Skewness & Kurtosis Lecture 16

 Skewness & Kurtosis 

Lecture 16

Symmetrical Distribution

A symmetric distribution is a type of distribution where the left side of the distribution mirrors the right side.

In symmetrical distribution:

Skewness

Any departure from or lack of symmetry is called skewness.

Practically, a distribution will be skewed if 

There are two types of skewness

1. Positive Skewness

2.  Negative Skewness

Positive Skewed Curve

If the left tail of a curve is longer than its right rail, the curve is said to be a positive-skewed curve.

In a positive skewed curve:

Skewed Curve

If the right tail of a curve is longer than its left rail, the curve is said to be a negatively skewed curve.

In a negative skewed curve:

Measure of Skewness

1. Karl Pearson Coeffeicient of Skewness

Karl Pearson’s proposed the following coefficient for the measurement of skewness:

Where:

Sk lies between - 1 and + 1.

When

Sk = 0, the distribution will be symmetrical.

Sk > 0, the distribution will be positively skewed.

Sk < 0, the distribution will be negatively skewed.

Sometimes the mode is ill-defined, then we use the following empirical relation:

Where:

Sk lies between -3 and + 3.

When

Sk = 0, the distribution will be symmetrical.

Sk>0, the distribution will be positively skewed.

Sk < 0, the distribution will be negatively skewed.

Bowley’s Coefficient of Skewness

A British statistician, Arthur Bowley, proposed the following formula for the measurement of skewness:

Where:

SB lies between -1 and +1.

When

SB = 0, the distribution will be symmetrical.

SB>0, the distribution will be positively skewed.

SB < 0, the distribution will be negatively skewed.

Fisher’s Coefficient of skewness

Sir Irving Fisher proposed the following formula for the measurement of skewness:

Example 4.18: Find the skewness in the following data, using

1. Pearsonian Method

2. Bowley’s Method  

Weight	Frequency
59.5 – 62.5	5
62.5 – 65.5	18
65.5 – 68.5	42
68.5 – 71.5	27
71.5 – 74.5	8

Solution:

Karl Pearson Coefficient of Skewness:

As Sk > 0, the distribution is positively skewed.

Bowley’s Coefficient of Skewness

Weight	Frequency	Cumulative Frequency
59.5 – 62.5	5	5
62.5 – 65.5	18	23
65.5 – 68.5	42	65
68.5 – 71.5	27	92
71.5 – 74.5	8	100
	100

SB > 0, the distribution is positive skewed.

Kurtosis

Kurtosis is the distribution of observed data around the means and describes the peak of a curve of a frequency distribution. It is measured by the second-moment ratio. 

Moments' Ratios

Moment’s ratios are used to describe the nature and shape of distribution.

Example 4.19: Compute the first four moments of the following frequency distribution and calculate the moment’s ratios given below:

Class	0 – 2	2 – 4	4 – 6	6 – 8	8 – 10	10 – 12
Frequency	8	5	6	9	4	2

Also discuss the shape of the distribution.

Solution:

Coefficient of Skewness:

Comments:

As γ1 = 0.05 > 0, the distribution is positive skewed.

β2 =1.93 < 0, the distribution is platykurtic. 

Example 4.20: The first four moments about point 4 of a distribution are -1.5, 17, -30, and 108. Calculate b1 and b2 and state that the distribution is leptokurtic or platykurtic.

Solution:

The four moments about 4 are given below:

Using the following relation to obtain mean moments:

The distribution is platykurtic.

• Read More: introduction to probability