{"id":2108,"date":"2023-07-23T15:52:04","date_gmt":"2023-07-23T15:52:04","guid":{"rendered":"https:\/\/statorials.org\/pl\/pozostalosc-pearsona\/"},"modified":"2023-07-23T15:52:04","modified_gmt":"2023-07-23T15:52:04","slug":"pozostalosc-pearsona","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/pozostalosc-pearsona\/","title":{"rendered":"Co to s\u0105 reszty pearsona? (definicja \u2013 przyk\u0142ad)"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\"><strong>Reszty Pearsona<\/strong> wykorzystuje si\u0119 w <a href=\"https:\/\/statorials.org\/pl\/test-niezaleznosci-chi-kwadrat\/\" target=\"_blank\" rel=\"noopener\">te\u015bcie niezale\u017cno\u015bci chi-kwadrat<\/a> w celu analizy r\u00f3\u017cnicy mi\u0119dzy obserwowan\u0105 liczb\u0105 kom\u00f3rek a oczekiwan\u0105 liczb\u0105 kom\u00f3rek w tabeli kontyngencji.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Wz\u00f3r na obliczenie <strong>reszty Pearsona<\/strong> jest nast\u0119puj\u0105cy:<\/span><\/p>\n<p> <span style=\"color: #000000;\">r <sub>ij<\/sub> = (O <sub>ij<\/sub> \u2013 E <sub>ij<\/sub> ) \/ \u221a <span style=\"border-top: 1px solid black;\">mi <sub>ij<\/sub><\/span><\/span><\/p>\n<p> <span style=\"color: #000000;\">Z\u0142oto:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>r <sub>ij<\/sub><\/strong> : Reszta Pearsona dla kom\u00f3rki i <sup>-tej<\/sup> kolumny i j- <sup>tego<\/sup> rz\u0119du<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>O <sub>ij<\/sub><\/strong> : Warto\u015b\u0107 zaobserwowana dla kom\u00f3rki i <sup>-tej<\/sup> kolumny i j- <sup>tego<\/sup> wiersza<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>E <sub>ij<\/sub><\/strong> : Oczekiwana warto\u015b\u0107 kom\u00f3rki i- <sup>tej<\/sup> kolumny i j- <sup>tego<\/sup> wiersza<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Podobn\u0105 miar\u0105 jest <strong>standaryzowana (skorygowana) reszta Pearsona<\/strong> , kt\u00f3r\u0105 oblicza si\u0119 w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n<p> <span style=\"color: #000000;\">r <sub>ij<\/sub> = (O <sub>ij<\/sub> \u2013 E <sub>ij<\/sub> ) \/ \u221a <span style=\"border-top: 1px solid black;\">mi <sub>ij<\/sub> (1-n <sub>ja+<\/sub> )(1-n <sub>+j<\/sub> )<\/span><\/span><\/p>\n<p> <span style=\"color: #000000;\">Z\u0142oto:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>r <sub>ij<\/sub><\/strong> : Reszta Pearsona dla kom\u00f3rki i <sup>-tej<\/sup> kolumny i j- <sup>tego<\/sup> rz\u0119du<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>O <sub>ij<\/sub><\/strong> : Warto\u015b\u0107 zaobserwowana dla kom\u00f3rki i <sup>-tej<\/sup> kolumny i j- <sup>tego<\/sup> wiersza<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>E <sub>ij<\/sub><\/strong> : Oczekiwana warto\u015b\u0107 kom\u00f3rki i- <sup>tej<\/sup> kolumny i j- <sup>tego<\/sup> wiersza<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>p <sub>i+<\/sub><\/strong> : Suma wiersza podzielona przez sum\u0119 ca\u0142kowit\u0105<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>p <sub>+ j<\/sub><\/strong> : Suma kolumny podzielona przez sum\u0119 ca\u0142kowit\u0105<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Standaryzowane reszty Pearsona maj\u0105 rozk\u0142ad normalny ze \u015bredni\u0105 0 i odchyleniem standardowym 1. Ka\u017cda standaryzowana reszta Pearsona o warto\u015bci bezwzgl\u0119dnej powy\u017cej pewnych prog\u00f3w (np. 2 lub 3) wskazuje na brak dopasowania.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Poni\u017cszy przyk\u0142ad pokazuje, jak w praktyce obliczy\u0107 reszty Pearsona.