Using R for Introductory Statistics : John Verzani

Univariate data 49> var(test.scores.b) # larger, asanticipated[1] 394.2> sd(test.scores)[1] 4.967Quantiles, quintiles, percentiles, and moreThe standard deviation, like the mean, can be skewed when an exceptionally large orsmall value is in the data. More resistant alternatives are available. A conceptually simpleone (the IQR) is to take the range of the middle 50% of the data. That is, trim off 25% ofthe data from the left and right, and then take the range of what is remaining.To be precise, we need to generalize the concept of the median. The median splits thedata in half—half smaller than the median and half bigger. The quantiles generalize this.The pth quantile is at position 1+p(n−1) in the sorted data. When this is not an integer, aweighted average is used. † This value essentially splits the data so 100p% is smaller and100(1−p)% is larger. Here p ranges from 0 to 1. The median then is the 0.5 quantile.The percentiles do the same thing, except that a scale of 0 to 100 is used, instead of 0to 1. The term quartiles refers to the 0,25, 50,75, and 100 percentiles, and the termquintiles refers to the 0, 20,40, 60, 80, and 100 percentiles.The quantile () function returns the quantiles. This function is called with the datavector and a value (or values) for p. We illustrate on a very simple data set, for which theanswers are easily guessed.> x = 0:5 # 0,1,2,3,4,5> length(x)[1] 6> sum(sort(x)[3:4])/2 # the median the hard way[1] 2.5> median(x) # easy way. Clearly themiddle[1] 2.5> quantile(x,.25)25%1.25> quantile(x,c(0.25,0.5,0.75)) # more than 1 at a time† There are other definitions used for the pth quantile implemented in the quantile() function. Thesealternatives are specified with the type= argument. The default is type 7. See ?quantile for thedetails.25% 50% 75%1.25 2.50 3.75> quantile(x) # default gives quartiles0% 25% 50% 75% 100%0.00 1.25 2.50 3.75 5.00■ Example 2.7: Executive pay The exec.pay (UsingR) data set contains compensation toCEOs of 199 U.S. companies in the year 2000 in units of $10,000. The data is not