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Hack 22. Avoid the Axis of Evil

Hack 22. Avoid the Axis of Evil

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mislead.

Typicalgraphshavetwoaxes,becausetheydescribetwo

differentvariables.Axesarethelinesalongthebottom,called

theX-axis,andalongtheside,calledtheY-axis.



YoucanrememberthattheverticalaxisiscalledtheY-axisbecausethe

cutelittleletterYisreachingitscutelittlehandsup,vertically,toward

thesky.Getit?(Welcometothecreativeworldofstatisticseducation.)



Thesortofgraphthatisappropriate(andnondeceptive)for

showingthevariablesyouhavemeasureddependsonthelevel

ofmeasurementofyourvariables[Hack#7].Youcanchoose

fromthreecommontypesofgraphs,andonlyonewillbethe

rightoneforyourvariables:



Barchart

InFigure2-8,theX-axisrepresentscategoriesorgroups,

suchasmalesandfemales.TheY-axisiscontinuous:the

tallerthebars,thehigherthevalueonvariableY.



Figure2-8.Barchart



Histogram

InFigure2-9,theX-axisrepresentscontinuousvalues.A

histogramisoftenusedwhentheX-axisrepresents

commoncategoriesthatreflectanunderlyingcontinuous

variable,suchasmonthsoftheyearorsomeother

distinctivesetofgroupingsthatcanbeplacedina

meaningfulorder.Theselooklikebarcharts,exceptthatthe

barsarepushedtogetherwithnospacesbetweenthem.



Figure2-9.Histogram



Linechart

InFigure2-10,boththeX-andY-axisarecontinuous

variables;inthisexample,they'retimeandvalue.The

higherthelineatanypoint,thegreaterthequantityas

representedbytheY-axis.



Figure2-10.Linechart



Topicktherightkindofgraph(i.e.,theonewiththeformat

thatistheleastdeceptiveandthemostintuitive),identifythe

typesofXvariableyouareusing(noticethatYiscontinuousin

alloftheseformats):

IfXrepresentsdifferentcategoriesandYiscontinuous,use

abarchart.

IfXcanbeconceivedofascategories,butthereisalso

somemeaningfulorderamongthemandYiscontinuous,



useahistogram.

IfXandYarebothcontinuous,usealinechart.



GraphicViolence

Acommonerroringraphing,eitherintentionalornot,hastodo

withsettingthescalefortheX-axis.Here'swhythisisa

problemandhowyoucanavoidit.

Graphswithtwovariablesinvitecomparisonsacrosscategories

ortimeoracrossdifferentvaluesofonevariable.Picturesare

worthathousandwords,astheysay,andagraphcanbevery

persuasiveevidence.Anytimelinesorbarsareusedtocompare

values,thecomparisonisaccurateonlywhentheheightofthe

lineorthelengthofthebarisjudgedagainstsomestandard

minimumvalue.Thatminimumvalueisoftenzero.Ifthegraph

isnotcalibratedtosomereasonablebasevalue,small

differenceslookhuge.

ComparethetwographsshowninFigure2-11,forexample.

Bothconveyexactlythesamedata,andyetyourinterpretation

ofeachmightbewildlydifferent.Thehistograminthetopleft

reflectsperformanceoftheU.S.stockmarketoverthelastfive

days.Noticearatherfrightening-lookingdropondayfive.No

doubt,earth-shakingnewshitneartheendofday4.Youmight

alsonoticethattheY-axis(theDowJonesIndex)doesnot

beginatzero;itbeginsat9,900,avaluethatislowenoughto

containthetopofallfivebars,butthatisotherwisenot

meaningful.



Figure2-11.ThepoweroftheY-axis



LookmorecloselyatthesecondhistograminFigure2-11,on

thebottomright.Bothchartspresentthesamedata,butthe

secondgraphuseszeroasthestartingpoint.Theinterpretation

ofthedataaspresentedinthisgraphshowsverylittle

fluctuationacrossthelastfivedays,andthefrighteningdropat

day5isbarelyahiccup.

Whichdisplayisthecorrectone?Bothreflectadropof2.8

percentinstockmarketvaluefromday4today5.Itreally

dependsontheintentofthegraphconstructorandthe

intendedaudience.Whennumbercountsareinvolved,or

money,themostmeaningfulandfaireststartingpointisusually

nothing.Manynewspapersprovidedailystockinformationin

theformatasshowninthefirsthistogram.Theybelievetheir

readersareinterestedinsmallchanges,sotheysetaY-axis

startingvaluethatisashighaspossiblebutlowenoughto

containalldatapointsontheX-axis.

