# Bowley’s Coefficient of Skewness using R with examples

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

## Bowley'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.

For a symmetric distribution, the two quartiles namely $Q_1$ and $Q_3$ are equidistant from the median (i.e. $Q_2$).

### Symmetric Distribution

If the distance of $Q_3$ from $Q_2$ is equal to the distance of $Q_2$ from $Q_1$, the distribution is symmetric (or not skewed) (i.e., $Q_3-Q_2 = Q_2-Q_1$).

### Positively Skewed Distrbiution

If the distance of $Q_3$ from $Q_2$ is greater than the distance of $Q_2$ from $Q_1$, the distribution is positively skewed (i.e., $Q_3-Q_2 > Q_2-Q_1$).

### Negatively Skewed Distrbiution

If the distance of $Q_3$ from $Q_2$ is less than the distance of $Q_2$ from $Q_1$, the distribution is positively skewed (i.e., $Q_3-Q_2 < Q_2-Q_1$).

The absolute measure of skewness is $(Q_3-Q_2)-(Q_2-Q_1)= Q_3+Q_1-2*Q_2$.

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

$$S_b = \dfrac{Q_3+Q_1 - 2Q_2}{Q_3 -Q_1}$$

where,

• $Q_1$ is the first quartile,
• $Q_2$ is the second quartile,
• $Q_3$ is the third quartile,

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

## Bowley's Coefficient of Skewness Interpretation

• If $S_b < 0$, i.e., $Q_3-Q_2 < Q_2-Q1$ then the distribution is negatively skewed.
• If $S_b = 0$, i.e., $Q_3-Q_2 = Q_2-Q1$ then the distribution is Symmetric or not skewed.
• If $S_b > 0$, i.e., $Q_3-Q_2 > Q_2-Q1$ then the distribution is positively skewed.

## Numerical Problem Bowley's Skewness Using R

### Example 1 : Bowley'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 Bowley'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)
# box-plot of the data
boxplot(x,horizontal = TRUE,col="lightblue")
Q <- quantile(x,probs=c(0.25,0.50,0.75),type=6,names = FALSE)
Q
[1] 132.25 137.00 145.50

From the above box-plot it is clear that the distance of $Q_2$ from $Q_1$ is $Q_2 - Q_1 = 137 - 132.25 = 4.75$ and the distance of $Q_2$ from $Q_3$ is $Q_3 - Q_2 = 145.5 - 137 = 8.5$.

As $Q_3 - Q_2 > Q_2 - Q_1$, the data is positively skewed.

The Bowley's coefficient of Skewness is

S_b <- (Q[3]+Q[1]-2*Q[2])/(Q[3]-Q[1])
S_b
[1] 0.2830189

 \begin{aligned} S_b &= \dfrac{Q_3+Q_1 - 2Q_2}{Q_3 -Q_1}\\ &=\frac{145.5+132.25-2*137}{145.5- 132.25}\\ &=0.2830189 \end{aligned}

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

### Example 2: Bowley'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 Bowley'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)
boxplot(DBP,horizontal = TRUE,col="lightpink")
Q_BP <- quantile(DBP,probs=c(0.25,0.50,0.75),type=6,names = FALSE)
Q_BP
[1] 65.75 73.50 78.25

From the above box-plot it is clear that the distance of $Q_2$ from $Q_1$ is $Q_2 - Q_1 = 73.5 - 65.75 = 7.75$ and the distance of $Q_2$ from $Q_3$ is $Q_3 - Q_2 = 78.25 - 73.5 = 4.75$.

As $Q_3 - Q_2 < Q_2 - Q_1$, the data is negatively skewed.

The Bowley's coefficient of Skewness is

S_b <- (Q_BP[3]+Q_BP[1]-2*Q_BP[2])/(Q_BP[3]-Q_BP[1])
S_b
[1] -0.24

 \begin{aligned} S_b &= \dfrac{Q_3+Q_1 - 2Q_2}{Q_3 -Q_1}\\ &=\frac{78.25+65.75-2*73.5}{78.25- 65.75}\\ &=-0.24 \end{aligned}

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

## Endnote

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