Math A160 - CooleyIntroduction to StatisticsOCCSECTION 3.2 – Measures of DispersionDispersion– The degree to which the data are spread out.Range– Therange,R, of a variable is the difference between the largest and the smallest data value. That is,Range =R= largest data value – smallest data value.Standard DeviationThestandard deviationmeasures variation by indicating how far, on average, the observations are from themean. For a data set with a large amount of variation, the observations will, on average, be far from the mean; sothe standard deviation will be large. For a data set with a small amount of variation, the observations will, onaverage, be close to the mean; so the standard deviation will be small.Population Standard Deviation,Thepopulation standard deviation,(lowercase Greek sigma) of a variable is the square root of the sum ofsquared deviations about the population mean divided by the number of observations in the population,N. Thatis, it is the square root of the mean of the squared deviations about the population mean.222212NixxxxNNwhere12,,...,Nxxxare theNobservations in the population andis the population mean.Conceptual Formula & Computational FormulaThere are two ways to calculate the population standard deviation,.Conceptual FormulaComputational Formula2ixN22iixxNNSample Standard Deviation,sThesample standard deviation,s, of a variable is the square root of the sum of squared deviations about thepopulation mean divided byn– 1, wherenis the sample size.222212NixxxxNNwhere12,,...,Nxxxare thenobservations in the sample andxis the sample mean.