The difference between kurtosis and excess kurtosis is that excess kurtosis measures the tailedness of a distribution relative to a normal distribution, while kurtosis is the absolute measure of tailedness.
Understanding Kurtosis and Tailedness
Kurtosis is a statistical measure that quantifies the "tailedness" of a probability distribution. As the provided reference states, "Kurtosis is a measure of the tailedness of a distribution."
What is tailedness? The reference clarifies, "Tailedness is how often outliers occur." So, kurtosis essentially tells us about the shape of the tails of a distribution – specifically, how heavy or light they are compared to a distribution with medium tails, and thus, how frequently extreme values (outliers) appear.
A distribution with high kurtosis has heavier tails and a sharper peak, indicating more frequent extreme outliers than a distribution with low kurtosis. Conversely, a distribution with low kurtosis has lighter tails and a flatter peak, meaning extreme outliers are less common.
Excess Kurtosis: The Relative View
While kurtosis gives an absolute measure of tailedness, excess kurtosis provides a relative perspective. The reference explicitly states, "Excess kurtosis is the tailedness of a distribution relative to a normal distribution."
A standard normal distribution has a kurtosis of exactly 3. Excess kurtosis is calculated by subtracting 3 from the calculated kurtosis of a distribution:
- Excess Kurtosis = Kurtosis - 3
This makes the normal distribution the baseline with an excess kurtosis of 0.
Interpreting Excess Kurtosis Values
Using excess kurtosis makes it easier to compare the tails of any distribution directly against the familiar normal distribution:
- Excess Kurtosis = 0: The distribution has similar tailedness to a normal distribution. The reference mentions, "Distributions with medium kurtosis (medium tails) are mesokurtic." These distributions have a kurtosis of 3, hence an excess kurtosis of 0.
- Excess Kurtosis > 0 (Positive): The distribution has heavier tails than a normal distribution (leptokurtic). This means more frequent extreme outliers.
- Excess Kurtosis < 0 (Negative): The distribution has lighter tails than a normal distribution (platykurtic). This means fewer extreme outliers.
Key Differences Summarized
Here's a breakdown of the core differences:
| Feature | Kurtosis | Excess Kurtosis |
|---|---|---|
| Definition | Absolute measure of tailedness/outlier frequency | Tailedness relative to a normal distribution |
| Baseline | No inherent baseline value of 0 | Normal distribution (Kurtosis = 3) serves as baseline |
| Normal Dist. | Has a kurtosis of 3 | Has an excess kurtosis of 0 |
| Calculation | Based on the fourth moment of the data | Kurtosis - 3 |
| Interpretation | Measures the amount of tailedness | Measures if tails are heavier or lighter than normal |
Understanding both kurtosis and excess kurtosis helps analysts describe the shape of a distribution, particularly focusing on the presence and frequency of extreme values, which is crucial in fields like finance (risk assessment), quality control, and scientific research.
