Kurtosis Calculator
Kurtosis Calculator computes the sample kurtosis of a dataset, measuring the “tailedness” of its distribution. Positive kurtosis indicates heavier tails (leptokurtic), while negative indicates lighter tails (platykurtic).
Enter comma-separated values (e.g., 1,2,3,4):
Methodology Used in Kurtosis Calculator
The calculator computes sample kurtosis using:
1. Kurtosis: \\( K = \frac{n (n+1) \sum (x_i – \bar{x})^4}{(n-1)(n-2)(n-3) s^4} – \frac{3 (n-1)^2}{(n-2)(n-3)} \\)
Where:
- \\( K \\): Sample kurtosis
- \\( n \\): Sample size
- \\( x_i \\): Data points
- \\( \bar{x} \\): Sample mean
- \\( s^2 \\): Sample variance
Measures tailedness relative to a normal distribution.
Example Calculation
Sample Input
Dataset = 1, 2, 3, 4, 5
Step 1: Mean: \\( \bar{x} = \frac{1+2+3+4+5}{5} = 3 \\)
Step 2: Variance: \\( s^2 = \frac{\sum (x_i – 3)^2}{4} = \frac{10}{4} = 2.5 \\)
Step 3: Fourth moment: \\( \sum (x_i – 3)^4 = 34 \\)
Step 4: Kurtosis: \\( K = \frac{5 \cdot 6 \cdot 34}{4 \cdot 3 \cdot 2 \cdot 2.5^4} – \frac{3 \cdot 4^2}{3 \cdot 2} = -1.2 \\)
Result: Kurtosis = -1.2 (platykurtic).