Skewness and Kurtosis

In our previous blog entry, we discussed how a probability distribution can be described using the first four moments which are mean, variance/standard deviation, skewness, and kurtosis. Mean and standard deviation were described and now we will talk about skewness and kurtosis.

Skewness: The measurement of the asymmetry of a random variable is a dimensionless quantity called skewness (b₁)^0.5, which is calculated from the second and third moments about the mean of a distribution. If (b₁)^0.5 < 0 the distribution is negatively skewed (tail to left) and if (b₁)^0.5 > 0 the distribution is positively skewed (tail to the right). The equations for calculating skewness from data are:

Distributions that are: (a) positively skewed with tail to the right (b₁)^0.5  > 0, (b) centered (b₁)^0.5  = 0, and (c) negatively skewed with tail to the left (b₁)^0.5  < 0. Red lines are means.

Kurtosis: A dimensionless quantity that characterizes the peakedness of a random variable is called kurtotsis b₂, which is calculated from the fourth and second moments of the distribution about the mean. If b₂ >> 3, the distribution has a high peak and for b₂ =1.8, the distribution becomes flat. At b₂=3, the distribution is normal

(a) distribution with a high peak (b₂  >> 3), (b) normal distribution (b₂  = 3), (c) flat (uniform) distribution with (b₂  =1.8).

In Excel, the functions SKEW and KURT can be used to calculate skewness and kurtosis, respectively. Be warned -- the Excel KURT function calculates the "Excess of Kurtosis" relative to a normal distribution.  In Excel, KURT will return 0 for a normal distribution rather than the value of 3 found in statistics textbooks.  We suggest that you add 3 to any to value returned by the KURT function when presenting data.

=KURT(A1:A10) + 3