Question

Daily wages of 110 workers, obtained in a survey, are tabulated below:

Daily wages (in Rs.)	100-120	120-140	140-160	160-180	180-200	200-220	220-240
Number of workers	10	15	20	22	18	12	13

Compute the mean daily wages and modal daily wages of these workers.

Answer 1

Hint: We will calculate the mean of the $\overline x = A + \dfrac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }} \times h$, where $A$ is the assumed mean and $h$ is the size of interval. Next, we will find the mode of the grouped data using the formula, $l + \dfrac{{{f_1} - {f_0}}}{{2{f_i} - {f_0} - {f_2}}} \times h$, where $l$ is the lower limit of modal class, $h$ is the size of interval, ${f_1}$ is the frequency of modal class, ${f_0}$ is the frequency of previous class and ${f_2}$ is the frequency of succeeding the modal class.

Complete step-by-step answer:
We will first find the middle value of the given class.
That is, the half of sum of lower and upper limit of a class interval.
Let the assumed mean be corresponding to the class interval 160-180.
We will denote assumed mean by $A$

Class interval	${f_i}$	${x_i}$	${d_i} = {x_i} - A$	${u_i} = \dfrac{{{x_i} - A}}{{20}}$	${f_i}{u_i}$
100-120	10	110	$ - 60$	$ - 3$	$ - 30$
120-140	15	130	$ - 40$	$ - 2$	$ - 30$
140-160	20	150	$ - 20$	$ - 1$	$ - 20$
160-180	22	170($A$)	0	0	0
180-200	18	190	20	1	18
200-220	12	210	40	2	24
220-240	13	230	60	3	39
	110				1

Now, apply the formula to find mean $\overline x = A + \dfrac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }} \times h$, where h is calculated by subtracting lower limit from upper limit.
$h = 120 - 100 = 20$
On substituting the values, we get,
$
  \overline x = 170 + \dfrac{1}{{110}} \times 20 \\
   \Rightarrow \overline x = \dfrac{{1870 + 2}}{{11}} \\
   \Rightarrow \overline x = 170.2 \\
$
Therefore, the mean class of the given data is 170.2
Now, we will calculate the modal class of the given data.
Let ${f_1} = 22$ corresponding to class 160-180 as it is the highest value of frequency, then the previous frequency will be ${f_0} = 20$ and ${f_2} = 18$.
The formula for calculating mode will be $l + \dfrac{{{f_1} - {f_0}}}{{2{f_i} - {f_0} - {f_2}}} \times h$, where $l$ corresponds to the lower limit of modal class.
On substituting the values, we will get,
\[160 + \dfrac{{22 - 20}}{{2\left( {22} \right) - \left( {20} \right) - \left( {18} \right)}} \times 20 = 160 + \dfrac{{20}}{3} = \dfrac{{500}}{3} = 166.67\]
Hence, the modal class is 166.67.
Note: Here, we have continuous frequency distribution as there are no gaps between the consecutive classes. We cannot calculate mode in group data by just looking the maximum number of times a number is repeated. Students must learn the formula and how to substitute values in the formula.