Standard Deviation for PDG Mass Data

Transcription

1 4 Dec 06 Stadard Devato for PDG Mass Data M. J. Gerusa Retred, 47 Clfde Road, Worghall, HP8 9JR, UK. phoe: +(44) Abstract Ths paper aalyses the data for the asses of eleetary partcles provded by the Partcles Data Group (PDG). It fds evdece that the best ass estates are ot based solely o statstcs but also o overall cosstecy, that soetes results sewed u ad axu ass lts. The paper also pots out to soe other qurs that result u ad axu ass lts whch are far fro the statstcal stadard devato. A statstcal ethod s proposed to copute the stadard devato such cases ad whe PDG does ot provde ay lts. Keywords: PDG, eleetary partcles, ass estates, statstcs, stadard devato, stadard error. Itroducto A recet paper by the author preseted a ew theory of eleetary partcles [] that provdes a forula for the relatoshp betwee sp ad ass. The forula ay be tested usg the exstg ass data provded by the Partcle Data Group (PDG) [], but there are dosycrases ad a sgfcat uber of partcles are ot assged ad ax lts. Ths paper provdes a ethod for estatg the stadard devato cases where PDG estates are ssg or for soe reaso are usutable. Each partcle for whch credble data have bee publshed has ts ow secto PDG Partcle Lstgs. Partcles for whch ore accurate forato has bee copled, usually cludg: best ass estate, u ass, axu ass, sp ad party appear also the Partcles Suary. The ass estates are based o the publshed easureets cluded the Lstgs. The estates are ot derved by purely statstcal eas, but also other cosderatos such as the overall cosstecy. For exaple, the values for partcles tae to cosderato the values of partcles. The result of such adustets soetes the best ass s ot be the average of the ad ax lts, hece the lts are ot the stadard devato. I soe extree cases a ass lt cocdes wth the best ass, as for exaple the case of (830) where the best ass = ax ass. Whe the lts are sewed t causes probles wth the relatve predcto error (RPE),.e. the rato of the absolute predcto error dvded by PDG + or error as requred. I the exaple gve above ths leads to a fte RPE. I cases where the ass lts are less sewed, RPE s fte but ay be equally uacceptable. I such cases we wsh to copute the stadard devato usg the publshed easureets appearg the PDG Lstgs. We ay also wsh to do so whe there are o publshed PDG ass lts, whch apples to a vast aorty of the partcles ot cluded PDG Suary. Each author usually provdes the best ass estate, the ad ax lts ad a orty of cases the uber of evets o whch the estates are based. It appears that for the partcles where each author provdes the uber of evets PDG soetes uses the stadard error stead of the stadard devato, but ot always. Whe the stadard error s used the for cosstecy we wsh to copute the stadard devato, gorg the uber of evets observed by dvdual authors, because the stadard error s saller tha the stadard devato by the factor / N, where N s the sze of the saple. The ass easureets for a gve partcle use dfferet reactos, soe authors use better equpet tha others ad tag all ths to cosderato t wll be assued here that the ost ubased ethod s to tae all easureets at ther face value gorg all factors etoed above.

2 4 Dec 06. Probablty ad statstcs Our obectve ths paper s to use the ass estates for a gve eleetary partcle publshed by dfferet authors that are cluded the PDG Lstgs to obta the collectve estate of the ass ea ad the stadard devato. We start wth the basc prcples of probablty ad statstcs [3]. The ea ad varace of cotuous probablty dstrbuto are gve by: ea = x p( x) dx (.) varace = ( x ) p( x) dx (.) stadard devato = (.3) We ca use the forulas above to derve ad for dfferet probablty dstrbutos, e.g. Noral, Beta, Gaa, Studet s t, etc. Soe basc propertes of ad for two depedet varables x ad y volvg costats a, b are ( ax by) a( x) b( y) (.4) ( ax by) a ( x) b ( y) (.5) If we have a saple of values x, x, x the the best ubased estators of ad are (.6) x ( x ) (.7) There s frequetly soe cofuso betwee the above estator of the populato varace ad the quatty ow as the Saple Varace whch s usually deoted by s ad s defed s ( x ) (.8) Ths s used soe statstcal procedures ad the dstcto betwee the two s portat. The Saple Varace s a based estator of the populato varace ad we shall ot use t ths paper.

