Question

Consider the following distribution of daily wages of a factory.

Daily wages (in Rs.)	500 – 520	520 – 540	540 – 560	560 – 580	580 – 560
Number of workers	12	14	8	6	10

Find the mean daily wages of the workers of the factory by using the appropriate method.

Answer 1

Hint: Here we are given the daily wages and number of workers falling in that range. To find mean we will directly use a formula to find mean. But we also have to find the middle value of the daily wages range. This middle value acts as a representative of the other frequencies falling in that class.

Complete step-by-step solution:
Now we will use the direct method for calculation. Let’s tabulate the data.
The middle value or class mark of a frequency range is given by,
Mid-value $ = \dfrac{{upper\,limit + lower\,limit}}{2}$
The frequency table is,

Daily Expense	No. of households (${f_i}$)	Mid value (${x_i}$)	${f_i}{x_i}$
500 – 520	12	510	6120
520 – 540	14	530	7420
540 – 560	8	550	4400
560 – 580	6	570	3420
580 – 600	10	590	5900
Total	$\sum {{f_i}} = 50$		$\sum {{f_i}{x_i}} = 27260$

We know that the general formula to find the mean value is,
Mean $ = \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{x_i}} }}$
Now, we will substitute the value for the sum of the product of frequency and midpoint and the value for the sum of total frequency.
$ \Rightarrow $ Mean $ = \dfrac{{27260}}{{50}}$
Divide numerator by the denominator,
$\therefore $ Mean $ = 545.2$

Hence the daily wages of workers is $545.2$.

Note: Here in this problem data given of classes are in grouped form. No direct numbers are given so do find the middle values or class mark of the range. And then proceed for calculations. Always tabulate these types of problems.
In the mean formula, while computing $\sum {fx} $, don’t take the sum of $f$ and $x$ separately and then multiply them. It will be difficult. Students should carefully make the frequency distribution table; there are high chances of making mistakes while copying and computing data.