Measures of Asymmetry
5. Kurtosis – exercise

Instruction

As we already know, kurtosis is given by the formula:

Kurtosis=\dfrac{\frac{1}{n}((x_1-\overline{x})^4+(x_2-\overline{x})^4+\ldots+(x_n-\overline{x})^4)}{(\frac{1}{n}((x_1-\overline{x})^2+(x_2-\overline{x})^2+\ldots+(x_n-\overline{x})^2)^2}-3

Let's calculate the kurtosis for the following set:

1,2,3,3,4,4,4,5

The arithmetic mean \overline{x} for this set is 3.25

x_i x_i-\overline{x} (x_i-\overline{x})^2 (x_i-\overline{x})^4
1 -2.25 5.0625 25.62890625
2 -1.25 1.5625 2.44140625
3 -0.25 0.0625 0.00390625
3 -0.25 0.0625 0.00390625
4 0.75 0.5625 0.31640625
4 0.75 0.5625 0.31640625
4 0.75 0.5625 0.31640625
5 1.75 3.0625 9.37890625
SUM 0 11.5 38.40625
Kurtosis=\dfrac{\frac{1}{8}\cdot38.40625}{(\frac{1}{8}\cdot11.5)^2}-3=-0.6767486

This histogram is platykurtic.