Question

Calculate Bowley’s coefficient of skewness for the following distribution of weekly wage of workers.

Wages	Number of workers
Below 300	5
300-400	8
400-500	18
500-600	35
600-700	27
Above 700	7

Answer 1

Hint: To find the Bowley’s coefficient of skewness, we first need to calculate three quartiles at $\dfrac{N}{4}$, $\dfrac{N}{2}$ and $\dfrac{3N}{4}$ respectively. Then, we can calculate the Bowley’s coefficient of skewness using the formula $S{{K}_{B}}=\dfrac{{{Q}_{3}}+{{Q}_{1}}-2{{Q}_{2}}}{{{Q}_{3}}-{{Q}_{1}}}$.

Complete step-by-step answer:
To calculate the Bowley’s coefficient of skewness, we must first form the following table,

Class	Frequency	Cumulative frequency
Below 300	5	5
300-400	8	13
400-500	18	31
500-600	35	66
600-700	27	93
Above 700	7	100

Here, we can see that N = 100 and class interval, h = 100.
We know that ${{Q}_{1}}$ = value of ${{\left( \dfrac{N}{4} \right)}^{\text{th}}}$ observation.
Thus, ${{Q}_{1}}$ = value of ${{25}^{\text{th}}}$ observation.
From the cumulative frequency column, we can see that the ${{25}^{\text{th}}}$ observation belongs to the class 400-500.
For this class, the frequency f = 18, cumulative frequency of previous class, CF = 13 and the lower limit for this class, L = 400.
Thus, we can write
${{Q}_{1}}=L+\dfrac{h\left( \dfrac{N}{4}-CF \right)}{f}$
Putting, the values, we get
${{Q}_{1}}=400+\dfrac{100\left( 25-13 \right)}{18}$
$\Rightarrow {{Q}_{1}}=466.67$
We know that ${{Q}_{2}}$ = value of ${{\left( \dfrac{N}{2} \right)}^{\text{th}}}$ observation.
Thus, ${{Q}_{1}}$ = value of ${{50}^{\text{th}}}$ observation.
From the cumulative frequency column, we can see that the ${{50}^{\text{th}}}$ observation belongs to the class 500-600.
For this class, the frequency f = 35, cumulative frequency of previous class, CF = 31 and the lower limit for this class, L = 500.
Thus, we can write
${{Q}_{2}}=L+\dfrac{h\left( \dfrac{N}{2}-CF \right)}{f}$
Putting, the values, we get
${{Q}_{2}}=500+\dfrac{100\left( 50-31 \right)}{35}$
$\Rightarrow {{Q}_{2}}=554.29$
We know that ${{Q}_{3}}$ = value of ${{\left( \dfrac{3N}{4} \right)}^{\text{th}}}$ observation.
Thus, ${{Q}_{3}}$ = value of ${{75}^{\text{th}}}$ observation.
From the cumulative frequency column, we can see that ${{75}^{\text{th}}}$ observation belongs to the class 600-700.
For this class, the frequency f = 27, cumulative frequency of previous class, CF = 66 and the lower limit for this class, L = 600.
Thus, we can write
${{Q}_{3}}=L+\dfrac{h\left( \dfrac{3N}{4}-CF \right)}{f}$
Putting, the values, we get
${{Q}_{3}}=600+\dfrac{100\left( 75-66 \right)}{27}$
$\Rightarrow {{Q}_{3}}=633.33$
We know that Bowley’s coefficient of skewness is defined as
$S{{K}_{B}}=\dfrac{{{Q}_{3}}+{{Q}_{1}}-2{{Q}_{2}}}{{{Q}_{3}}-{{Q}_{1}}}$
\[\Rightarrow S{{K}_{B}}=\dfrac{633.33+466.67-2\left( 554.29 \right)}{633.33-466.67}\]
\[\Rightarrow S{{K}_{B}}=\dfrac{1100-1108.58}{166.67}\]
\[\Rightarrow S{{K}_{B}}=-0.05\]
Thus, the Bowley’s coefficient of skewness is -0.05

Note: We must remember during the calculation of ${{Q}_{1}}$, ${{Q}_{2}}$ and ${{Q}_{3}}$ that the cumulative frequencies of their previous classes are taken into account. If the skewness is positive, it is said to be positively skewed, and if the skewness is negative, it is said to be negatively skewed.