Skewness.

Nature of curve.

There are two methods of statistical analysis in research which is based on the nature of curve and distribution. They are.

1. Skewness.
2. Kurtosis

Skewness.

Skewness is the measure of description statistical which is used to measure the shape of the curve drawn from the frequency distribution to measure the direction of variation. It is the lack of symmetry. Based on this observation there are two types of Skewness. They are.

1. No Skewness. (or Symmetrical).
2. Positive Skewness.
3. Negative Skewness.

A short description of the types of skewness is summarised as below.

1. No skewness or Symmetrical.

In this case, A distribution of data is said to be no skewness if the curve drawn from the data is neither elongated more to the left side nor to the right side. In this case, the curve is equally elongated to the right as well as to the left side. In other words, it is said to have no skewness if mean=median=mode.

2. Positive skewness.

In this case, a distribution of data is said to be positive skewness or right skewed if the curve drawn from the data is more elongated to the right side.

In other words, it is said to have positive Skewness or right skewed if mean<median<mode.

3. Negative skewness.

In this case, a distribution is said to be negative skewness or left skewed if the curve is drawn from the data is more elongated to the left side. In other words, it is said to have negative skewness or left skewed if mean>median>mode.

Bowley’s measure of Skewness.

The absolute measure of Skewness is

$$Skewness=(Q_3-Md)-(Md-Q_1)=Q_3+Q_1-2Md$$

$$Where\,,Q_1\,,Q_3\,,and\,Md\,have\,usual\,meaning$$

The relative Bowley’s measure of skewness is known as Bowley’s coefficient of Skewness and is given by.

$$S_1(B)=\frac{(Q_3-Md)-(Md-Q_1)}{(Q_3-Md)+(Md-Q_1)}$$

$$S_1(B)=\frac{Q_3+Q_1-2Md}{Q_3-Q_1}$$

This coefficient of Skewness is also known as quartile coefficient of Skewness.

Note.

Bowley's coefficient of skewness generally lies between -1 and +1 i,e -1≤S_k(B)≤+1.For the distribution with open-ended classes or distribution having ill-defined mode and distribution with extreme observation, this measure is particularly useful.

Interperationof calculating value of Bowley's coefficient of Skewness.

$$If\,S_k(B)=0\,,then\,this\,shows\,that\,the\,distribution\,is\,symmetrical$$

$$If\,S_k(B)>0\,,then\,this\,shows\,that\,the\,distribution\,is\,Positively\,skewed$$

$$If\,S_k(B)<0\,,then\,this\,shows\,that\,the\,distribution\,is\,Negative\,skewed$$

Karl Pearson’s Measure of Skewness.

The absolute measure of skewness are.

1. Skewness=mean-mode.
2. Skewness=mean-median.

The absolute measures of skewness are not widely used because they are expressed in terms of the original unit of data. The relative measure of skewness is known as the coefficient and used more frequently.

The coefficient of Skewness (S_k) defined by Karl Pearson is given by.

$$S_k(P)=\frac{Mean\,-Mode}{Standard\,deviation}$$

$$=\frac{\overline{X}-M_0}{σ}......(1)$$

If the mode is ill-defined or in a terminal, then the coefficient of skewness becomes.

$$S_k(P)=\frac{3(Mean\,-Mode)}{Standard\,deviation}$$

$$=\frac{\overline{3(X}-M_0)}{σ}......(2)$$

Note.

The pearson's coeficient of skewness obtained from (1) generally lies between -1to +1 i,e -1≤ Sk(p)≤+1. But the coeficient obtained from 2 generally lies between -3to +3 i,e -3≤Sk(p)≤+3

Interperationof calculating value of coefficient of Skewness.

$$If\,S_k(P)=0\,,then\,this\,shows\,that\,the\,distribution\,is\,symmetrical$$

$$If\,S_k(P)>0\,,then\,this\,shows\,that\,the\,distribution\,is\,Positively\,skewed$$

$$If\,S_k(P)<0\,,then\,this\,shows\,that\,the\,distribution\,is\,Negative\,skewed$$

A measure of Skewness Based on Moment.

Karl Pearson defined the following coefficient of skewness based on moments. These coefficient are also known as moment coefficient of skewness.

$$β_1=\frac{µ_3^2}{µ_2^3}$$

Ifµ₃=0, the distribution is symmetrical which implies thatβ₁=0

Ifµ₃>0 , then β1>0, then the distribution no doubt positively skewed.

If µ₃<0, then β1>0, however, the distribution is negative skewed. At this point, this formula has a strong limitation. The limitation is that for having negatively skewed distribution,β₁shows positive skewness.

Forµ₃<β₁ show positive skewness, which is not true. In order to overcome this limitation, Karl Pearson gave another coefficient of skewness based on moments as below.

$$γ_1=±{\sqrtβ _1}$$

$$=\sqrt\frac{µ_3^2}{µ_2^3}$$

This coefficient is equally applicable for any value of µ₃ (i,e., forµ₃ <0,µ₃>0 andµ₃=0).

Interpretation.

If µ_₃=0 i,e ,β_1 =0 i,e γ_1=0, then the distribution is symmetry

If µ_₃ <0 i,e γ_1<0, then the distribution is negatively skewed.

If µ__{3 >}0 i,e γ_1>0, then the distribution is positively skewed

Reference.

Kerlinger, F.N. Foundation of Behavioural Research. New Delhi: Surjeet Publication, 2000.

Kothari, C.R. Research Methodology. India: Vishwa Prakashan, 1990.

Singh, M.L. and J.M Singh. Understanding Research Methodology. 1998.

Singh, Mrigendra Lal. Understanding Research Methodology. Nepal: National Book centre, 2013.