2.4 Skewness and Kurtosis

Skewness and kurtosis are key measures of distribution shape. Skewness shows asymmetry, with positive skew having a longer right tail and negative skew a longer left tail. Kurtosis measures peakedness and tailedness, affecting the concentration of data around the mean.

These measures impact statistical method selection. For skewed distributions, the median is more robust than the mean. Kurtosis affects variability measures, with standard deviation less accurate for kurtotic distributions. Both influence hypothesis testing and confidence intervals, potentially violating normality assumptions in parametric tests.

Measures of Distribution Shape

Skewness and kurtosis in distributions

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• Skewness measures the asymmetry of a probability distribution indicates the degree and direction of deviation from a symmetric distribution
  • Positive skewness has a tail on the right side that is longer or fatter than the left side (income distribution)
  • Negative skewness has a tail on the left side that is longer or fatter than the right side (exam scores with a difficult test)
  • Zero skewness represents a perfectly symmetric distribution (normal distribution)
• Kurtosis measures the tailedness and peakedness of a probability distribution indicates the presence of outliers and the concentration of data around the mean
  • Leptokurtic distributions have a higher and sharper central peak, fatter tails, and excess kurtosis > 0 (financial returns)
  • Platykurtic distributions have a lower and broader central peak, thinner tails, and excess kurtosis < 0 (uniform distribution)
  • Mesokurtic distributions have a normal distribution shape and excess kurtosis = 0 (t-distribution with high degrees of freedom)

Calculation of skewness and kurtosis

• Skewness calculation uses the sample skewness formula: $\frac{\sum_{i=1}^{n}(x_i-\bar{x})^3}{(n-1)s^3}$, where $\bar{x}$ is the sample mean and $s$ is the sample standard deviation
  • Positive values indicate right-skewness, negative values indicate left-skewness, and values close to zero suggest symmetry
• Kurtosis calculation uses the sample excess kurtosis formula: $\frac{\sum_{i=1}^{n}(x_i-\bar{x})^4}{(n-1)s^4}-3$, where $\bar{x}$ is the sample mean and $s$ is the sample standard deviation
  • Positive values indicate leptokurtosis, negative values indicate platykurtosis, and values close to zero suggest mesokurtosis

Types of skewed distributions

• Positively skewed distributions have mean > median > mode and a right tail that is longer or fatter than the left tail (income distribution, house prices)
• Negatively skewed distributions have mean < median < mode and a left tail that is longer or fatter than the right tail (exam scores with a difficult test, asset returns during a financial crisis)
• Symmetric distributions have mean = median = mode and left and right tails that are mirror images of each other (normal distribution, t-distribution with high degrees of freedom)

Impact on statistical method selection

• Skewness and kurtosis affect the choice of central tendency measures
  • Skewed distributions: median is a more robust measure than the mean
  • Symmetric distributions: mean and median are similar and can be used interchangeably
• Kurtosis impacts the choice of variability measures
  • Kurtotic distributions: standard deviation may not accurately capture the variability due to the presence of outliers
  • Alternative measures include interquartile range and median absolute deviation
• Skewness and kurtosis affect hypothesis testing and confidence intervals
  • Skewed and kurtotic distributions may violate the normality assumption of some parametric tests (t-test, ANOVA)
  • Alternative approaches include non-parametric tests (Mann-Whitney U test, Kruskal-Wallis test), bootstrapping, or data transformations to achieve normality