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 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 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.*