Study on the Normal and Skewed Distribution of Isometric Grouping

Transcription

1 Open Journ of Sttstcs Pubshed Onne October ( Study on the orm nd Skewed Dstrbuton of Isometrc Groupng Zhensheng J Wenk J Schoo of Economcs nd Mngement Chongqng Three Gorges Unversty Chongqng Chn Students Affrs Deprtment Chongqng Three Gorges Unversty Chongqng Chn Em: kx76@sn.com eceved Juy 5 ; revsed August ; ccepted August 3 ABSTACT Becuse of thnkng ony the number of numbers but not fttng functon t woud be dequte to tke further fed when ccutng group numbers wth emprc formu. We hve proved the three theorems bsed on studyng the norm dstrbuton nd then rech the concuson tht there s better method to do the sme work. The method s smper nd more prctc thn emprc method nd so works we wth ny skewed dstrbuton. Keywords: Men Vue; Vrnce; Ft Functon; nge; One-Qurter nge. Introducton In vrous books on sttstcs when dscussng the sometrc group the emprc formu s treted s ony reference or put sde smpy. It s ttrbuted tht the css nterv s ony reevnt to the number of numbers not the shpe of ft functon n the emprc formu. Ths pper nyzes the sometrc group whe conformng to the norm dstrbuton of seres nd derves smpe nd prctc method to fnd css nterv. Furthermore the sme formu so works we wth ny skewed dstrbuton.. Hstogrm nd the Upper Bound of Css Interv.. Fnd Out Theorem ) Observe x x x nd fnd out the mnmum vue x nd mxmum vue x. And seect proper c tht sghty ess thn x nd d tht sghty greter thn x nd dvde ( cd ) nto ntervs. d c Ech nterv hs the sme ength s et d c t s esy to see. s rnge normy nd then dvde nto fve or more groups. Denote d cs d c we c s cosed rnge []. The seres s be-shped dstrbuted n symmetrc wy. Set the xs of symmetry x ts ft densty x functon s f x e ccordng to the sm- π pe numbers dvde t nto groups nd the scope of ech group s 3 nd denote t s m whch s the number of smpes n A x p A f x dx. Drwng dgrm s m m m Ths dgrm s ced frequency dstrbuton of csses of smpe numbers. Obvousy m S m m m. Copyrght Sces.

2 Z. S. JIA W. K. JIA Set terv x s bse e s heght n ech n- π mke rectnge S e nd then the hstogrm s mde π out (Fgure ). And S P A S S e S e π π π S e π s s s equs (Tbe ) S s s π e e e π H m S m( s s ) m e e π π dh d π () m e e π G e e ) Set when we get the mnmum of G. π S μ Fgure. The formton process of hstogrm. x Tbe. orm dstrbuton of dt (Exmpe ) ) When d H we hve the turnng pont d ow m e e 9 3e e 5e when the mxmum of H s nd S s the prt sum of H. Accordng to the eference [3] 6 so s the upper bound of css nterv []... Mjor Theorem Theorem. If group of numbers shows norm dstrbuton ts ft densty s e [3]. x π When we dscuss the sometrc group s the upper bound of ( s the css nterv). ) Defntons Defnton We dvde nto four equ prts Ther ponts re Q Q Q Q3 Q Q Q s the md-rnge. Q3 Q Q Q= s the qurte. Q- Q qunte of order eght. Defnton The mdpont of nterv (or the symmetrc nterv) s ced one-qurter rnge nd denoted s Defnton 3 If numbers re dstrbuted n then. ndcted the verge dstnce ech number shres. Copyrght Sces.

