[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571679-105":53,"doc-detail-571679-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","statistics-cheat-sheet-probability-and-descriptiveinferential-statistics","Statistics Cheat Sheet - Probability and Descriptive/Inferential Statistics","","Statistics Cheat Sheet covering core probability axioms and key properties, linking descriptive methods to inferential reasoning. Defines population, sample, discrete vs continuous random variables, sample mean/median, trimmed mean, variability measures (range, sample variance, standard deviation), and box plot construction. Includes probability foundations: sample spaces and events, probability axioms, addition and inclusion–exclusion formulas, product rule, conditional probability, Bayes’ theorem, and independence. Also summarizes common distributions, pmf conditions, cdf interpretation, and expected value as a weighted average.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/statistics-cheat-sheet-probability-and-descriptiveinferential-statistics/571679/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/statistics-cheat-sheet-probability-and-descriptiveinferential-statistics/571679.png","ImageObject",442,249,{"name":88,"@type":89},"DoDO","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-21","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",6,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What is the difference between population and sample in this cheat sheet?","Question",{"text":109,"@type":110},"A population is a well-defined collection of objects, while a sample is a subset of that population used for analysis.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How are discrete and continuous random variables distinguished?",{"text":114,"@type":110},"A numerical variable is discrete if its set of possible values is at most countable, and continuous if its set of possible values is uncountable.",{"name":116,"@type":107,"acceptedAnswer":117},"What do Bayes’ theorem and conditional probability relate?",{"text":118,"@type":110},"Conditional probability gives the probability of A given that B has occurred, and Bayes’ theorem rewrites probabilities using the partition of the sample space into disjoint exhaustive events.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},571679,1792382271,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":101,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":73},962093270745,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Statistics Cheat Sheet  \nAxioms & Properties of Probability:  \n1. 8A 2 S ; 0 􀀔 P (A) 􀀔 1  \nCh 1: Overview & Descriptive Stats Populations, Samples and Processes  \nPopulation: well-de􀀌ned collection of objects  \nSample: a subset of the population  \nDescriptive Stats: summarize & describe features of data Inferential Stats: generalizing from sample to population Probability: bridge btwn descriptive & inferential techniques. In probability, properties of the population are assumed known & questions regarding a sample taken from the population are posed and answered.  \n2. P (S) = 1  \n3. If A1 ; A2 ; : : : is an in􀀌nite collection of disjoint events, P (A1 [ A2 [ 􀀁 􀀁 􀀁 ) = P P (Ai)  \n4. P (;) = 0  \n5. 8A; P(A) + P(A0 ) = 1 from which P (A) = 1 􀀀 P (A0 )  \n6. For any two events A; B 2 S ;  \nP (A [ B) = P (A) + P (B) 􀀀 P (A \\ B)  \n7. For any three events  \nA; B; C 2 S; P(A [ B [ C) = P (A) + P (B) + P (C) 􀀀 P (A \\ B) 􀀀 P (A \\ C) 􀀀 P (B \\ C) + P (A \\ B \\ C)  \nDiscrete and Continuous Variables: A numerical variable is discrete if its set of possible values is at most countable. A numerical value is continuous if its set of possible values isan uncountable set.  \nProbability: pop ! sample  \nStats: sample ! pop  \nMeasures of Location  \nFor observations x 1; x 2 ; 􀀁 􀀁 􀀁 ; xn Sample Mean  = Pn~~1~~~~1 ~~x~~i~~  \nSample Median  = ( ~~n+~~2~~1~~ )nth observation  \nTrimmed Mean btwn  and , compute by removing smallest and largest observations  \nMeasures of Variability  \nRange  \nSample Variance, 􀀛 2 S xx  \nSample Standard Deviation, 􀀛  \n=lgst-smllst observation = P (xi 􀀀 ))2 = S xx  \nn􀀀1 n􀀀1  \n= Px2i 􀀀 (Pxn~~i~~)2  \n= p􀀛2  \nBox Plots  \nOrder the n observations from small to large. Separate the smallest half from the largest (If n is odd then  is in both halves) . The lower fourth is the median of the smallest half (upper fourth..largest..) . A measure of the spread that is resistant to outliers is the fourth spread fs given by fs = upper fourth-lower fourth. Box from lower to upper fourth with line at median. Whiskers from smallest to largest xi .  \nCh 2: Probability  \nSample Space and Events  \nExperiment Sample Space (S) Event  \nactivity with uncertain outcome the set of all possible outcomes any collection of outcomes in S  \nAxioms, Interpretations and Properties of Probability  \nGiven an experiment and a sample space S, the objective probability is to assign to each event A a number P (A), called the probability of event A, which will give a precise measure of the chance that A will occur. Behaves very much like norm.  \nEqually Likely Outcomes : P (A) = NN(A)  \nCounting Techniques  \nProduct Rule for Ordered k-Tuples: If the 􀀌rst element can be selected in n 1 ways, the second in n 2 ways and so on, then there are n 1n 2 􀀁 􀀁 􀀁 nk possible k-tuples.  \nPermutations: An ordered subset. The number of permutations of size k that can be formed from a set of n elements is Pk;n  \nPk;n = (n)(n 􀀀 1) 􀀁 􀀁 􀀁 (n 􀀀 k + 1) = (n~~!~~​k)!  \nCombinations: An unordered subset.  \n􀀀nk􀀁 = Pkk~~ ~~;!n = k!(~~n~~n􀀀~~! ~~k)!  \nConditional Probability  \nP (Aj B) is the conditional probability of A given that the event B has occurred. B is the conditioning event.  \nP (A \\ B)  \nP (Aj B) =  \nP (B)  \nMultiplication Rule: P (A \\ B) = P (Aj B) 􀀁 P(B)  \nBaye's Theorem  \nLet A1; A 2; : : : ; Ak be disjoint and exhaustive events (that partition the sample space) . Then for any other event BP (B) = P (Bj A1)P (A1) + 􀀁 􀀁 􀀁 + P(B jAk)P (Ak)  \n= Pki=1 P (BjAi)P (Ai)  \nIndependence  \nTwo events A and B are independent if P (Aj B) = P (A) and are dependent otherwise.  \nA and B are independent i􀀋 P (A \\ B) = P (A) 􀀁 P(B) and this can be generalized to the case of n mutually independent events.  \nRandom Variables  \nRandom Variable: any function X : 􀀊 ! R  \nProb Dist .: describes how the probability of 􀀊 is  \ndistributed along the range of X  \nDiscrete rv: rv whose domain is at most countable  \nContinuous rv: rv whose domain is uncountable  \nand where 8c 2 R; P(X = c) = 0  \nBernoulli r","cbCaibMPZatbTrhB","https://ap.wps.com/l/cbCaibMPZatbTrhB","pdf",342724,"English","# Axioms & Properties of Probability\n## Overview & Descriptive Stats\n## Probability and Conditional Probability\n# Discrete and Continuous Variables\n## Measures of Location\n## Measures of Variability\n## Box Plots\n# Probability\n## Sample Space and Events\n## Counting Techniques\n## Independence\n## Random Variables and Distributions\n## CDF and Expected Value","[{\"question\":\"What is the difference between population and sample in this cheat sheet?\",\"answer\":\"A population is a well-defined collection of objects, while a sample is a subset of that population used for analysis.\"},{\"question\":\"How are discrete and continuous random variables distinguished?\",\"answer\":\"A numerical variable is discrete if its set of possible values is at most countable, and continuous if its set of possible values is uncountable.\"},{\"question\":\"What do Bayes’ theorem and conditional probability relate?\",\"answer\":\"Conditional probability gives the probability of A given that B has occurred, and Bayes’ theorem rewrites probabilities using the partition of the sample space into disjoint exhaustive events.\"}]","Statistics Cheat Sheet - Probability and Descriptive/Inferential Statistics | PDF",1791923093]