[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-311373-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-311373-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","ap-statistics-2021-free-response-questions-formulas-for-ap-statistics-descriptive-statistics","AP® Statistics 2021 Free-Response Questions - Formulas for AP Statistics - Descriptive Statistics","","AP Statistics 2021 Free-Response Questions provide reference formulas and problem prompts for descriptive statistics, probability and distributions, and inferential methods. The document includes key relationships for mean, variance, conditional probability, common distributions (including binomial and geometric), and sampling distributions for proportions, means, and simple linear regression. It also contains exam instructions for solving questions with clear methods and justification, plus a detailed Question 1 scenario involving hospital length of stay with a dotplot and outlier analysis using the 1.5 IQR rule and the 2 standard deviations rule.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/ap-statistics-2021-free-response-questions-formulas-for-ap-statistics-descriptive-statistics/311373/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/ap-statistics-2021-free-response-questions-formulas-for-ap-statistics-descriptive-statistics/311373.png","ImageObject",300,407,{"name":42,"@type":43},"Arica Lee","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-04","2026-09-20",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What descriptive statistics and formulas are included for AP Statistics?","Question",{"text":63,"@type":64},"The document lists core descriptive statistics relationships involving x-bar and variance/standard deviation concepts used throughout AP Statistics.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Which outlier identification methods are used in Question 1, and how are they defined?",{"text":68,"@type":64},"Method A uses the 1.5 × IQR rule, and Method B uses the 2 standard deviations rule to flag potential outliers in the length-of-stay data.",{"name":70,"@type":61,"acceptedAnswer":71},"How does the document define confidence intervals in inferential statistics?",{"text":72,"@type":64},"A confidence interval is given in the form statistic ± (critical value)(standard error of statistic).","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},311373,1790203092,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145,"preview_page_count":4},8796096645457,"https://ap-avatar.wpscdn.com/avatar/800003749518d68ffe3?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345340919836971","2021  \nAP® Statistics  \nFree-Response Questions  \n© 2021 College Board. College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered trademarks of College Board. Visit College Board on the web: [collegeboard.org](collegeboard.org).  \n[AP Central is the official online home for the AP Program: apcentral.collegeboard.org](AP Central is the official online home for the AP Program: apcentral.collegeboard.org).  \nFormulas for AP Statistics  \nI. Descriptive Statistics  \nx~~ ~~ = 1n~~ ~~ ∑ xi = ∑nxi~~ ~~  \n = a + bx  \nr = n~~ ~~~~ ~~1 ∑ 􀂩􀂨􀂧 ~~ ~~xi~~ ~~s−x~~ ~~x~~  ~~ 􀂷􀂹􀂸 􀂩􀂨􀂧 ~~ ~~yi~~ ~~s−y~~ ~~y~~ ~~􀂷􀂹􀂸 II. Probability and Distributions  \nP ( A ∪ B) = P ( A) + P( B ) − P( A ∩ B)  \nsx = ~~ ~~n~~ ~~~~ ~~1 ∑ ( xi − x~~ ~~ )2 = y = a + bx  \nb = r ~~ ~~ssy~~ ~~  \nx  \nP ( A | B ) = P~~ ~~(AP~~(~~∩B~~)~~B~~ ~~)  \n| Probability Distribution | Mean | Standard Deviation |\n| --- | --- | --- |\n| Discrete random variable, X | µX = E ( X ) = ∑ xiP ( xi ) | σX = ∑ ( xi − µX )2 P ( xi ) |\n| If 􀜺 has a binomial distribution with parameters n and p, then:\u003Cbr>P ( X = x ) = 􀂧􀂩􀂨 nx􀂷􀂹􀂸 px ( 1 − p )n − x where x = 0, 1, 2, 3, 􀀡 , n | µX = np | σX = np ( 1 − p ) |\n| If 􀜺 has a geometric distribution with parameter p, then:\u003Cbr>P ( X = x ) = ( 1 − p )x −1 p where x = 1, 2, 3, 􀀡 | µX = 1p |  1 − p σ =\u003Cbr>X p |\n\nIII. Sampling Distributions and