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LABORATORY STATISTICS Handbook of Formulas and Terms LABORATORY STATISTICS Handbook of Formulas and Terms by A K NDERS ALLNER Assoc.Professor(R) KarolinskaUniv.Laboratories StockholmSWEDEN AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON (cid:129) NEW YORK (cid:129) OXFORD PARIS (cid:129) SAN DIEGO (cid:129) SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO Elsevier 225WymanStreet,Waltham,MA02451,USA 525BStreet,Suite1800,SanDiego,CA92101-4495,USA Firstedition Copyright©2014.ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproduced,storedinaretrievalsystem ortransmittedinanyformorbyanymeanselectronic,mechanical, photocopying,recordingorotherwisewithoutthepriorwrittenpermission ofthepublisher. PermissionsmaybesoughtdirectlyfromElsevier’sScience&Technology RightsDepartmentinOxford,UK:phone(þ44)(0)1865843830;fax(þ44) (0)1865853333;email:[email protected] submityourrequestonlinebyvisitingtheElsevierwebsiteat http://elsevier.com/locate/permissions,andselectingObtaining permissiontouseElseviermaterial. Notice Noresponsibilityisassumedbythepublisherforanyinjuryand/ordamage topersonsorpropertyasamatterofproductsliability,negligenceor otherwise,orfromanyuseoroperationofanymethods,products, instructionsorideascontainedinthematerialherein.Becauseofrapid advancesinthemedicalsciences,inparticular,independentverificationof diagnosesanddrugdosagesshouldbemade. LibraryofCongressCataloging-in-PublicationData Kallner,Anders. Laboratorystatistics:handbookofformulasandterms/by AndersKallner.–Firstedition. pagescm Includesbibliographicalreferencesandindex. ISBN978-0-12-416971-5 1.Mathematicalnotation.2.Mathematics–Terminology.I.Title. QA41.K322014 519.503–dc23 2013019706 BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ForinformationonallElsevierpublications visitourwebsiteatstore.elsevier.com PrintedandboundinUSA 14 15 16 17 18 10 9 8 7 6 5 4 3 2 1 ISBN:978-0-12-416971-5 Acknowledgment Extracts from the International Vocabulary of Metrology— BasicandGeneralConceptsandAssociatedTerms(VIM),3rdedi- tion, JCGM 200:2012 are published with permission of the DirectoroftheInternationalBureauofWeightsandMeasures (BIPM),inhisfunctionsasChairmanoftheJCGM.Themem- ber organizations of the JCGM also retain full internationally protected right on their titles, slogans, and logos included in theJCGM’spublications.TheJCGMdoesnotacceptanyliabil- ity for the relevance, accuracy, completeness, or quality of reproduced information and materials. The only official version is the original version of the document published by the JCGM. vii Introduction Whenwehavetoadd,ormultiply,evenbignumberseverything goesalmostmechanically.Thisisaroutinework,...,thetruemath- ematicalthinkingbeginswhenonehastosolvearealproblem,thatis to say, to identify a mathematical structure that would match the conditionsoftheproblem,tounderstandprinciplesofitsfunctioning, to grasp connections with other mathematical structures, and to deduce the consequences implied by the logic of the problem. Such manipulationsofstructuresarealwaysimmersedintovariouscalcu- lations since calculations form a natural language of mathematical structures. Michel Heller (2008) This present “compendium” is for those who like me are engagedinpracticallaboratoryworkanddonothaveamajor in statistical analysis and feel somewhat uncomfortable with the statistical jargon. We frequently face the need to analyze largeamountsofdataofvariousorigins,collectedforvarious purposesinroutineorresearchwork,andhavediscoveredthe power of spreadsheet programs in calculations and general data analysis. Commercial statistical “packages” provide many of the analysisusedinthelaboratory.Bynecessity,theorganization of the data in these packages has to accommodate many dif- ferentrequirementsandisperhapsnotoptimalforaparticular practical purpose. Laboratorians often desire to visualize their results graphically and interactively. The availability of spreadsheet programs has eliminated much of problems and hassle with calculations in statistics, provided simple under- standable formulas are available. Indeed, simple spreadsheet programming can satisfy most of the necessary calculations and offer simple, efficient, and customized solutions. This present compendium is not meant to be a “short course”instatisticsbutasourceofaquickreference,repetition orexplanationofformulasandconcepts,andencouragedevel- opmentofstatisticaltoolsandroutinesintheresearchandrou- tine laboratories. ix x INTRODUCTION Specialattentionhasbeengiventoexpressionsthatcantake differentformatsbut,ofcourse,givethesameresults.Exposing formulasin differentformats maytosomeextentexplaintheir origin, relation to other procedures, and their usage. We have tried to align formulas regarding style and terminology and group them in a logical order. Some formulas in the collection