Histogram

Firstdescribedby

KarlPearson

Purpose Toroughlyassesstheprobabilitydistributionofagivenvariablebydepictingthefrequenciesofobservationsoccurringincertainrangesofvalues

HistogramFromWikipedia,thefreeencyclopedia

Ahistogramisagraphicalrepresentationofthedistributionofdata.Itisanestimateoftheprobabilitydistributionofacontinuousvariable(quantitativevariable)andwasfirstintroducedbyKarlPearson.[1]Toconstructahistogram,thefirststepisto"bin"therangeofvaluesthatis,dividetheentirerangeofvaluesintoaseriesofsmallintervalsandthencounthowmanyvaluesfallintoeachinterval.Arectangleisdrawnwithheightproportionaltothecountandwidthequaltothebinsize,sothatrectanglesabouteachother.Ahistogrammayalsobenormalizeddisplayingrelativefrequencies.Itthenshowstheproportionofcasesthatfallintoeachofseveralcategories,withthesumoftheheightsequaling1.Thebinsareusuallyspecifiedasconsecutive,nonoverlappingintervalsofavariable.Thebins(intervals)mustbeadjacent,andusuallyequalsize.[2]Therectanglesofahistogramaredrawnsothattheytoucheachothertoindicatethattheoriginalvariableiscontinuous.[3]

Histogramsgivearoughsenseofthedensityofthedata,andoftenfordensityestimation:estimatingtheprobabilitydensityfunctionoftheunderlyingvariable.Thetotalareaofahistogramusedforprobabilitydensityisalwaysnormalizedto1.Ifthelengthoftheintervalsonthexaxisareall1,thenahistogramisidenticaltoarelativefrequencyplot.

Ahistogramcanbethoughtofasasimplistickerneldensityestimation,whichusesakerneltosmoothfrequenciesoverthebins.Thisyieldsasmootherprobabilitydensityfunction,whichwillingeneralmoreaccuratelyreflectdistributionoftheunderlyingvariable.Thedensityestimatecouldbeplottedasanalternativetothehistogram,andisusuallydrawnasacurveratherthanasetofboxes.

AvariablebinwidthhistogramwasintroducedbyDenbyandMallows(2009)(http://pubs.research.avayalabs.com/pdfs/ALR2007003paper.pdf).ExamplesofthisaredisplayedonCensusbureaudatabelow.

Anotheralternativeistheaverageshiftedhistogram(http://www.stat.rice.edu/~scottdw/stat550/HW/hw4/c05.pdf)whichisfasttocompute,andgetsasmoothcurveestimateofthedensitywithoutusingkernels.

Thehistogramisoneofthesevenbasictoolsofqualitycontrol.[4]

Histogramsareoftenconfusedwithbarcharts.Abarchartisaplotofcategoricalvariables,andthediscontinuityshouldbeindicatedbyhavinggapsbetweentherectangles.Oftenthisisneglectedwhichmayleadtoabarchartbeingconfusedforahistogram.

Anexamplehistogramoftheheightsof31BlackCherrytrees.

Contents

1Etymology2Examples3Mathematicaldefinition

3.1Cumulativehistogram3.2Numberofbinsandwidth

4Seealso5References6Furtherreading7Externallinks

Etymology

Theetymologyofthewordhistogramisuncertain.SometimesitissaidtobederivedfromtheGreekhistos'anythingsetupright'(asthemastsofaship,thebarofaloom,ortheverticalbarsofahistogram)andgramma'drawing,record,writing'.ItisalsosaidthatKarlPearson,whointroducedthetermin1891,derivedthenamefrom"historicaldiagram".[5]

Examples

Thisisatoyexample

Bin Count3.5 92.5 321.5 1090.5 1800.5 1321.5 342.5 43.5 9

Thelanguageusedtodescribethepatternsinahistogramaresymmetric,skewedleftorright,unimodal,bimodalormultimodal.

Symmetric,unimodal

Skewedright

Skewedleft

Bimodal

Multimodal

Symmetric

Itisagoodideatoplotyourdataonseveraldifferentbinwidthstolearnmoreaboutit.Hereisanexampleontipsgiveninarestaurant.