Inferential Statistics  \n\n| Standardized test statistic: ~~ ~~stanstatidardstic~~ ~~−~~ ~~paerror~~ ~~ofrtameterhe~~ ~~statistic |\n| --- |\n| Confidence interval: statistic ± ( critical value )( standard error of statistic) |\n\nChi-square statistic: χ 2 = ∑ ~~ ~~(~~ ~~observedexp−~~ ~~expectedected~~ ~~)2~~ ~~  \nIII. Sampling Distributions and Inferential Statistics (continued) Sampling distributions for proportions:  \n\n| Random\u003Cbr>Variable | Parameters of\u003Cbr>Sampling Distribution |  |  |  | Standard Error* of Sample Statistic |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| For one population:\u003Cbr>ˆ\u003Cbr>p | µ = p | σ = |  | | s = |  | |\n| For two populations:\u003Cbr>1 − 2 | µ1 − 2 = p1 − p2 | σ1 − 2 = | \u003Cbr>\u003Cbr>p1 ( 1 − p1 ) p2 ( 1 − p2 )+\u003Cbr>n n\u003Cbr>1 2 |  | s1 − 2 = | |  |\n|  |  |  |  |  | When p1 = p2 is assumed:\u003Cbr>s1 − 2 = c ( 1 − c ) 􀂧􀂩􀂨 1n1 + ~~ ~~1n2~~ ~~􀂷􀂹􀂸 where c = Xn11~~ ~~~~ ~~Xn22 |  |  |\n\nSampling distributions for means:  \n\n| Random\u003Cbr>Variable | Parameters of Sampling Distribution |  |  | Standard Error* of Sample Statistic |  |\n| --- | --- | --- | --- | --- | --- |\n| For one population:\u003Cbr>X | µX = µ | σ   =  σ \u003Cbr>X\u003Cbr>~~ ~~ n\u003Cbr>|  | s   =  s  X\u003Cbr>~~ ~~ n\u003Cbr>|  |\n| For two populations:\u003Cbr>X − X\u003Cbr>1 2 | µX − X = µ1 − µ2 1 2 | σ ~~ ~~ ~~ ~~ = X − X\u003Cbr>1 2 | | s ~~ ~~ ~~ ~~ = X − X\u003Cbr>1 2 | |\n\nSampling distributions for simple linear regression:  \n\n| Random\u003Cbr>Variable | Parameters of Sampling Distribution |  | Standard Error* of Sample Statistic |  |\n| --- | --- | --- | --- | --- |\n| For slope: b | µb = β | σb = σ~~ ~~σ~~ ~~n~~  ~~ , x\u003Cbr>where σ =\u003Cbr>x\u003Cbr>\u003Cbr>| \u003Cbr>s\u003Cbr>sb = ,\u003Cbr>s n − 1\u003Cbr>x\u003Cbr>where s =\u003Cbr>∑ ( yi − i )2 n − 2 |  |\n|  |  |  | and s = x | |\n\n*Standard deviation is a measurement of variability from the theoretical population. Standard error is the estimate of the standard deviation. If the standard deviation of the statistic is assumed to be known, then the standard deviation should be used instead of the standard error.  \nAP® Statistics 2021 Free-Response Questions  \nBegin your response to QUESTION 1 on this page.  \nSTATISTICS  \nSECTION II  \nTotal Time—1 hour and 30 minutes  \n6 Questions  \nPart A  \nQuestions 1-5  \nSpend about 1 hour and 5 minutes on this part ofthe exam.  \nDirections: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations.  \n1. The length of stay in a hospital after receiving a particular treatment is of interest to the patient, the hospital, and insurance providers. Of particular interest are unusua","cbCaiddQyxBVtrKS","https://ap.wps.com/l/cbCaiddQyxBVtrKS","pdf",4508878,22,"English","# Formulas for AP Statistics\n## I. Descriptive Statistics\n## II. Probability and Distributions\n## III. Sampling Distributions and Inferential Statistics\n## Sampling distributions for proportions\n## Sampling distributions for means\n## Sampling distributions for simple linear regression\n# AP® Statistics 2021 Free-Response Questions\n## Question 1: Hospital length of stay","[{\"question\":\"What descriptive statistics and formulas are included for AP Statistics?\",\"answer\":\"The document lists core descriptive statistics relationships involving x-bar and variance/standard deviation concepts used throughout AP Statistics.\"},{\"question\":\"Which outlier identification methods are used in Question 1, and how are they defined?\",\"answer\":\"Method A uses the 1.5 × IQR rule, and Method B uses the 2 standard deviations rule to flag potential outliers in the length-of-stay data.\"},{\"question\":\"How does the document define confidence intervals in inferential statistics?\",\"answer\":\"A confidence interval is given in the form statistic ± (critical value)(standard error of statistic).\"}]","AP® Statistics 2021 Free-Response Questions - Formulas for AP Statistics - Descriptive Statistics | PDF",1789894210,55]