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Przyk\u0142ad: Obliczanie reszt Pearsona<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Za\u0142\u00f3\u017cmy, \u017ce badacze chc\u0105 zastosowa\u0107 test niezale\u017cno\u015bci chi-kwadrat, aby ustali\u0107, czy p\u0142e\u0107 jest powi\u0105zana z preferencj\u0105 partii politycznej.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Decyduj\u0105 si\u0119 pobra\u0107 prost\u0105 losow\u0105 pr\u00f3b\u0119 500 wyborc\u00f3w i zapyta\u0107 ich o preferencje dotycz\u0105ce partii politycznych.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Wyniki badania przedstawia poni\u017csza tabela kontyngencji:<\/span><\/p>\n<table width=\"379\">\n<tbody>\n<tr>\n<td width=\"84\"><\/td>\n<td style=\"text-align: center;\" width=\"74\"> <span style=\"color: #000000;\"><strong>Republika\u0144ski<\/strong><\/span><\/td>\n<td style=\"text-align: center;\" width=\"74\"> <span style=\"color: #000000;\"><strong>Demokrata<\/strong><\/span><\/td>\n<td style=\"text-align: center;\" width=\"110\"> <span style=\"color: #000000;\"><strong>Niezale\u017cny<\/strong><\/span><\/td>\n<td width=\"37\"> <span style=\"color: #000000;\"><strong>Ca\u0142kowity<\/strong><\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>M\u0119\u017cczyzna<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">120<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">90<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">40<\/span><\/td>\n<td> <span style=\"color: #000000;\">250<\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>Kobieta<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">110<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">95<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">45<\/span><\/td>\n<td> <span style=\"color: #000000;\">250<\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>Ca\u0142kowity<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">230<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">185<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">85<\/span><\/td>\n<td> <span style=\"color: #000000;\">500<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p> <span style=\"color: #000000;\">Przed obliczeniem reszt Pearsona musimy najpierw obliczy\u0107 oczekiwane zliczenia dla ka\u017cdej kom\u00f3rki w tabeli kontyngencji. W tym celu mo\u017cemy skorzysta\u0107 z nast\u0119puj\u0105cej formu\u0142y:<\/span><\/p>\n<p> <span style=\"color: #000000;\">Oczekiwana warto\u015b\u0107 = (suma wierszy * suma kolumn) \/ suma tabeli.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Na przyk\u0142ad oczekiwana warto\u015b\u0107 dla m\u0119\u017cczyzn z Partii Republika\u0144skiej wynosi: (230*250) \/ 500 = <strong>115<\/strong> .<\/span><\/p>\n<p> <span style=\"color: #000000;\">Mo\u017cemy powt\u00f3rzy\u0107 t\u0119 formu\u0142\u0119, aby uzyska\u0107 oczekiwan\u0105 warto\u015b\u0107 dla ka\u017cdej kom\u00f3rki tabeli:<\/span><\/p>\n<table width=\"379\">\n<tbody>\n<tr>\n<td width=\"84\"><\/td>\n<td style=\"text-align: center;\" width=\"74\"> <span style=\"color: #000000;\"><strong>Republika\u0144ski<\/strong><\/span><\/td>\n<td style=\"text-align: center;\" width=\"74\"> <span style=\"color: #000000;\"><strong>Demokrata<\/strong><\/span><\/td>\n<td style=\"text-align: center;\" width=\"110\"> <span style=\"color: #000000;\"><strong>Niezale\u017cny<\/strong><\/span><\/td>\n<td width=\"37\"> <span style=\"color: #000000;\"><strong>Ca\u0142kowity<\/strong><\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>M\u0119\u017cczyzna<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">115<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">92,5<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">42,5<\/span><\/td>\n<td> <span style=\"color: #000000;\">250<\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>Kobieta<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">115<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">92,5<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">42,5<\/span><\/td>\n<td> <span