### 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