isiuios rnt oo n, Arn Gphl Unn, 3 31-5 Eas IS (197 Geea geomoomey eiaies o aiue a ae W (1959 Eames o eaue om e Gaussia i geomoic esciie saisics i Sptl Anl n Grphl, e aaysis Arn rnl f Sn, 57 5- Coey 17-9 oo ae W (19 umeica comaiso o geomoic sames Sn, Eas IS Caea a W i (1975 Spf trnfrtn 131 155- r nr. Cesus eseac Ui Uiesiy o uam Wokig ae ayo (1971 isaces wii saes a ioucio o a amiy o iie equecy isiuios Grf Annlr, 53 -53 Gaie (1971 A aiage esiy ma o amoo rntn f th vnhr Atn, 13 17- i (197 Stttl thd fr th rth ntt: n ntrdtn. oo Gaie (1973 Uiaiae isiuioa caaceisics o some moomeic aiaes Grf Annlr, 5A 17-53 Waig E a W We (1975 Saia aiaio o ie wae quaiy a suey o e ie Ee rntn f th Inttt f rth Grphr, 5 155-71 5 64 C Oe woks eee o i e e ikey (1977 O quick coice o owe asomaio Appld Sttt, 7-9 Aews icke ame ue W oges a W ukey (197 bt Ett f tn. iceo ogg (197 Aaie ous oceues a aia eiew a some suggesios o uue aicaios a eoy rnl f th Ascome (1953 iscussio oowig a ae y oeig Arn Stttl Atn, 9 99-7 rnl l Stttl St, Sr , 15 9 owe WE (195 Cou seeig agais e 19 oug i e o- ae MS (197 e use o asomaios tr, 3 39-5 Eas rnl f Appld Mtrl, 553-9 o GE (1953 o-omaiy a ess o aiaces tr, ue (197 ous saisics a eiew Annl f Mthtl 31-35 Sttt, 3 11-7 o GE a S Aese (1955 emuaio eoy i e eiaio usege (191 Elnt f Stttl Infrn. oso o ous cieia a e suy o eaues om assumio rnl f th l Stttl St, Sr , 17 1-3 Kea MG a W ucka (1971 A tnr f Stttl r. (rd. Edtn. Eiug o GE a Co (19 A aaysis o asomaios rnl f Kig CAM (19 oo th l Stttl St, Sr , 11-5 hn n Grphl. Kowaski C (197 e eomace o some oug ess o iaiae aey (19 trbtnfr tttl tt. Egewoo Cis omaiy eoe a ae cooiae asomaio o omaiy Cayes (195 e og-oma isiuio o e eemes; a iscussio hntr, 1 517- Gh. Ch. At, 119- Kowaski C (197 O e eecs o o-omaiy o e isiuio Ci A a K O (1973 Sptl Atrrltn. oo o e same ouc-mome coeaio coeicie Appld 1 1-1 aigo (197 Is kuosis eay eakeess? Arn Sttt, Stttn, 19- Kuska (19 Saisica aaysis asomaio o aa i ai A O aey a ES easo (195 e isiuio o e Intrntnl Enlpd f th Sl Sn. Cicago aio i a sige oma same o age o saa eiaio Maia K (197 Measues o muiaiae skewess a kuosis wi tr, 1 -93 aicaios tr, 57 519-3 uo W a A uy (197 A comue ogam o eemiig oima McCammo (19 Eiciecies o eceie measues o esciig aa asomaios miimisig skew hvrl rh Mthd e mea sie a soig o seimeay aices rnl f nd Intrnt, - Gl, 7 53-5 iuca M (19 A oe o kuosis rnl f th l Stttl Mice (1975 Wa o successie equecy isiuios sow? St, 111- Arltrl En rh, 7 11- ise A a Yaes (19 Stttl bl, Eiug Mie (1955 Eemes o symmey i e skewe icome cue ok a WC Wa (1957 aos ie a a suy i e rnl f th Arn Stttl Atn, 5 55-71 sigiicace o gai sie aamees rnl f Sdntr Mice (1971 A comaiso o Ci-Squae a Komogoo-Smio trl, 7 3- ess Ar, 3 37-1 eema M a W ukey (195 asomaios eae o e agua Mice (197 ee aoaces o esoig oems aisig om a e squae oo Annl f Mthtl Sttt, 1 7-11 assumio ioaio uig saisica aaysis i geogaica Geay C (1935 e aio o e mea eiaio o e saa eiaio eseac Chr d Grph d Qb, 1 57-3 as a es o omaiy tr, 7 31-55 Muee CG (199 umeica asomaios i e aaysis o eeimea aa Geay C (193 Momes o e aio o e mea eiaio o e hll lltn, 19-3 saa eiaio o oma sames tr, 95-37 ois C a em (191 o-omaiy a ouc-mome coeaio 9 1-7 Gegoy K a Gaie (1975 aiage esiy a cimae rnl f Exprntl Edtn, thrft fr Grphl, 19 7-9 oges A (197 Stttl nl f ptl dprn: th drt oo ee W (197 e imac o socasic ocess eoy uo saia thd. aaysis i uma geogay rr n Grph, 91-1 Sma S (197 e aaysis o aiage ewok comosiio Erth Srf r, 3 19-7 ikey (1975 O owe asomaio o symmey tr, 11-11 Smi E Aeey EE a ewaie (195 A cou seeig eeime i ew Ega Ausaia rnl f Appld Mtrl, 33-1 7 ukey W (1957 O e comaaie aaomy o asomaios Annals of , -3 Was E (19 Handbook of . 3 os iceo Waso GS (1957 e gooess-o-i es o oma isiuios Biometrika, 33- Wigey (1973 An introduction to the use of logit models in geography. CAMOG o 1 Geo Asacs owic