style=\"color: #000000;\">250<\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>Ca\u0142kowity<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">230<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">185<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">85<\/span><\/td>\n<td> <span style=\"color: #000000;\">500<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p> <span style=\"color: #000000;\">Nast\u0119pnie mo\u017cemy obliczy\u0107 <strong>reszt\u0119 Pearsona<\/strong> dla ka\u017cdej kom\u00f3rki w tabeli.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Na przyk\u0142ad reszt\u0119 Pearsona dla kom\u00f3rki zawieraj\u0105cej republikan\u00f3w mo\u017cna by obliczy\u0107 w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">r <sub>ij<\/sub> = (O <sub>ij<\/sub> \u2013 E <sub>ij<\/sub> ) \/ \u221a <span style=\"border-top: 1px solid black;\">mi <sub>ij<\/sub><\/span><\/span><\/li>\n<li> <span style=\"color: #000000;\">r <sub>ij<\/sub> = (120 \u2013 115) \/ \u221a <span style=\"border-top: 1px solid black;\">115<\/span><\/span><\/li>\n<li> <span style=\"color: #000000;\">r <sub>ij<\/sub> = 0,466<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Mo\u017cemy powt\u00f3rzy\u0107 t\u0119 formu\u0142\u0119, aby uzyska\u0107 reszt\u0119 Pearsona dla ka\u017cdej kom\u00f3rki w tabeli:<\/span><\/p>\n<table width=\"379\">\n<tbody>\n<tr>\n<td width=\"84\"><\/td>\n<td style=\"text-align: center;\" width=\"74\"> <span style=\"color: #000000;\"><strong>Republika\u0144ski<\/strong><\/span><\/td>\n<td style=\"text-align: center;\" width=\"74\"> <span style=\"color: #000000;\"><strong>Demokrata<\/strong><\/span><\/td>\n<td style=\"text-align: center;\" width=\"110\"> <span style=\"color: #000000;\"><strong>Niezale\u017cny<\/strong><\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>M\u0119\u017cczyzna<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">0,446<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">-0,259<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">-0,383<\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>Kobieta<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">-0,446<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">0,259<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">0,383<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p> <span style=\"color: #000000;\">Nast\u0119pnie mo\u017cemy obliczy\u0107 <strong>standaryzowan\u0105 reszt\u0119 Pearsona<\/strong> dla ka\u017cdej kom\u00f3rki w tabeli.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Na przyk\u0142ad standaryzowan\u0105 reszt\u0119 Pearsona dla kom\u00f3rki zawieraj\u0105cej republikan\u00f3w mo\u017cna by obliczy\u0107 w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">r <sub>ij<\/sub> = (O <sub>ij<\/sub> \u2013 E <sub>ij<\/sub> ) \/ \u221a <span style=\"border-top: 1px solid black;\">mi <sub>ij<\/sub> (1-p <sub>ja+<\/sub> )(1-p <sub>+j<\/sub> )<\/span><\/span><\/li>\n<li> <span style=\"color: #000000;\">r <sub>ij<\/sub> = (120 \u2013 115) \/ \u221a <span style=\"border-top: 1px solid black;\">115(1-250\/500)(1-230\/500)<\/span><\/span><\/li>\n<li> <span style=\"color: #000000;\">r <sub>ij<\/sub> = 0,897<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Mo\u017cemy powt\u00f3rzy\u0107 ten wz\u00f3r, aby otrzyma\u0107 standaryzowan\u0105 reszt\u0119 Pearsona dla ka\u017cdej kom\u00f3rki w tabeli:<\/span><\/p>\n<table width=\"379\">\n<tbody>\n<tr>\n<td width=\"84\"><\/td>\n<td style=\"text-align: center;\" width=\"74\"> <span style=\"color: #000000;\"><strong>Republika\u0144ski<\/strong><\/span><\/td>\n<td style=\"text-align: