Kelly's coefficient of Skewness using R with examples

In this tutorial, you will learn about what is Kelly's coefficient of skewness and how to calculate Kelly's coefficient of skewness in R.

Kelly's Coefficient of Skewness

Skewness is a measure of symmetry. The meaning of skewness is "lack of symmetry". Skewness gives us an idea about the concentration of higher or lower data values around the central value of the data.

Kelly's coefficient of skewness is based on deciles or percentiles of the data. The Bowley's coefficient of skewness is based on the middle 50 percent of the observations of data set. It means the Bowley's coefficient of skewness leaves the 25 percent observations in each tail of the data set.

Kelly suggested a measure of skewness which is based on middle 80 percent of the observations of data set

For a symmetric distribution, the $1^{st}$ decile $D_1$ and $9^{th}$ decile $D_{9}$ is equidistant from $5^{th}$ decile $D_5$ (i.e., median).

Symmetric Distribution

If the distance of $D_9$ from $D_5$ is equal to the distance of $D_5$ from $D_1$ , the distribution is symmetric (or not skewed) (i.e., $D_9-D_5 = D_5-D_1$ ).

Positively Skewed Distrbiution

If the distance of $D_9$ from $D_5$ is greater than the distance of $D_5$ from $D_1$ , the distribution is positively skewed (i.e., $D_9-D_5 > D_5-D_1$ ).

Negatively Skewed Distrbiution

If the distance of $D_9$ from $D_5$ is less than the distance of $D_5$ from $D_1$ , the distribution is positively skewed (i.e., $D_9-D_5 < D_5-D_1$ ).

The absolute measure of skewness is $(D_9-D_5)-(D_5-D_1)= D_9+D_1-2*D_5$ .

Kelly's coefficient of skewness is the relative measure of skewness. It is denoted by $S_k$ and is defined as

$$S_k = \dfrac{D_9+D_1 - 2D_5}{D_9 -D_1}$$

where,

$D_1$ is the first decile,
$D_5$ is the fifth decile,
$D_9$ is the ninth decile,

To know more about deciles, check the tutorial on how to compute quantiles using R with examples.

Kelly's Coefficient of Skewness Interpretation

If $S_k < 0$, i.e., $D_9-D_5 < D_5-D_1$ then the distribution is negatively skewed.
If $S_k = 0$, i.e., $D_9-D_5 = D_5-D_1$ then the distribution is Symmetric or not skewed.
If $S_k > 0$, i.e., $D_9-D_5 > D_5-D_1$ then the distribution is positively skewed.

Numerical Problem Pearson's Skewness Using R

Example 1 : Kelly's Coefficient of Skewness using R

The following data are the heights, correct to the nearest centimeters, for a group of children:
126, 129, 129, 132, 132, 133, 133, 135, 136, 137,
137, 138, 141, 143, 144, 146, 147, 152, 154, 161
Find Kelly's coefficient of skewness and interprete the result.

# create a data vector
x <- c(126, 129, 129, 132, 132, 133, 133, 135, 136, 137,
       137, 138, 141, 143, 144, 146, 147, 152, 154, 161)

D <- quantile(x,probs=c(0.10,0.50,0.90),type=6,names = FALSE)
D

[1] 129.0 137.0 153.8

# box-plot of the data
boxplot(x,horizontal = TRUE,col="lightblue")
segments(y0=0.5,x0=D[1],y1=1.5,x1=D[1],col="red",lwd=2)
segments(y0=0.5,x0=D[3],y1=1.5,x1=D[3],col="blue",lwd=2)

From the above box-plot it is clear that the distance of $D_5$ from $D_1$ is $D_5 - D_1 = 137 - 129 = 8$ and the distance of $D_5$ from $D_9$ is $D_9 - D_5 = 153.8 - 137 = 16.8$.

As $D_9 - D_5 > D_5 - D_1$, the data is positively skewed.

The Kelly's coefficient of Skewness is

S_k <- (D[3]+D[1]-2*D[2])/(D[3]-D[1])
S_k

[1] 0.3548387

$$ \begin{aligned} S_k &= \dfrac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ &=\frac{153.8+129-2*137}{153.8- 129}\\ &=0.3548387 \end{aligned} $$

The Kelly's coefficient of skewness $S_k > 0$. The distribution of height (in cm) is $\text{positively skewed}$.

Example 2: Kelly's Coefficient of Skewness using R

Diastolic blood pressure (in mmHg) of a sample of 18 patients admitted to the hospitals are as follows:
65,76,64,73,74,80,71,68,66,
81,79,75,70,62,83,63,77,78.
Find Kelly's coefficient of skewness and interprete the result.

DBP <-c(65,76,64,73,74,80,71,68,66,81,79,75,70,62,83,63,77,78)

D_BP <- quantile(DBP,probs=c(0.10,0.50,0.90),type=6,names = FALSE)
D_BP

[1] 62.9 73.5 81.2

boxplot(DBP,horizontal = TRUE,col="lightpink")
segments(y0=0.5,x0=D_BP[1],y1=1.5,x1=D_BP[1],col="red",lwd=2)
segments(y0=0.5,x0=D_BP[3],y1=1.5,x1=D_BP[3],col="blue",lwd=2)

From the above box-plot it is clear that the distance of $D_5$ from $D_1$ is $D_5 - D_1 = 73.5 - 62.9 = 10.6$ and the distance of $D_5$ from $D_9$ is $D_9 - D_5 = 81.2 - 73.5 = 7.7$.

As $D_9 - D_5 < D_5 - D_1$, the data is negatively skewed.

The Kelly's coefficient of Skewness is

S_k <- (D_BP[3]+D_BP[1]-2*D_BP[2])/(D_BP[3]-D_BP[1])
S_k

[1] -0.1584699

$$ \begin{aligned} S_k &= \dfrac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ &=\frac{81.2+62.9-2*73.5}{81.2- 62.9}\\ &=-0.1584699 \end{aligned} $$

The Kelly's coefficient of skewness $S_k < 0$. The distribution of Diastolic Blood Pressure is $\text{negatively skewed}$.

Endnote

In this tutorial you learned about what is Kelly's coefficient of Skewness and how to calculate Kelly's coefficient using R.

To learn more about descriptive statistics using R, please refer to the following tutorials:

Hopefully you enjoyed learning this tutorial on how to compute Kelly's coefficient of skewness using R.

Kelly’s coefficient of Skewness using R with examples