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Numericaldescriptionsandgraphicaldisplaysshouldbeusedtogether.   MEASURESOFCENTER  h  Thecenterofadistributionisusuallythemostimportantaspectto e  describe.Measuresofcenterareoftenusedtodescribeorrepresentwhat   the typicalvalueis.   Wewilltalkaboutthreemeasuresofcenter:mean,median,andmode. v  Mean  s The mean or average ofasetofnumbersistheirsumdividedbyhowmany * numbersthereare.Themeanie  1!b `@E   i.Thenotationn W isusedforthenumberofobservationsinadataset. / Thenotationie1!b `@E ,oo ,oi߀isreservedforthesamplemean.TheGreekletter__Ԁisused , todenotethepopulationmean.Mostoften,adatasetcontainssampledata  insteadofdataontheentirepopulationofinterest.  $ Forexample,supposetheAmericanMedicalAssociationwishestofindthe !#~  averageweightforadultwomenwithheight5feet,1inch,intheagegroup N$! 3550years.TheAMAwouldliketoknow__Ԁ=averageweight,butitwillbe {%" impossibletoobserveweightsforeveryindividualinthepopulationof &$ interest.TheAMAsettlesforestimating__Ԁbasedonasample.Thatis, '2% theyestimatethepopulationmean__Ԁbythesamplemeanie1!b `@EN)ooN)oi. )_&   p*'  @(2   Median  Z Anotherusefulmeasureofcenteristhemedian.Themedianisthemiddle W valueofasetofnumberswhenthenumbersarearrangedinorder.When  thereareanevennumberofobservations,themedianistheaverageofthe   twomiddleobservations.  ; Themedianissuchthathalfoftheobservationsarelessthanitandhalfof 8  theobservationsarelargerthanit.Thispropertyofmediandoes not hold e  forthemean.    Example  I  Asampleof10adultswasaskedtoreportthenumberofhourstheyspent F ontheinternetthepreviousmonth.Findthemeanandmedian. s @AA0,7,12,5,33,14,8,0,9,22 * Mean=110/10=11.0 ' Formedian,ordertheobservationsfirst.  @AA0,0,5,7,8,9,12,14,22,33 8 Median=(8+9)/2=8.5 5! BEHAVIOROFMEANANDMEDIAN #  Supposetherespondentwhoreported33hoursontheinternetactually %F# reported133hours. 's$ Thehighoutlierpullsthemeaninternetusagefrom11.0to21.0, butthe p)&! medianstaysthesame .Themeanispulledinthedirectionofextreme *'" observationswhilethemedianstaysthesame. +')#  Q/,& @(3  Wesaythemedianisa resistant measureofcenter,becauseitcanresist Z theinfluenceofoutliers.Themean,ontheotherhand,cannotresistthe * influenceofoutliers.Themeanishighlyinfluencedbyextremevalues. W Whenadatasetcontainsextremevalues,themedianwillusuallybemore   representativeofwhatis typical.  ; MEAN,MEDIAN,SYMMETRY,ANDSKEWNESS 8   (-y3"3"   ,X3z2-`3  0    Themeanandmedianofasymmetricdistributionareclosetogether.3z݌ (#(# Ќ  "3"  32-`3  0    Inarightskeweddistribution,large(high)observations pullthemean I  rightofthemedian.Themeanispulledtowardthelongtail.3݌v(#(# Ќ  "3"  32-`3  0    Inaleftskeweddistribution,small(low)observations pullthemeanleft s ofthemedianandtowardthelongtail.3݌(#(# Ќ  Example W <( 4 <DL,<XX\909594939690928782868186867171 ' 757577707575777562696162615656 > 5853514020213 U ̀nMeanMedian  w$C3/b~0e `. @E (! 2\w3769.5175.00#\XXߎ#   @(4   Mode  Z Modeisanothermeasureofcenter,butcanbemisleading.Themodeofa W datasetisthevalueoftheobservationthatoccursmostfrequently.  Fortheinternetusagedata,  ; @AA0,0,5,7,8,9,12,14,22,33 8  themodeis0,andisnotagoodrepresentationofwhatis typical.   Therecanbemorethanonemodeornomode.How? I  Forcategoricaldatawheretheorderingofthecategoriesisnotrelevant, F meanandmedianarenotappropriatemeasuresofcenter,butmodecanbe s used.  