3.1.3 Measures of Shape

An important aspect of the description of a variable is its shape, which indicates the frequency of values from different ranges of the variable. One is typically interested to know how well the distribution can be approximated by the normal distribution.

Skewness

Skewness is a measure of the asymmetry of the distribution. The normal distribution is symmetrical; its skewness is equal to zero. A distribution with a significant positive skewness has a long right tail. A distribution with a significant negative skewness has a long left tail. For a skewed distribution, the mean tends to lie on the same side of the mode as the longer tail.

Skewness is a dimensionless quantity:

To avoid computing the mode, the following empirical formula is used:

A skewness coefficient is considered significant if the absolute value of the ratio (skewness/ standard error of skewness) is greater than 2. The standard error of skewness = [6/N]2.

Kurtosis

Kurtosis is the degree of peakedness in a distribution, usually taken relative to a normal distribution. The peakedness property means that there is an excess frequency at the center of the distribution. The index of kurtosis is given by:

d = E {(x-m) 4} / s 4 -3                                            

where  E { (x-m) 4 }  is  the  fourth  moment  and  s  is  the  standard deviation of  the  distribution. Since  the value  of E {(x-m) 4}  is equal to 3 for a normal distribution, this index measures kurtosis relative to a normal distribution. Positive values of d indicate longer thicker tails than a normal distribution, whereas the negative values of d indicate shorter thinner tails. A distribution with positive kurtosis is called Leptokurtic, and a distribution with negative kurtosis is called Platokurtic. When d =0, the distribution is called Mesokurtic.