Variance

Dispersion of Data: Introducing Variance

Consider the three data sets \(X\), \(Y\) and \(Z\)

\[X= \{900,925,950,975,1025,1050,1075,1100 \}\] \[Y=\{900,905,910,920,1080,1090,1095,1100\}\] \[Z=\{900,985,990,995,1005,1010,1015,1100\}\]

For each of the data sets, the following statements can be verified

• The mean of each data set is 1000
• There are 8 elements in each data set
• The minima and maxima are 900 and 1100 for each set
• The range is 200.

From the plot below, notice how different the three data sets are in terms of dispersion around the mean value.

            X=c(900,925,950,975,1025,1050,1075,1100)
            Y=c(900,905,910,920,1080,1090,1095,1100)
            Z=c(900,985,990,995,1005,1010,1015,1100)
            # y-values for graph
            Z.y = rep(3,8)
            Y.y = rep(4,8)
            X.y = rep(5,8)
            
            plot(Z,Z.y,pch=16,col="red",ylim=c(2.5,5.5),main=c("Variance") )
            
            points(Y,Y.y,pch=16,col="blue" )
            points(X,X.y,pch=16,col="green" )
            points(c(1000,1000,1000),c(3,4,5),pch=18,cex=1.2)
            lines(c(1000,1000),c(2.75,5.25),lty=3)

var(X)

## [1] 5357.143

sd(X)

## [1] 73.19251

var(Y)

## [1] 9578.571

sd(Y)

## [1] 97.87018

var(Z)

## [1] 2957.143

sd(Z)

## [1] 54.37962