[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571218-105":53,"doc-detail-571218-en":126},{"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":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","list-of-formulae-and-tables-of-the-normal-distribution","List of Formulæ and Tables of the Normal Distribution","","A mathematics reference compiling essential formulae and statistical tables for advanced-level examinations. It organizes pure mathematics topics including algebra, trigonometry, differentiation, integration, and vectors, then extends to mechanics relationships and probability and statistics results. The document also provides the normal distribution function Φ(z) and a detailed standard normal table for values of z, supporting direct lookups in calculations.",{"@graph":63,"@context":118},[64,80,101],{"@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":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/list-of-formulae-and-tables-of-the-normal-distribution/571218/",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/list-of-formulae-and-tables-of-the-normal-distribution/571218.png","ImageObject",442,249,{"name":88,"@type":89},"BAd apple","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What does the document provide for statistics calculations?","Question",{"text":108,"@type":109},"It compiles key summary-statistics formulas for ungrouped and grouped data, distributions such as binomial and Poisson, and sampling/testing results, plus the standard normal function Φ(z).","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How is the normal distribution table used?",{"text":113,"@type":109},"For each z, the table gives Φ(z)=P(Z≤z). For negative z, it specifies using Φ(−z)=1−Φ(z) to obtain the needed value.",{"name":115,"@type":106,"acceptedAnswer":116},"Which subjects are covered under the pure mathematics section?",{"text":117,"@type":109},"It covers algebra (including series and binomial expansion), trigonometry (identities and principal values), differentiation and integration rules, and vector dot-product relationships.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},571218,1792066571,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"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":76},19241458193282,"https://us-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","List MF9  \nCAMBRIDGE INTERNATIONAL EXAMINATIONS  \nGeneral Certificate of Education Advanced Level  \nGeneral Certificate of Education Advanced Subsidiary Level Advanced International Certificate of Education  \nMATHEMATICS (8709, 9709) HIGHER MATHEMATICS (8719)  \nSTATISTICS (0390)  \nLIST OF FORMULAE  \nAND  \nTABLES OF THE NORMAL DISTRIBUTION  \nPURE MATHEMATICS  \nAlgebra  \nFor the quadratic equation ax2 + bx + c = 0 :  \n− b ± √ (b2 − 4ac) x = ~~ ~~  \n2a  \nFor an arithmetic series:  \nun = a + (n −1)d , Sn = 12 n(a + l) = 12 n{2a + (n −1)d } For a geometric series:  \nun = arn−1 , Sn = a(11−~~ ~~rrn~~ ~~) (r ≠ 1) , S∞ = 1~~ ~~~~ ~~r~~ ~~ (  r  \u003C 1)  \nBinomial expansion:  \n(a + b) n = an + n1 􀀍􀀌􀀋an−1b + n2 􀀍􀀌􀀋an−2b2 + 􀀐􀀏􀀎 n3􀀍􀀌􀀋an−3b3 + 􀀕 + bn , where n is a positive integer and 􀀎nr 􀀋 = r!~~(~~nn−!~~ ~~r~~)!~~​  \n(1 + x) n = 1 + nx + n(n2−!~~ ~~1) x2 + n(n~~ ~~−~~ ~~13)!(n~~ ~~−~~ ~~2) x3 + 􀀕 , where n is rational and  x  \u003C 1 Trigonometry  \nArc length of circle = rθ (θ in radians)  \nArea of sector of circle = 12 r2θ (θ in radians) tan θ ≡ scθ~~θ~~  \ncos2 θ + sin2 θ ≡ 1 , 1 + tan2 θ ≡ sec2 θ , cot2 θ + 1 ≡ cosec2 θ  \nsin(A ± B) ≡ sinAcosB ± cosAsin B  \ncos(A ± B) ≡ cosAcos B \\# sinAsinB  \ntan(A ± B) ≡ 1ta\\#ntAantantanBB  \nsin 2A ≡ 2 sinAcos Acos 2A ≡ cos2 A − sin2 A ≡ 2 cos2 A − 1 ≡ 1 − 2sin2 Atan 2A ≡  2 tanA   \n1 − tan2 A  \nPrincipal values:  \n− 12 π ≤ sin−1 x ≤ 12 π  \n0 ≤ cos−1 x ≤ π  \n− 12 π \u003C tan−1 x \u003C 12 π  \nDifferentiation  \nf(x)  \nnx  \nln x  \nex sin x cos x  \ntan x  \nuv  \nu  \nv  \nf′(x)  \nn−1 nx  \n1  \nx  \nex  \ncos x  \n− sin x  \n2 sec x  \ndu + u  dv v dx dx du dv v dx − u dx  \n2 v  \nIf x = f(t) and y = g(t) then dy~~d~~x = dy~~d~~t ÷ d~~d~~xt  \nIntegration  \nf(x)  \nnx  \n1 x ex  \nsin x cos x  \n2 sec x  \n⌡􀀘⌠ u dv~~d~~x dx = uv − ⌡􀀘⌠ v ~~ ~~du~~d~~x dx ⌡􀀘⌠ f~~f~~′~~(~~~~)~~) dx = ln  f(x) + c  \n∫ f(x) dx  \nxn11 + c (n ≠ −1) ln  x  + c  \nex + c  \n− cos x + c sin x + c tan x + c  \nVectors  \nIf a = a1i + a2j + a3k and b = b1i + b2j + b3k then  \na.b = a1b1 + a2b2 + a3b3 =  a b  cos θ  \nNumerical integration  \nTrapezium rule:  \n∫abf(x) dx ≈ 12 h {y0 + 2(y1 + y2 + 􀀕 + yn−1) + yn } , where h = b~~ ~~−n~~ ~~a  \nMECHANICS  \nUniformly accelerated motion  \nv = u + at , s = 12 (u + v)t , s = ut + 12 at2 , v2 = u2 + 2as  \nMotion of a projectile  \nEquation of trajectory is:  \ny = x tanθ − 2V2gx2cos2~~ θ~~  \nElastic strings and springs  \nT =  λx  E =  λx2  \nl , 2l  \nMotion in a circle  \nFor uniform circular motion, the acceleration is directed towards the centre and has magnitude  \n2  \nv  \n2  \nω r or  r   \nCentres of mass of uniform bodies  \nTriangular lamina: 23 along median from vertex  \nSolid hemisphere of radius r: 38 r from centre  \nHemispherical shell of radius r: 12 r from centre  \nCircular arc of radius r and angle 2α : r~~ ~~sα from centre  \nCircular sector of radius r and angle 2α : 2rs3α from centre  \nSolid cone or pyramid of height h: 34 h from vertex  \nPROBABILITY AND STATISTICS  \nSummary statistics For ungrouped data:  \nx =  \nFor grouped data:  \nΣx  \nn  \n,  \n Σxf   \nΣf , x =  \nDiscrete random variables  \nstandard deviation = Σ(x − x)2 = Σx2 − x~~ ~~ 2  \nn n  \nstandard deviation = Σ(x − x)2 f = Σx2f − x~~ ~~ 2  \n Σf  Σf  \nE(X) = Σxp  \nVar(X) = Σx2p − {E(X)}2  \nFor the binomial distribution B(n, p) :  \npr = 􀀐􀀏􀀎 nr􀀍􀀌􀀋 pr(1 − p) n−r , µ = np , σ 2 = np(1 − p)  \nFor the Poisson distribution Po(a) :  \nr  \na  \n−a 2  \npr = e  r! , µ = a , σ = a  \nContinuous random variables  \nE(X) = ∫x f(x) dx Var(X) = ∫x2 f(x) dx − {E(X)}2  \nSampling and testing  \nUnbiased estimators:  \nx = Σnx , s2 = n~~ ~~~~ ~~1 Σx2 − (Σxn)2~~ ~~ 􀀍􀀌􀀋 Central Limit Theorem:  \nX ~ N􀀐􀀎µ, σ 2  􀀍􀀋  \n􀀏 n 􀀌  \nApproximate distribution of sample proportion:  \nN􀀎􀀐􀀏p, p(1~~ ~~p) 􀀍􀀌􀀋  \nTHE NORMAL DISTRIBUTION FUNCTION  \nIf Z has a normal distribution with mean 0 and variance 1 then, for each value of z, the table gives the value of Φ(z) , where  \nΦ(z) = P(Z ≤ z) . For negative values of z use Φ(−z) = 1 − Φ(z) .  \n| z | 0 1 2 3 4 5 6 7 8 9 |  |  |  | 1 2 3 4 5 6 7 8 9 ADD |  |  |\n| ---","cbCaipJQlH9VZEUK","https://ap.wps.com/l/cbCaipJQlH9VZEUK","pdf",39496,8,"English","# Pure Mathematics\n## Algebra\n## Trigonometry\n## Differentiation\n## Integration\n## Vectors\n# Mechanics\n## Uniformly accelerated motion\n## Projectile motion\n## Motion in a circle\n## Centres of mass\n# Probability and Statistics\n## Summary statistics\n## Discrete random variables\n## Continuous random variables\n## Sampling and testing\n# The Normal Distribution Function","[{\"question\":\"What does the document provide for statistics calculations?\",\"answer\":\"It compiles key summary-statistics formulas for ungrouped and grouped data, distributions such as binomial and Poisson, and sampling/testing results, plus the standard normal function Φ(z).\"},{\"question\":\"How is the normal distribution table used?\",\"answer\":\"For each z, the table gives Φ(z)=P(Z≤z). For negative z, it specifies using Φ(−z)=1−Φ(z) to obtain the needed value.\"},{\"question\":\"Which subjects are covered under the pure mathematics section?\",\"answer\":\"It covers algebra (including series and binomial expansion), trigonometry (identities and principal values), differentiation and integration rules, and vector dot-product relationships.\"}]","List of Formulæ and Tables of the Normal Distribution | PDF",1791904159]