Example:0 ` Supposeasampleof100individualscontains15left_handers_,80 W right_handers_,and5ambidextrousindividuals.Thedataentryis ' codedasfollows:1=righthanded,2=lefthanded,3= T ambidextrous.` (#` (# 0  0` (#(#Thesamplemeanis1.25!meaningless!8` (#` (# 0  0` (#(#Themodeis1=righthanded. e` (#` (# Comment:0 ` Whenthegraphicaldisplayofadistributionshowstwopeaks,it b" isoftendescribedasbimodal.Abimodaldistributionmightbe #  indicativeoftwonaturalgroupingsinthedata.Inabimodal $" distribution,numericalmeasuresofcenteroftendonotdescribe %F# the typicalvalue.'s$` (#` (#   C(%   B.\BB.@--@@,,MeasuresofCenterWorksheet   ! ,XXX!1.0  Arealestateagentnotesthatthemeanhousingpriceforanareais$125,780andconcludesthat $ halfofthehousesintheareacostmorethanthat.Explainwhytheagentsconclusionisnot @ necessarilytrue.(#(# 2.0  Acompanyexecutiveconcludesthatanaccountantmusthavemadeamistakebecauseshe 0 preparedareportstatingthat90%ofthecompanysemployeesearnlessthanthemeansalary.  Explainhowitispossiblethattheaccountantsreportiscorrect.Q(#(# 3.0  Arestaurantownerdecidesthatmorethanhalfofhercustomerspreferchocolateicecream $ becausechocolateisthemodewhencustomersareofferedtheirchoiceofchocolate,vanilla,and A% strawberry.Explainwhytherestaurantownersconclusionisnotnecessarilytrue. ]&(#(# @YY(2  4.0  Ifyouhaddataforallstudentsinyourschoolontheamountofmoneyspentinthepreviousyear = onovernightstaysinahospital,probablythemedianandmodewouldbe0butthemeanwould  bepositive.Explainwhy.Makereferencetothedefinitionsofmedian,mode,andmeaninyour ^ explanation.(Youcanuseanexampletoexplain,ifyouwant.)z(#(# 5.0  Asampleof99distanceshasameanof24feetandamedianof24.5feet.Unfortunately,ithas  justbeendiscoveredthatanobservationwhichwaserroneouslyrecordedas 30"actuallyhada J valueof 35.Ifwemakethiscorrectiontothedata,then: f(#(#   0 ` A)0 ` (#` (#themeanremainsthesame,butthemedianisincreased. (# (#   0 ` B)0 ` (#` (#themeanandmedianremainthesameb (# (#   0 ` C)0 ` (#` (#themedianremainsthesame,butthemeanisincreased  (# (#   0 ` D)0 ` (#` (#themeanandmedianarebothincreased.^" (# (#   0 ` E)0 ` (#` (#wedonotknowhowthemeanandmedianareaffectedwithoutfurther $ calculations;butthestandarddeviationisincreased.>% (# (# 6.0  Whentestingwaterforchemicalimpurities,resultsareoftenreportedas bdl,whichmeans { (  belowdetectionlimit.Thefollowingareninemeasurementsoftheamountofleadinaseriesof :!) watersamplestakenfrominnercityhouseholds(measuredinpartspermillion,_ppm_).!V*(#(#    ` 0 50 (# (#70h(#(#120h(#h(#bdl0(#(#100p(#(#80p(#p(#bdl0 (#(#200x (# (#6w# ,x(#x(# 0  0` (#(#Whichofthefollowingiscorrect?$R".` (#` (# @  0 ` A)0 ` (#` (#Themeanleadlevelinthewaterisabout10_ppm_.s� (# (#   0 ` B)0 ` (#` (#Themeanleadlevelinthewaterisabout8_ppm_.O(%2 (# (#   0 ` C)0 ` (#` (#Themedianleadlevelinthewateris7_ppm_.+*'4 (# (#   0 ` D)0 ` (#` (#Themedianleadlevelinthewateris8_ppm_.,d)6 (# (# @  0 ` E)0 ` (#` (#Neitherthemeannorthemediancanbecomputedbecausesomevaluesare -+8 unknown.s.+9 (# (#  2/,: @YY(3  7.0  Forthefollowinghistogram,whatistheproperorderingofthemean,median,andmode?!~(#(# @wCC3/b~0   `@Eg(  N w (#(#0  0` (#A)0 ` ` (#I=mean,II=median,III=mode  (#     `     h        0 ` B)0 ` ` (#I=mode,II=median,III=mean{  (#     0 ` C)0 ` ` (#I=median,II=mean,III=modeW   (#     0 ` D)0 ` ` (#I=mode,II=mean,III=median3    (#     0 ` E)0 ` ` (#I=mean,II=mode,III=median#B ##\!#