Afterall,toanavidinvestorwhochangesherportfoliooften

andbuysandsellsfrequently,adropof2.8percentisserious

business.Agraphdesignedtomakesmallchangeslookserious

mightbethemostvalidforthatreader.Ifaninvestorisoneof

those"initforthelonghaul"types,arelativelysmallchangeis



meaningless,however.

Togetthemostmeaningoutofgraphslikethese,alwayscheck

thebottomvalueontheY-axis.Thisway,youcangetasense

oftherealdifferencesontheX-axisasyoucrawlfrombarto

bar.Ifyouaremakinggraphslikethese,thinkaboutthemost

honestwaytopresenttheinformation.Youwanttoinform,not

deceive(probably).



SeeAlso

Thebookthatfirstpointedouttothegeneralpublichow

chartscandeceive,especiallyinadvertising,wasHowtoLie

WithStatistics.Huff,D.(1954).NewYork:Nortonand

Company.







Chapter3.MeasuringtheWorld

Hacks23-34

Thereisgreatvalueinunderstandingphenomenabyhanginga

quantityonit.Thoughsometimesasomethingimportantislost

inthetranslationfromideatonumber,creatingscoresto

representwhateverweareinterestedindoesallowforalevelof

precisioninunderstanding,anditalsoallowsforcomparison.

Thesehacksallinvolvemeasurementandinterpretationof

scores.

Awholefamilyofhacksreliesonthenormaldistribution[Hack

#23]anditspresenceeverywherewelook.Withthenormal

curve,youcantellwhereyoustandcomparedtoeveryoneelse

[Hack#24],knowhowyouarelikelytoperformonatest

beforeyoueventakeit[Hack#25],andunderstandyourtest

resultsatadeeperlevel[Hacks#26and#27].

Speakingoftesting,you'lllearnhowtoproduceagoodsetof

questions[Hack#28]andmakeaqualitytest[Hacks#31and

#32].Youcanidentifybaditems,worthlessquestions,anddo

wellonatestwithoutknowingtheanswers[Hack#29].You

canalsoimproveyourtestperformancewithoutcrackinga

singlebook[Hack#30].

Finally,bylearningacoupleofsolidmeasurementprinciples,

youcandeterminethelifespanofanera,person,orbusiness

[Hack#33]andalsolearnhowtousemedicalinformation

[Hack#34]tomaybeincreaseyourownlifespan.

Measurebymeasure,hereisawholechapterfullof

measurementhacks.







Hack23.SeetheShapeofEverything



Almosteverythinginthenaturalworldisdistributedin

thesameway.Aslongasyoucanmeasurethething,

whateveritis,andscoresareallowedtovary,ithasa

well-defined"normaldistribution."Ifyouknowthe

specificsabouttheshapeofthisnormalcurve,youcan

makeveryaccuratepredictionsaboutperformance.

Thereareafewmiraclesintheworldofstatistics.Thereareat

leastthreetoolsthreediscoveriesthataresocoolandmagical

thatoncestudentsofstatisticslearnaboutthemandbeginto

comprehendtheirbeauty,theyfrequentlyexplode.

Well,maybeIamexaggeratingabit,butherearethreedandy

toolsforunderstandingtheworld:

Thecorrelationcoefficient[Hack#11]

TheCentralLimitTheorem[Hack#2]

Thenormalcurve

Sincewe'vediscussedtheusesofthefirsttwomiraclesinother

hacks,let'sspendourtimenowgettingtoknowtheshapeand

usesofthethird:thenormalcurve.Iampleasedtopresentthe

normalcurve,thenormaldistribution,thebell-shapedcurve,

thewholeworld,asshowninFigure3-1.



Figure3-1.Thenormalcurve



ApplyingAreasUndertheNormalCurve

Statisticianshavedefinedthenormalcurveveryspecifically.

Usingbothcalculusandhundredsofyearsofreal-worlddata

collection,thetwomethodshavereachedthesamesetof

conclusionsabouttheexactshapeofthenormaldistribution.

Figure3-2showstheimportantcharacteristicsofthenormal

curve.Themeanisinthemiddle,andthereisroomforfewer

andfewerscoresasyoumoveawayfromthatcenter.



Figure3-2.Areasunderthenormalcurve



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Hack 22. Avoid the Axis of Evil

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