Laboratory Statistics: Handbook of Formulas and Terms

146 Pages·2013·3.259 MB·English

by  Anders Kallner

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
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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
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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 ver-
sion 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 sim-
ple 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.