center;\" width=\"74\"> <span style=\"color: #000000;\"><strong>Demokrata<\/strong><\/span><\/td>\n<td style=\"text-align: center;\" width=\"110\"> <span style=\"color: #000000;\"><strong>Niezale\u017cny<\/strong><\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>M\u0119\u017cczyzna<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">0,897<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">-0,463<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">-0,595<\/span><\/td>\n<\/tr>\n<tr>\n<td> <span style=\"color: #000000;\"><strong>Kobieta<\/strong><\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">-0,897<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">0,463<\/span><\/td>\n<td style=\"text-align: center;\"> <span style=\"color: #000000;\">0,595<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p> <span style=\"color: #000000;\">Widzimy, \u017ce \u017cadna ze standaryzowanych reszt Pearsona nie ma warto\u015bci bezwzgl\u0119dnej wi\u0119kszej ni\u017c 3, co wskazuje, \u017ce \u017cadna z kom\u00f3rek nie przyczynia si\u0119 do znacz\u0105cego braku dopasowania.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Je\u015bli u\u017cyjemy <a href=\"https:\/\/statorials.org\/pl\/kalkulator-testu-niezaleznosci-chi-kwadrat\/\" target=\"_blank\" rel=\"noopener\">tego kalkulatora internetowego<\/a> do wykonania testu niezale\u017cno\u015bci chi-kwadrat, oka\u017ce si\u0119, \u017ce warto\u015b\u0107 p testu wynosi <strong>0,649198<\/strong> .<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\" wp-image-19624 aligncenter\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/ence1.png\" alt=\"\" width=\"443\" height=\"447\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Poniewa\u017c ta warto\u015b\u0107 p jest nie mniejsza ni\u017c 0,05, nie mamy wystarczaj\u0105cych dowod\u00f3w, aby stwierdzi\u0107, \u017ce istnieje zwi\u0105zek mi\u0119dzy p\u0142ci\u0105 a preferencjami partii politycznych.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Dodatkowe zasoby<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Poni\u017csze samouczki wyja\u015bniaj\u0105, jak przeprowadzi\u0107 test niezale\u017cno\u015bci chi-kwadrat przy u\u017cyciu r\u00f3\u017cnych program\u00f3w statystycznych:<\/span><\/p>\n<p> <a href=\"https:\/\/statorials.org\/pl\/test-niezaleznosci-chi-kwadrat\/\" target=\"_blank\" rel=\"noopener\">Wprowadzenie do testu niezale\u017cno\u015bci chi-kwadrat<\/a><br \/> <a href=\"https:\/\/statorials.org\/pl\/test-niezaleznosci-chi-kwadrat-excel\/\" target=\"_blank\" rel=\"noopener\">Jak wykona\u0107 test niezale\u017cno\u015bci chi-kwadrat w programie Excel<\/a><br \/> <a href=\"https:\/\/statorials.org\/pl\/test-niezaleznosci-chi-kwadrat-w-r\/\" target=\"_blank\" rel=\"noopener\">Jak wykona\u0107 test niezale\u017cno\u015bci chi-kwadrat w R<\/a><br \/> <a href=\"https:\/\/statorials.org\/pl\/kalkulator-testu-niezaleznosci-chi-kwadrat\/\" target=\"_blank\" rel=\"noopener\">Test chi-kwadrat kalkulatora niezale\u017cno\u015bci<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Reszty Pearsona wykorzystuje si\u0119 w te\u015bcie niezale\u017cno\u015bci chi-kwadrat w celu analizy r\u00f3\u017cnicy mi\u0119dzy obserwowan\u0105 liczb\u0105 kom\u00f3rek a oczekiwan\u0105 liczb\u0105 kom\u00f3rek w tabeli kontyngencji. Wz\u00f3r na obliczenie reszty Pearsona jest nast\u0119puj\u0105cy: r ij = (O ij \u2013 E ij ) \/ \u221a mi ij Z\u0142oto: r ij : Reszta Pearsona dla kom\u00f3rki i -tej kolumny i [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-2108","post","type-post","status-publish","format-standard","hentry","category-przewodnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Co to s\u0105 reszty Pearsona? 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