The formula to calculate the Coefficient of Kurtosis (K) is:
\[ K = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum \left( \frac{(x_i - \mu)^4}{\sigma^4} \right) - \frac{3(n-1)^2}{(n-2)(n-3)} \]
Where:
Let's say the data points are 2, 4, 6, 8, and 10. Using the formula:
\[ K = \frac{5(5+1)}{(5-1)(5-2)(5-3)} \sum \left( \frac{(x_i - \mu)^4}{\sigma^4} \right) - \frac{3(5-1)^2}{(5-2)(5-3)} \]
We get:
\[ K \approx 2.62 \]
So, the coefficient of kurtosis is approximately 2.62.
Definition: The percentile coefficient of kurtosis measures the "tailedness" of the probability distribution of a real-valued random variable.
Formula: \( K = \frac{P_{90} - P_{10}}{2(P_{75} - P_{25})} \)
Example: \( K = \frac{30 - 10}{2(25 - 15)} \)
Definition: Kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable.
Formula: \( K = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum \left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)} \)
Example: \( K = \frac{5(5+1)}{(5-1)(5-2)(5-3)} \sum \left(\frac{10 - 8}{2}\right)^4 - \frac{3(5-1)^2}{(5-2)(5-3)} \)
Definition: The measure of kurtosis indicates the extent to which data points in a distribution are tailed.
Formula: \( K = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum \left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)} \)
Example: \( K = \frac{6(6+1)}{(6-1)(6-2)(6-3)} \sum \left(\frac{12 - 9}{3}\right)^4 - \frac{3(6-1)^2}{(6-2)(6-3)} \)
