12 Measures of Kurtosis
12.1 Kurtosis
Kurtosis (Karl Pearson, 1905) measures the “tailedness” of the distribution or the peakedness. It indicates how much of the data is concentrated in the tails and the peak of the distribution relative to a normal distribution.
- Mesokurtic (Kurtosis = 3): Indicates a distribution with kurtosis similar to that of a normal distribution. It is referred to as mesokurtic.
- Leptokurtic (Kurtosis > 3): Indicates a distribution that is more peaked than a normal distribution, with fatter tails. Such distributions have more extreme values (outliers).
- Platykurtic (Kurtosis < 3): Indicates a distribution that is flatter than a normal distribution with thinner tails. Such distributions have fewer extreme values.
12.1.1 Formula for Kurtosis:
\[ Kurtosis = \frac{N(N+1) \sum (X_i - \overline{X})^4}{(N-1)(N-2)(N-3)S^4} - \frac{3(N-1)^2}{(N-2)(N-3)} \]
Where:
- The symbols represent the same quantities as in the skewness formula.
Kurtosis measures the “tailedness” of the distribution:
- Kurtosis > 3 → Leptokurtic (Heavy tails)
- Kurtosis = 3 → Mesokurtic (Normal distribution)
- Kurtosis < 3 → Platykurtic (Light tails, flat distribution)
12.1.2 Example of Kurtosis:
Consider a dataset representing the heights of a group of people. If most people are of average height, with few very short or very tall people, the distribution might be leptokurtic, indicating a peaked distribution with fat tails.
Application in Real Life
- Finance: Skewness and kurtosis are used to analyze the distribution of returns for an investment, helping to understand the risk and the likelihood of extreme outcomes.
- Quality Control: In manufacturing, these measures help in identifying the deviation from the process standards.
- Environmental Science: Analyzing rainfall or temperature data to understand the distribution and the occurrence of extreme weather conditions.
Graphical Summaries
Graphical representations are integral to descriptive statistics, visually summarizing data through various charts and plots.
- Histograms: Illustrate the distribution of data, helping identify its shape.
- Box plots: Visualize the minimum, first quartile, median, third quartile, and maximum, revealing dispersion and outliers.
- Scatter plots: Explore relationships and trends between two variables.
Summary
| Concept | Description |
|---|---|
| Core Idea | |
| Kurtosis Definition | A measure of the tailedness and peakedness of a distribution relative to the normal reference |
| Kurtosis Types | |
| Mesokurtic | Kurtosis equal to three, matching the tail behaviour of the normal distribution |
| Leptokurtic | Kurtosis greater than three, indicating heavier tails and more frequent extreme values |
| Platykurtic | Kurtosis less than three, indicating lighter tails and fewer extreme values |
| Computation | |
| Kurtosis Formula | Sample kurtosis combines a fourth power sum and a bias-correction term based on N |
| moments::kurtosis() | Computes sample kurtosis in R using the moments package |
| scipy.stats.kurtosis() | Computes excess kurtosis in Python via the SciPy statistics module |
| Real-World Applications | |
| Finance Application | Used to assess the probability of extreme returns and tail risk in investment data |
| Quality Control Application | Used in manufacturing to detect non-standard variation in process outputs |
| Environmental Science Application | Used to evaluate the frequency of extreme weather events in temperature or rainfall data |
| Graphical Summaries | |
| Histogram | A chart of binned counts that reveals the overall shape of a distribution |
| Box Plot | A five-number summary plot that highlights quartiles, median, and outliers |
| Scatter Plot | A plot of paired observations used to reveal relationships and trends between two variables |