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.