Tipsusinga$1binwidth,skewedright,unimodal

Tipsusinga10cbinwidth,stillskewedright,multimodalwithmodesat$and50camounts,indicatesrounding,alsosomeoutliers

Hereareacouplemoreexamples.

PricesofhousessoldinAmesin2009,exhibitssomerightskew.

Acesbyplayersinagrandslamtennistournament,facettedbygender.Therearemoreacesinthemensgame.

TheU.S.CensusBureaufoundthattherewere124millionpeoplewhoworkoutsideoftheirhomes.[6]Usingtheirdataonthetimeoccupiedbytraveltowork,Table2belowshowstheabsolutenumberofpeoplewhorespondedwithtraveltimes"atleast30butlessthan35minutes"ishigherthanthenumbersforthecategoriesaboveandbelowit.Thisislikelyduetopeopleroundingtheirreportedjourneytime.Theproblemofreportingvaluesassomewhatarbitrarilyroundednumbersisacommonphenomenonwhencollectingdatafrompeople.

Histogramoftraveltime(towork),US2000census.Areaunderthecurveequalsthetotalnumberofcases.ThisdiagramusesQ/widthfromthetable.

Databyabsolutenumbers

Interval Width Quantity Quantity/width

0 5 4180 836

5 5 13687 2737

10 5 18618 3723

15 5 19634 3926

20 5 17981 3596

25 5 7190 1438

30 5 16369 3273

35 5 3212 642

40 5 4122 824

45 15 9200 613

60 30 6461 215

90 60 3435 57

Thishistogramshowsthenumberofcasesperunitintervalastheheightofeachblock,sothattheareaofeachblockisequaltothenumberofpeopleinthesurveywhofallintoitscategory.Theareaunderthecurverepresentsthetotalnumberofcases(124million).Thistypeofhistogramshowsabsolutenumbers,withQinthousands.

Histogramoftraveltime(towork),US2000census.Areaunderthecurveequals1.ThisdiagramusesQ/total/widthfromthetable.

Databyproportion

Interval Width Quantity(Q) Q/total/width

0 5 4180 0.0067

5 5 13687 0.0221

10 5 18618 0.0300

15 5 19634 0.0316

20 5 17981 0.0290

25 5 7190 0.0116

30 5 16369 0.0264

35 5 3212 0.0052

40 5 4122 0.0066

45 15 9200 0.0049

60 30 6461 0.0017

90 60 3435 0.0005

Thishistogramdiffersfromthefirstonlyintheverticalscale.Theareaofeachblockisthefractionofthetotalthateachcategoryrepresents,andthetotalareaofallthebarsisequalto1(thefractionmeaning"all").Thecurvedisplayedisasimpledensityestimate.Thisversionshowsproportions,andisalsoknownasaunitareahistogram.

Inotherwords,ahistogramrepresentsafrequencydistributionbymeansofrectangleswhosewidthsrepresentclassintervalsandwhoseareasareproportionaltothecorrespondingfrequencies:theheightof

Anordinaryandacumulativehistogramofthesamedata.Thedatashownisarandomsampleof10,000pointsfromanormaldistributionwithameanof0andastandarddeviationof1.

eachistheaveragefrequencydensityfortheinterval.Theintervalsareplacedtogetherinordertoshowthatthedatarepresentedbythehistogram,whileexclusive,isalsocontiguous.(E.g.,inahistogramitispossibletohavetwoconnectingintervalsof10.520.5and20.533.5,butnottwoconnectingintervalsof10.520.5and22.532.5.Emptyintervalsarerepresentedasemptyandnotskipped.)[7]

Mathematicaldefinition

Inamoregeneralmathematicalsense,ahistogramisafunctionmithatcountsthenumberofobservationsthatfallintoeachofthedisjointcategories(knownasbins),whereasthegraphofahistogramismerelyonewaytorepresentahistogram.Thus,ifweletnbethetotalnumberofobservationsandkbethetotalnumberofbins,thehistogrammimeetsthefollowingconditions:

Cumulativehistogram

Acumulativehistogramisamappingthatcountsthecumulativenumberofobservationsinallofthebinsuptothespecifiedbin.Thatis,thecumulativehistogramMiofahistogrammjisdefinedas:

Numberofbinsandwidth

Thereisno"best"numberofbins,anddifferentbinsizescanrevealdifferentfeaturesofthedata.GroupingdataisatleastasoldasGraunt'sworkinthe17thcentury,butnosystematicguidelinesweregiven[8]untilSturges'sworkin1926.[9]

Usingwiderbinswherethedensityislowreducesnoiseduetosamplingrandomnessusingnarrowerbinswherethedensityishigh(sothesignaldrownsthenoise)givesgreaterprecisiontothedensityestimation.Thusvaryingthebinwidthwithinahistogramcanbebeneficial.Nonetheless,equalwidthbinsarewidelyused.