havebeeneditedtofacilitateapplyinginspreadsheetprograms. Theselectionofformulasinthecompendiumhasdeveloped during several courses in applied statistics for laboratorians and scientists with experimental projects. The number of worked examples is extensive and regularly enhanced by tablesandfigures.Wheneverfeasiblethetextmakesreference ® to functions and routines in Microsoft EXCEL . Formulashavebeencollectedandcomparedfrommanydif- ferentsources,scientificliterature,commontextbooks,andthe Internet. It is all out there, cast in different forms and shapes but may be difficult to find. An idea with this compendium istohavemostofthestatisticalproceduresusedinthelabora- tory collected in one source and described in a standardized but not compressed format. References to individual sources are not given but a list of contemporary literature. A threat with preprogrammed routines is that, unless simple rules are violated and thus prevented from use, they will always produce an answer. The process of programming and calculating statistical routines has proved to deepen the understandingoftheproceduresandhopefullydiminisherro- neous use of established procedures. However, the author takes no responsibility for any erroneous decisions based on calculations using formulas in this compendium. Acomprehensivelistofcontentsandanindexfacilitatethe access of the desired concept or procedure. VOCABULARY AND CONCEPTS IN METROLOGY Many organizations have invested heavily in formulating internationally acceptable, clear, comprehensive, and under- standabledefinitionsoftermsinmetrology.Superficially,this xi INTRODUCTION maynot seem to have any bearing onstatistics. Basically, sta- tisticsisonewayofformulatingandexpressingmathematical relationships,butwealsoneedtoagreeonandusedefinitions of common concepts. The most extensive and internationally recognizedlistofconceptsandtheirdefinitionsisthatcreated bythejointBIPM,ISO,IEC,IFCC,IUPAC,IUPAP,OIML,and ILAC document International Vocabulary of Metrology—Basic andGeneralConceptsand AssociatedTerms(VIM),downloadable at http://www.bipm.org/ (accessed 2013-06-30). ThedefinitionsarereproducedinextensofromtheVIM,but somenoteshavebeendeletedwhenpertainingtopuremetro- logical problems. Theauthorisgratefulfortheinterestandmanyexcellentsug- gestions from students and other users of previous editions of the compendium. In particular, Professor Elvar Theodorsson, Department of Clinical Chemistry, University of Linko¨ping, Sweden, has provided healthy critics. Anders Kallner ([email protected]) Some Notes on Nomenclature Mathematicalformulasmaybedifficulttodecipherbutare in fact unambiguous and comprehensive. Inthiscompendium,theformulasarenotascompressedas they may be and therefore easier to understand. A few rules may help: The number of items is abbreviated n or N. > < (cid:1) isread“largerthan,” “smallerthan,” “largerthanor (cid:3) equal to,” “smaller than or equal to.” (cid:4) (cid:5) is read “much larger than,” “much smaller than.” (cid:6) Fractions (division), a/b; multiplication a b. (cid:6) Multiplications in the body of the text are written , i.e., (cid:6) a b. pffiffiffi pffiffiffi Square root: a or explicit 2 a, which allows for higher order roots. X þ þ þ(cid:7)(cid:7)(cid:7)þ n Sum: a a a a is abbreviated: a. 1 2 3 n i¼1 i XSumofsquares:(a1)2þ(a2)2þ(a3)2þ(cid:7)(cid:7)(cid:7)þ(an)2isabbreviated n a2. (cid:3)X (cid:4) Ai¼1sqiuared sum (a þa þa þ(cid:7)(cid:7)(cid:7)þa )2 is n a 2. 1 2 3 n i¼1 i Absolute value: |a|, i.e., disregarding any sign. Standarddeviationofasamplexiss(x),s(X),orsifthereis no risk for misunderstanding. ð(cid:1)Þ C(cid:5)on(cid:6)sequently, thestandarderrorofthemean(SEM) iss x (cid:1) or s X , as appropriate. The abbreviation SEM is also used. The period (full stop) “.” is used as the decimal sign and a comma “,” as the 1000 separator. Additional abbreviations as appropriate are explained in the text. xiii xiv SOMENOTESONNOMENCLATURE SomeGreeklettersusedforcertainpurposes(smallandcapital): Alpha:aandΑ,Beta:bandΒ,Gamma:gandG,Delta:dandD, Epsilon:eandΕ,Zeta:zandΖ,Eta:ZandΗ,Kappa:kandΚ, Lambda:landL,My:mandΜ,Xi;xandX,Pi:pandP,Rho:r andΡ,Sigma:sandS,Tau:tandΤ,Chi:wandΧ. Formulas BASICS Logarithms and Exponents The logarithm of a given number and a given base is the power to which the base must be raised to get the number. Ifbisthebaseandathegivennumber,thelogarithmisx.In “ ” many applications, the notation log refers to 10-logarithms (Briggs), i.e., the base 10 and ln refers to e-logarithms or “ ” ¼ natural logarithms with e 2.7183 as the base: If blogðaÞ ¼ x; thenantilogðxÞ ¼ a ¼ bx ð1Þ thusifelogðaÞ ¼ x; lnðaÞ ¼ x; thenantilnðxÞ ¼ a ¼ ex ð2Þ and if10logðaÞ ¼ x; thenantilogðxÞ ¼ a ¼ 10x ð3Þ (cid:2) ¼ ; ð Þþ ð Þ ¼ ð Þ; a b c log a log b log c a ¼ c; logðaÞ(cid:3) logðbÞ ¼ logðcÞ ð4Þ b (cid:1) (cid:3) log ab ¼ b(cid:2) logðaÞ ð5Þ clogðaÞ ¼ blogðaÞ ð6Þ clogðbÞ LaboratoryStatistics 1 ©2014ElsevierInc.Allrightsreserved.