Example 8.42: skewness and kurtosis and more moments (oh my!)

While skewness and kurtosis are not as often calculated and reported as mean and standard deviation, they can be useful at times. Skewness is the 3rd moment around the mean, and characterizes whether the distribution is symmetric (skewness=0). Kurtosis is a function of the 4th central moment, and characterizes peakedness, where the normal distribution has a value of 3 and smaller values correspond to thinner tails (less peakedness).

Some packages (including SAS) subtract three from the kurtosis, so that the normal distribution has a kurtosis of 0 (this is sometimes called excess kurtosis.

R

library(moments)
library(lattice)
ds = read.csv("http://www.math.smith.edu/r/data/help.csv")
ds$gender = ifelse(ds$female==1, "female", "male")
densityplot(~ cesd, data=ds, groups=gender, auto.key=TRUE)

We see that the distribution of CESD scores is skewed with a long left tail, and appears somewhat less peaked than a normal distribution. This is confirmed by the actual statistics:

> with(ds, tapply(cesd, gender, skewness))
    female       male 
-0.4906171 -0.2464390 
> with(ds, tapply(cesd, gender, kurtosis))    # kurtosis
  female     male 
2.748968 2.547061 
> with(ds, tapply(cesd, gender, kurtosis))-3  # excess kurtosis
    female       male 
-0.2510318 -0.4529394 

SAS
SAS includes much detail on the moments and other statistics in the output from proc univariate. As usual, the quantity of output can be off-putting for new users and students. Here we extract the moments we need with the ODS system. We also generate kernel density estimates roughly analogous to the densityplot() results shown above.

ods output moments = cesdmoments;
proc univariate data="c:\book\help.sas7bdat";
  class female;
  var cesd;
  histogram cesd / kernel;
run;

proc print data=cesdmoments; 
  where label1 = "Skewness";
  var female label1 nvalue1 label2 nvalue2;
run;

With the result:

Obs   FEMALE    Label1         nValue1    Label2         nValue2

  4     0      Skewness      -0.247513   Kurtosis      -0.442010
 10     1      Skewness      -0.497620   Kurtosis      -0.204928

We note that the default is to produce unbiased (REML) estimates, rather than the biased method of moments estimator produced by the kurtosis() function (and that SAS presents the excess kurtosis).