3 4 Dec Weghted ethod The artcles that publsh results of soe experets aed at deterg the ass of a partcle gve the best ass estate together wth the ass lts,.e. u ad axu. Ofte addto or stead of the ass lts the authors quote the errors ad. If for a gve partcle we have publshed refereces the we wsh to fd the collectve syetrcal lts ± that ay be cosdered to defe the oral dstrbuto. Oe way of estatg the collectve ea ad varace s the ethod of weghted ea ad varace as descrbed a PDG report [4] whch ples that PDG use ths ethod, f ot always the at least soe crcustaces. If the have dfferet ow varaces the the weghted ea s coputed usg the forula ( w (.) w where w ad w w (.) The stadard devato s (.3) w Ths ethod s ot sutable for our eeds aly because of the sall uber of easureets that are usually avalable for a partcle, whch s ostly less tha half a doze. Ths wll be deostrated o a partcular exaple,.e. eleetary partcle L(350). The PDG lstg for ths partcle provdes three ass easureets wth the followg values MeV: 370±50 365±0 358± 6 These easureets result the followg weghted ad = 359 = 6 We ote that = 359 s alost exactly the sae as the best ass estate 3 =358 ad = 6 s exactly the sae 3! Thus the frst two easureets wth greater errors have bee copletely gored! O these grouds we coclude that ths ethod s ot sutable for our eeds because t s ot uusual for partcles to have oly three or a hadful of easureets. 3

4 4 Dec Tw varace ethod Aother way of fdg s to use the u-weghted ea of the best ass estates. However, for we eed to tae to accout two factors, the spread of the best ass estates ad the spread of the values ( ) ad (ax ). Thus we have ( ad ( ) ( ) (3.) where ( (3.) ( ( ( )) (3.3) ad ( for =, (3.4a) I order to fd () we frst eed to fd () ax ( for = +, (3.4b) ( ) (3.5a) ( ) ( ( ) (ax ( ) where = - (3.5b) ( ) ( ( )) (3.6a) ( ) ( ( ( )) (ax ( ( )) (3.6b) I the PDG data reported by dvdual authors t s a excepto whe for a partcle easureet. Thus ( s close to the correspodg (ax (),.e. approxately. Therefore there s oe less degree of freedo ad to reflect ths the dvsor s used above stead of. For a sall uber of authors, for exaple N(060) wth three ass easureets, we obta whch leads to a saller ass tha the sallest ass the data, ad a greater ax ass tha the largest ax ass the data. Thus ths ethod caot be rght. The reaso why ths ethod s wrog s that forula (.5) for addg two varaces apples strctly oly to very large saples. Whe the saple s sall the forula s grossly accurate, as ca be easly verfed usg two rado varables wth three values each. 4

5 4 Dec Approxate forulas Fro the results the prevous secto we coclude that uless we have a very large uber of authors cotrbutg to a gve partcle we ust drop fro the forula (3.) for ether ( or (). Usg the PDG data there s a cosstet patter, aely that ( s usually saller tha () by a order of agtude ad occasoally eve saller. Thus, f we were to use ( we would obta ad ax lts slar to the weghted ethod. But ths also eas that droppg ( wll usually ot ae uch dfferece. Thus we eed to re-exae the forula for (). Let us therefore re-wrte forula (3.5b) for () a way that s easer to uderstad, aely ax ( ) ( (4.) We see that f for each publshed referece we have = the we have the best ass estates values = ( + ax)/ ad substtutg ths to (4.) above we get ( ) 0 (4.) There s a very sall percetage of cases whe ths codto s ot satsfed, hece () s always approxately zero. Substtutg the above to forula (3.6b) we fd ( ) ( ( ) (ax ( ) (4.3) If for a gve easureet the best ass does ot cocde wth the ea ass ( the we have ( () dfferet fro (ax () ad ther su s greater tha the su of ( ) ad (ax ) sce as a rule they are equal. Thus the forula gvg the u possble () s ( ) (4.4) There are occasos whe ost authors over-estate the accuracy of ther ow experets ad as a result dsagree aog theselves o the best ass estate. Ths results a urealstc () that s uch saller tha (. O such occasos we reserve the freedo to use ( ( ( ( )) (4.5) I cocluso, f t s ecessary to calculate the stadard devato for PDG ass data we wll use forulas (4.4) or (4.5) whchever gve greater as otherwse we ay obta slar results to the weghted ethod, whch s uacceptable. I practce t eas that we wll ostly use (4.4) ad oly very occasoally (4.5). 5

6 4 Dec 06 Acowledgeets I would le to tha y wfe Jo for her lovg care wthout whch ths wor would ot have bee possble. I tha God for everythg. Refereces [] Gerusa M J 06 Vr Theory of Eleetary Partcles vxra:6.036 [Hgh Eergy Partcle Physcs] [] Partcles Data Group (PDG) 05 [3] Cooper B E 969 Statstcs for Experetalsts (Pergao Press) p.9, 36, 46, 47, 50 [4] Cowa G 05 p.4 6