**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__

__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__

__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__

__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__

__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__

__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__

__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__

__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__

__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__

__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**