12  Measures of Kurtosis

Measures of Kurtosis

Kurtosis 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.

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)

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.

Calculation in R

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.