3 Z. S. JIA W. K. JIA 9 Defnton If there s n nterv whch mkes m then the nterv s ced smr nterv. ) Mjor theorem Lemm. Men vue theorem ntegrs. If f(x) n s consecutve nd symmetrc ccdenty functon there s pont mkes f x d x f. Theorem. There re t est two smr ntervs n normy dstrbuted seres [3]. Proof. By Lemm there s fnd m m mn. m m Let mn m then the ntervs n m re smr ntervs. By Lemm the smr ntervs re t est. Theorem 3. A normy dstrbuted seres hs smr 3 nterv n where s the css nterv [3]. 3. Work out the Wy to Fnd Css Interv by Theorem 3 ) Arrnge the dt n scendng order nd ccute the verge nd vrnce of the numbers. ) Fnd the cosed rnge nd ccute f ; to Expndng from ( to the two ponts). 3) Fnd out the mnmum smr ntervs s sn ; nd ther numbers re s s respectvey. Let t stsfy s Let s. n mn s - sn. s - be the mnmum vue thus et 3 ) Fx on the css nterv : s. 5) Groupng group of numbers nk the groups up crefuy. If t s done we t cn refect the over trend. The chn of numbers s b b bn bn where b b bn re ncuded n one group. How to nsert ths css nto the chn of dt? We set rue tht ths group shoud be ncuded n c where b b ; bn bn c 6) If t s skewed dstrbuton the bove method s so vbe but need to do twce referrng to the fowng exmpe. 3.. Exmpe By smpe survey of vng condtons of urbn househods we get the foowng numbers of per cpt monthy househod ncome (redy rrnged). The mnmum number s 6 nd the mxmum number s 3. It s mrm dstrbuton progresson. Men: c = 6 d = 6. ) Becuse t s not competey symmetrc we just consder the dt from 5 to. From 6 to there re 7 numbers nd the verge dstnce between 9 them: F ts coordnte s = 95; 5 ts coordnte s = 75; 675 ts coordnte s = ) There re ponts ncuded n tht s (75 75) the ength of ths nterv s 5. The ponts re nd When ccutng the verge dstnce of the ponts the nterv we shoud 5 consder s (Tbe ). It s suggested tht the smr nterv s coser to 3 thn to. It s esy to see tht there re sx numbers between nd whch re nd 97 nd F 33.3 whch s smr to 6 Copyrght Sces.

4 5 Z. S. JIA W. K. JIA F Let be the css nterv then = s the vuton of. We coud get the foowng dstrbuton seres fter further rrngement. 3.. Exmpe By smpe survey of vng condtons of urbn househods we get the foowng numbers of per cpt monthy househod ncome (redy rrnged). ) Tbe 3: Skewed dstrbuton of dt for Exmpe []. The mnmum number s nd the mxmum number s 3. And the verge s where c d the cosed rnge s 6 (Tbe ). ) Becuse t s not competey symmetrc we dvde nto two steps to fnsh t. From to 97. there re 7 numbers nd the verge dstnce between them s Tbe. orm dstrbuton of dt (Exmpe ) Tbe 3. Here choose C = 6 D = for Exmpe. Per Cpt Monthy Income (MB Yun) Househods Frequency (%) Tot 5. Tbe. Here choose C = D = for Exmpe 3 []. Per Cpt Monthy Income (MB Yun) Househods Frequency (%) Tot F The hf rnge s 697. ; 3.6 ts coordnte s =.6; 7.3 ts coord- nte s = ) There 3 re 3 ponts ncuded n tht s (97 33) the ength of ths nterv s 39. The 3 ponts re nd 3. When ccutng the verge dstnce of the 3 ponts the nterv we shoud consder s nd the verge dstnce s whch s greter thn F 5.. It suggests tht when seectng smr ntervs t s skewed to the rght. Thus we shoud deete 99 n the foowng dscusson. We hve the resut s the foowng We dd horzont ne between numbers. We fnd tht.6 s pont on the rght sde the dstnce s the verge dstnce of ponts on the eft s ; Copyrght Sces.

5 Z. S. JIA W. K. JIA 5 We coud further scertn tht there re two ponts on the rght sde of.6 t most whch re 6 nd. It s esy to fnd tht there re numbers n nterv ( ) (see the ffth ponts n the thrd prt) whch re nd. ) Snce F 5 5. F ( ) s smr nterv. And S = The Length of smr nterv s s the uton of. Mthemtcs dgrm Frequency dstrbuton of the per cpt monthy ncome vbe for vng expenses of urbn househods n certn cty. EFEECES then the nterv [] Q.. Xe nd Z. Z. Hn Prncpe of Sttstcs 6th By the concuson of ) nd ) s we coud get the foowng dstrbuton seres fter further rrngement. Edton Jnn Unversty Press Jnn 99 pp [] C. S. Wu Probbty nd Sttstcs Hgher Educton Press Bejng pp. -. [3] B. H. Qn nd L. W. Hung Sttstcs Schun Peope s Pubshng House Chengdu p. 7. [] Z. S. J A the Proof Are Pubshed n the Mthemtcs Prctces nd Theory Vo. o. pp. 3-. Copyrght Sces.