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]
Pearsonian's Coefficient of Skewness = β₁ = μ₃² / μ₂³; γ₁ = ± √β₁; (sign of μ₃)
Pearsonian's Coefficient of Kurtosis
β₂ = μ₄ / μ₂²
γ₂ = (β₂ - 3)
Part - 1: Skewness and Measures of Skewness
Concept Recap:
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
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]
Pearsonian's Coefficient of Skewness = β₁ = μ₃² / μ₂³; γ₁ = ± √β₁; (sign of μ₃)
Part - 2: Kurtosis and Measures of Kurtosis
Concept Recap:
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
Pearsonian's Coefficient of Kurtosis
β₂ = μ₄ / μ₂²
γ₂ = (β₂ - 3)
Part - 3: Solved Examples
Question: Find Karl Pearson's Coefficient of Skweness?
Given:
Mean = 100
Mode = 80
Standard Deviation = 20
Question: Find Karl Pearson's Coefficient of Skweness?
Given:
Mean = 60
Median = 75
Variance = 900
Question: Find Mode and Median?
Given:
Mean = 50
Variance = 400
Karl Pearson's Coefficient of Skweness = -0.4
Part - 4: Solved Examples
Question: Find Bowley's Coefficient of Skewness?
Given:
Lower Quartile for a distribution is 15
Upper Quartile for a distribution is 21
Median is 17
Question: Find the median?
Given:
Bowley's Coefficient of Skewness = -0.8
Q₁ = 44.1
Q₃ = 56.6
Question: Find Bowley's Coefficient of Skewness?
Given:
Q₃ - Q₂ = 100
Q₂ - Q₁ = 120
Part - 5: Solved Examples
Question: Find Pearsonian's Coefficient of Skewness (γ₁) for a distribution?
Given:
μ₂ = 25
μ₃ = 100
Question: Find μ₃ for a distribution?
Given:
Standard Deviation = 4
Pearsonian's Coefficient of Skewness (γ₁) = 1
Question: Find Pearsonian's Coefficient of Skewness (γ₁) for a distribution?
Given:
The first three moments about 2 are 1, 16 and -40 respectively.
Part - 6: Solved Examples
Question: Find Pearsonian's Coefficient of Kurtosis (γ₂) for a distribution?
Given:
μ₂ = 16
μ₄ = 1024
Question: Find μ₂ for a distribution?
Given:
The distribution is mesokurtic
μ₄ = 108
Question: Find Pearsonian's Coefficient of Kurtosis (γ₂) for a distribution?
Given:
The first four moments about 4 are 1, 4, 10 and 46 respectively.
Part - 7: Solved Example
Question: Find Karl Pearson's Coefficient of Skweness for a distribution?
Given:
Mean = 160
Mode = 157
Standard Deviation = 50
Part - 8: Solved Example
Question: Find the mode and median of a frequency distribution?
Given:
Mean = 40
Variance = 625
Pearsonian's Coefficient of Skewness (Sₖₚ) = -0.2
Part - 9: Solved Example
Question: Find the Coefficient of Skewness?
Given:
Sum of upper and lower quartiles is 200
Difference of upper and lower quartiles is 20
Median is 100