Sometheoreticianshaveattemptedtodetermineanoptimalnumberofbins,butthesemethodsgenerallymakestrongassumptionsabouttheshapeofthedistribution.Dependingontheactualdatadistributionandthegoalsoftheanalysis,differentbinwidthsmaybeappropriate,soexperimentationisusuallyneededtodetermineanappropriatewidth.Thereare,however,varioususefulguidelinesandrulesofthumb.[10]

Thenumberofbinskcanbeassigneddirectlyorcanbecalculatedfromasuggestedbinwidthhas:

Thebracesindicatetheceilingfunction.

Squarerootchoice

whichtakesthesquarerootofthenumberofdatapointsinthesample(usedbyExcelhistogramsandmanyothers).[11]

Sturges'formula

Sturges'formula[9]isderivedfromabinomialdistributionandimplicitlyassumesanapproximatelynormaldistribution.

Itimplicitlybasesthebinsizesontherangeofthedataandcanperformpoorlyifn

WikimediaCommonshasmediarelatedtoHistograms.

where isthesamplestandarddeviation.Scott'snormalreferencerule[14]isoptimalforrandomsamplesofnormallydistributeddata,inthesensethatitminimizestheintegratedmeansquarederrorofthedensityestimate.[8]

FreedmanDiaconis'choice

TheFreedmanDiaconisruleis:[15][8]

whichisbasedontheinterquartilerange,denotedbyIQR.Itreplaces3.5ofScott'srulewith2IQR,whichislesssensitivethanthestandarddeviationtooutliersindata.

ChoicebasedonminimizationofanestimatedL2[16]riskfunction

where and aremeanandbiasedvarianceofahistogramwithbinwidth , and.

Remark

Agoodreasonwhythenumberofbinsshouldbeproportionalto isthefollowing:supposethatthedataareobtainedas independentrealizationsofaboundedprobabilitydistributionwithsmoothdensity.Thenthehistogramremainsequallyruggedas tendstoinfinity.If isthewidthofthedistribution(e.g.,thestandarddeviationortheinterquartilerange),thenthenumberofunitsinabin(thefrequency)isoforder andtherelativestandarderrorisoforder .Comparingtothenextbin,the

relativechangeofthefrequencyisoforder providedthatthederivativeofthedensityisnonzero.Thesetwoareofthesameorderif isoforder ,sothat isoforder .

Thissimplecubicrootchoicecanalsobeappliedtobinswithnonconstantwidth.

Seealso

DatabinningDensityestimation

Kerneldensityestimation,asmootherbutmorecomplexmethodofdensityestimation

FreedmanDiaconisrule

ImagehistogramParetochartSevenBasicToolsofQualityVoptimalhistograms

References

1. ^Pearson,K.(1895)."ContributionstotheMathematicalTheoryofEvolution.II.SkewVariationinHomogeneousMaterial".PhilosophicalTransactionsoftheRoyalSocietyA:Mathematical,PhysicalandEngineeringSciences186:343414.Bibcode:1895RSPTA.186..343P(http://adsabs.harvard.edu/abs/1895RSPTA.186..343P).doi:10.1098/rsta.1895.0010(http://dx.doi.org/10.1098%2Frsta.1895.0010).

2. ^Howitt,D.andCramer,D.(2008)StatisticsinPsychology.PrenticeHall3. ^CharlesStangor(2011)"ResearchMethodsForTheBehavioralSciences".Wadsworth,CengageLearning.

ISBN9780840031976.4. ^NancyR.Tague(2004)."SevenBasicQualityTools"(http://www.asq.org/learnaboutquality/sevenbasic

qualitytools/overview/overview.html).TheQualityToolbox.Milwaukee,Wisconsin:AmericanSocietyforQuality.p.15.Retrieved20100205.

