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.
