Learning Objective:
What is Skewness?
What are measures of Skewness?
What does Kurtosis relate to?
How to measure Kurtosis?
Concept:
Skewness relates to asymmetry of a frequency distribution.
Skewness can be of two types.
If there is a positive tail in a frequency distribution, it is termed as postive skewness.
If there is a negative tail in a frequency distribution, is it termed as negative skeness.
Measures of Skewness can be Absolute or Relative.
Absolute measure of skewness cannot compare series with different units. It is not widely used.
Relative measure of skewness are of 4 types.
Karl Pearson's Coefficeint of Skewness based on Mean, Median, Mode and Standard Deviation.
Bowley's Coefficient of Skewness based on Quartiles.
Kelly's Coefficient of Skewenss based on upper and lower Deciles and Median.
Pearsonian's Coefficient of Skewness based on central Moments.
If the value of above 4 relative measures of skewness is 0, the distribution is symmetric.
If the value of above 4 relative measures of skewness is greater then 0, the distribution is positively skewed.
If the value of above 4 relative measures of skewness is less then 0, the distribution is negatively skewed.
In symmetric distribution, x̄ (Mean) = Median = Mode
In positive skewness, x̄ (Mean) > Median > Mode
In negative skewness, x̄ (Mean) < Median < Mode
Kurtosis relates to bulginess, peakedness, tailedness of a frequency distribution.
Leptokurtic frequency distribution is narrow with sharp peak and extended tails.
Mesokurtic frequency distribution is normal with less sharp peak and not much extended tails.
Platykurtic frequency distribution is like platypus without sharp peak and very less extended tails.
Kurtosis is measured with Pearsonian's Coefficient of Kurtosis which is based on central moments.
If γ₂ > 0, the frequency distribution is Leptokutic.
If γ₂ = 0, the frequency distribution is Mesokurtic.
If γ₂ < 0, the frequency distribution is Platukurtic.
Formulas:
Absolute Measure of Skewness = Sₖ = Mean - Mode
Karl Pearson's Coefficient of Skewness = [Mean - Mode] / Standard Deviation
Karl Pearson's Coefficient of Skewness = 3 x [Mean - Median] / Standard Deviation
Bowley's Coefficient of Skewness = [Q₃ + Q₁ - 2Q₂] / [Q₃ - Q₁]
Kelly's Coefficient of Skewness = [D₉ - D₁ - 2Q₂] / [D₉ - D₁]