5. ^M.EileenMagnello(December2006)."KarlPearsonandtheOriginsofModernStatistics:AnElasticianbecomesaStatistician"(http://www.rutherfordjournal.org/article010107.html).TheNewZealandJournalfortheHistoryandPhilosophyofScienceandTechnology.1volume.OCLC682200824(https://www.worldcat.org/oclc/682200824).

6. ^US2000census(http://www.census.gov/prod/2004pubs/c2kbr33.pdf).7. ^Dean,S.,&Illowsky,B.(2009,February19).DescriptiveStatistics:Histogram.Retrievedfromthe

ConnexionsWebsite:http://cnx.org/content/m16298/1.11/

8. ^abcScott,DavidW.(1992).MultivariateDensityEstimation:Theory,Practice,andVisualization.NewYork:JohnWiley.

9. ^abSturges,H.A.(1926)."Thechoiceofaclassinterval".JournaloftheAmericanStatisticalAssociation:6566.JSTOR2965501(https://www.jstor.org/stable/2965501).

10. ^e.g.5.6"DensityEstimation",W.N.VenablesandB.D.Ripley,ModernAppliedStatisticswithS(2002),Springer,4thedition.ISBN0387954570.

11. ^EXCEL2007:Histogram(http://cameron.econ.ucdavis.edu/excel/ex11histogram.html)12. ^OnlineStatisticsEducation:AMultimediaCourseofStudy(http://onlinestatbook.com/).ProjectLeader:David

M.Lane,RiceUniversity(chapter2"GraphingDistributions",section"Histograms")13. ^DoaneDP(1976)Aestheticfrequencyclassication.AmericanStatistician,30:18118314. ^Scott,DavidW.(1979)."Onoptimalanddatabasedhistograms".Biometrika66(3):605610.

doi:10.1093/biomet/66.3.605(http://dx.doi.org/10.1093%2Fbiomet%2F66.3.605).15. ^Freedman,DavidDiaconis,P.(1981)."Onthehistogramasadensityestimator:L2theory".Zeitschriftfr

WahrscheinlichkeitstheorieundverwandteGebiete57(4):453476.doi:10.1007/BF01025868(http://dx.doi.org/10.1007%2FBF01025868).

WikimediaCommonshasmediarelatedtoHistogram.

LookuphistograminWiktionary,thefreedictionary.

16. ^Shimazaki,H.Shinomoto,S.(2007)."Amethodforselectingthebinsizeofatimehistogram"(http://www.mitpressjournals.org/doi/abs/10.1162/neco.2007.19.6.1503).NeuralComputation19(6):15031527.doi:10.1162/neco.2007.19.6.1503(http://dx.doi.org/10.1162%2Fneco.2007.19.6.1503).PMID17444758(https://www.ncbi.nlm.nih.gov/pubmed/17444758).

Furtherreading

Lancaster,H.O.AnIntroductiontoMedicalStatistics.JohnWileyandSons.1974.ISBN0471512508

Externallinks

JourneyToWorkandPlaceOfWork

(http://www.census.gov/population/www/socdemo/journey.html)(locationofcensusdocumentcitedinexample)Smoothhistogramforsignalsandimagesfromafewsamples(http://www.mathworks.com/matlabcentral/fileexchange/30480histconnect)Histograms:Construction,AnalysisandUnderstandingwithexternallinksandanapplicationtoparticlePhysics.(http://quarknet.fnal.gov/toolkits/ati/histograms.html)AMethodforSelectingtheBinSizeofaHistogram(http://2000.jukuin.keio.ac.jp/shimazaki/res/histogram.html)Interactivehistogramgenerator(http://www.shodor.org/interactivate/activities/histogram/)Matlabfunctiontoplotnicehistograms(http://www.mathworks.com/matlabcentral/fileexchange/27388plotandcomparenicehistogramsbydefault)DynamicHistograminMSExcel(http://excelandfinance.com/histograminexcel/)Histogramconstruction(http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_ModelerActivities_MixtureModel_1)andmanipulation(http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs)usingJavaapplets,andcharts(http://www.socr.ucla.edu/htmls/SOCR_Charts.html)onSOCR

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Categories: Statisticalchartsanddiagrams Qualitycontroltools EstimationofdensitiesNonparametricstatistics

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