[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-36541-en":3,"doc-seo-36541-105":29,"detail-sidebar-cat-0-en-105":89},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},36541,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown","The article addresses the Kolmogorov-Smirnov test for checking whether observations follow a normal distribution when both the mean and variance are unknown and must be estimated from the same sample. Standard Kolmogorov-Smirnov critical tables fail under parameter estimation, so the note provides Monte Carlo–derived critical values for the adjusted statistic. It also compares the test’s power and discusses its conservativeness relative to tabulated results when parameters are estimated.","","cbCaircD9FVniI2b","https://ap.wps.com/l/cbCaircD9FVniI2b","pdf",267039,1,5,"English","en",105,"# On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown\n## Background: limitations of standard Kolmogorov-Smirnov tables\n## Proposed procedure using estimated parameters\n## Monte Carlo derivation of critical values\n## Power considerations and comparison with chi-square","[{\"question\":\"Why are the standard Kolmogorov-Smirnov critical tables not valid when mean and variance are estimated from the sample?\",\"answer\":\"Because when distribution parameters are estimated from the data, the usual Kolmogorov-Smirnov null distribution used by standard tables no longer applies, so the tabulated critical points become invalid.\"},{\"question\":\"What decision rule does the article propose for the normality test with unknown mean and variance?\",\"answer\":\"Compute D=maxx|F*(X)−Sn(X)|, where Sn is the sample cumulative distribution and F* is the normal CDF using the sample mean and sample variance; reject normality if D exceeds the Monte Carlo critical value from the provided table.\"},{\"question\":\"How were the critical values in the table obtained?\",\"answer\":\"They were derived by Monte Carlo simulation: for each sample size N, 1,000 or more samples were generated and the empirical distribution of D was estimated.\"}]",1782939829,13,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":15,"description":14,"schema_data":33,"social_meta":84,"head_meta":86,"extra_data":88,"updated_unix":27},"on-the-kolmogorov-smirnov-test-for-normality-with-mean-and-variance-unknown",{"@graph":34,"@context":83},[35,52,66],{"@type":36,"itemListElement":37},"BreadcrumbList",[38,42,46,49],{"item":39,"name":40,"@type":41,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":43,"name":44,"@type":41,"position":45},"https://docshare.wps.com/document/","Document",2,{"item":47,"name":12,"@type":41,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":41,"position":51},"https://docshare.wps.com/document/on-the-kolmogorov-smirnov-test-for-normality-with-mean-and-variance-unknown/36541/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":60,"encodingFormat":59,"isAccessibleForFree":61,"interactionStatistic":62},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":39,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-01",true,{"@type":63,"interactionType":64,"userInteractionCount":4},"InteractionCounter",{"@type":65},"ViewAction",{"@type":67,"mainEntity":68},"FAQPage",[69,75,79],{"name":70,"@type":71,"acceptedAnswer":72},"Why are the standard Kolmogorov-Smirnov critical tables not valid when mean and variance are estimated from the sample?","Question",{"text":73,"@type":74},"Because when distribution parameters are estimated from the data, the usual Kolmogorov-Smirnov null distribution used by standard tables no longer applies, so the tabulated critical points become invalid.","Answer",{"name":76,"@type":71,"acceptedAnswer":77},"What decision rule does the article propose for the normality test with unknown mean and variance?",{"text":78,"@type":74},"Compute D=maxx|F*(X)−Sn(X)|, where Sn is the sample cumulative distribution and F* is the normal CDF using the sample mean and sample variance; reject normality if D exceeds the Monte Carlo critical value from the provided table.",{"name":80,"@type":71,"acceptedAnswer":81},"How were the critical values in the table obtained?",{"text":82,"@type":74},"They were derived by Monte Carlo simulation: for each sample size N, 1,000 or more samples were generated and the empirical distribution of D was estimated.","https://schema.org",{"og:url":50,"og:type":85,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":87,"canonical":50},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":90},[91,95,99,103,107,112,117,120,125,128,132],{"id":20,"doc_module":4,"doc_module_name":44,"category_name":92,"show_sort_weight":93,"slug":94},"Story & Novel",90,"story-novel",{"id":45,"doc_module":4,"doc_module_name":44,"category_name":96,"show_sort_weight":97,"slug":98},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":44,"category_name":100,"show_sort_weight":101,"slug":102},"Exam",70,"exam",{"id":21,"doc_module":4,"doc_module_name":44,"category_name":104,"show_sort_weight":105,"slug":106},"Comic",60,"comic",{"id":108,"doc_module":4,"doc_module_name":44,"category_name":109,"show_sort_weight":110,"slug":111},6,"Technology",50,"technology",{"id":113,"doc_module":4,"doc_module_name":44,"category_name":114,"show_sort_weight":115,"slug":116},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":44,"category_name":12,"show_sort_weight":118,"slug":119},30,"research-report",{"id":121,"doc_module":4,"doc_module_name":44,"category_name":122,"show_sort_weight":123,"slug":124},9,"Religion & Spirituality",20,"religion-spirituality",{"id":123,"doc_module":4,"doc_module_name":44,"category_name":126,"show_sort_weight":123,"slug":127},"World Cup","world-cup",{"id":129,"doc_module":4,"doc_module_name":44,"category_name":130,"show_sort_weight":129,"slug":131},10,"Lifestyle","lifestyle",{"id":133,"doc_module":4,"doc_module_name":44,"category_name":134,"show_sort_weight":21,"slug":135},19,"General","general"]