Summarystatistics2Summarystatistics2Inthischapter:weintroducevariousmeasurestodescribethespreadordispersionofarangeofdata,includingtherange,varianceandstandarddeviation.weuseoxandwhiskerdiagramsasvisualdisplaysofsymmetryandspreadofdata.Summarystatistics2Summarystatistics2RangePerhapsthemostsimplemeasureofspreadisthedifferencebetweenthelargestandsmallestitemsofdatai.e.thedifferencebetweentheextremes.Thisistherange.Inthecaseofthemotorwaytolltimesthelongesttimerecordedwas118secondsandtheshortesttimewas9seconds,hencetherangeofthesedataisgivenby:range=118–9=109secondsThismeasureofspreaddosenottakeintoaccountanythingaboutthedistributionofthedataotherthantheextremes.Neitherisitveryreliableortypical.Why?Summarystatistics2Summarystatistics2QuartilespreadAmoretrustworthymeasureistherangeofthemiddlehalfofthedata.Toidentifythisrangeweneedtofindtheitemsofdatawhicharepositionedhalfwaybetweentheextremesandthemedian.Takethecaseofthedatafollowing:3456778881015medianlowerquartileupperquartileIngeneral,theitemsofdatalyingmidwaybetweenthemedianandtheextremesareknowasthequartiles.Itismoreusualtorefertothemasthefirstorlowerquartileandthethirdorupperquartile.Thedifferencebetweenthemiscalledtheinterquartilerange(IQR)orquartilespread(QS)Summarystatistics2Summarystatistics2StandarddeviationTheinterquartilerangemeasuresthespreadofthemiddlehalfofthedataandiscloselylinkedtothemedian.Wecandefineameasureofdispersion,takingintoaccountallthedata,whichislinkedinsteadtothemean.DeviationfromthemeanSupposetheitemsofdata:18,20,21,22,24.Themeanofthecollectiondatais21,hencethedeviationfromthemeanare:-3,-1,0,1,3,andtheaverage(ormean)ofthisdeviationsistheirsumdividedbythenumberofitems.Thiscomestozero,i.e.)(1xxnSummarystatistics2Summarystatistics2Squareddeviationfromthemean(1)Ignoringthesignofthedeviationseemsasoundprinciple,sincethedeviation18from21isasmuchasthedeviationof24from21.Asimilareffectcanbeachievedbysquaringthedeviationsfromthemean.Fortheitemsofdata:18,20,21,22,24,thisresultsinthefollo...
