Measures of Dispersion information sheet

MEASURESOFDISPERSIONPage1of3DISPERSION.Dispersionisameasureofhowspreadoutyourstatisticaldatais.ItrepresentshowVARIABLEorVARIEDyourdatais.HIGHDISPERSIONmeansVERYVARIABLEDATAorVERYINCONSISTENTDATA.LOWDISPERSIONmeansLESSVARIABLEDATAorMORECONSISTENTDATA.THERANGE.1.Therangeisthesimplestmeasureofdispersion.Forasetofdataitisdefinedasfollows.Range=LargestItem–Smallestitem.2.Itiseasilycalculatedbuttellsusnothingaboutthedistributionofdatawithinthesetofdata.Forexample:Rangeofmaleheights=35.5cmRangeoffemaleheights=28.6cmThissuggeststhatmaleheightsaremorevariablethanfemaleheights.Insomecircumstancestherangecangivearoughandreadymeasureofthespreadoutnessofdatabutwouldnotbeofsuchusewheredataisveryskewedduetohepresenceafewextremelylargeorsmallvaluesknownasoutliers.THEINTERQUARTILERANGEIfadistributionisskewedduetooutliers,inthismanner,itisbettertoexcludetheupperquarterandlowerquarterofthedistributionandfindtherangeofthemiddlehalf.ThisvalueiscalledtheINTERQUARTILERANGE.THESTANDARDDEVIATION.Thestandarddeviationisameasureofvariationmostoftenusedinstatistics.Itmeasuresvariationaroundthearithmeticmean.Thestandarddeviationisgivenbythefollowingformula:StandarddeviationS=nxxnii12)(,wherex=themean=nxnii1Wherethesetofdatais:(x1,x2,x3,x4,…xn),i.e.nitemsofdata.TheVarianceisthesquareofthestandarddeviation.Thisisamoreusefulmeasureofvariation.MEASURESOFDISPERSIONPage2of3VarianceS2=nxxnii12)(Question1:Usethetablebelowtofindthemeanandstandarddeviationix)(xxi2)(xxi457910Thesecondcolumnshowsthedeviationofeachdatavaluefromthemean.Ifthesedeviationsaretotalled,theresultwillalwaysbezeroorveryclosetozero.Thismethodofcalculatingthestandarddeviationlooksstraightforwardenoughwiththeeasynumbersofthisexample.Butifthemeanisnotawholenumber,subtractingitfromeveryvalueandsquaringwillinvolvedecimalsandroundingoff.Thereisaneasiermethod.Itcanbeshownthattheformulaforthestandarddeviationisalsogivenby:S=212)(xnxniiFromwhichitcanbeseenthatthevarianceisgivenby:VarianceS2=212)(xnxniiMEASURESOFDISPERSIONPage3of3Question2.i.Findthemeanii.Usethetablebelowtocalculatethestandarddeviationusingbothformulae.ix)(xxi2)(xxi2ix168.3175.2195.3163.0175.0Question3.Findthemean,standarddeviationandvarianceofthefollowingsetsofdata:a.5,3,4,4,2,6,7,1,2,6b.28,27,26,32,25,26,20,21,19,22,23,24,27,26,31,30c.16years8months,16yrs4months,17years,17year2month,15years7months,16years3months,17years.MicrosoftEquation3.0