Question

The daily wages (in rupees) of 100 workers in a factory are given below:

Daily wages (in Rs)	125	130	135	140	145	150	160	180
No. of workers	6	20	24	28	15	4	2	1

Find the median wages of a worker (in the nearest integer value).

Answer 1

Hint: The question consists of a discrete frequency distribution. Here, the number of workers is the frequency. First, we need to calculate cumulative frequency to get the total frequency. Since there are 100 workers in the factory, the total frequency will be 100. As total frequency is even,
\[\text{median}=\dfrac{{{\left( \dfrac{n}{2} \right)}^{\text{th}}}\text{observation}+{{\left( \dfrac{n}{2}+1 \right)}^{\text{th}}}\text{observation}}{2}\]
Here, n is the total frequency. The observation (daily wages) corresponding to cumulative frequency is taken.

Complete step-by-step answer:
First, construct a table with a column for daily wages, number of workers and cumulative
Frequency as follows:
The cumulative frequency is calculated by adding each frequency from a frequency distribution table to the sum of its predecessors. The last value will always be equal to the total for all observations, since all frequencies will already have been added to the previous total.
     

Therefore, N =100(even).
Median will be calculated as:
\[ \text{median}=\dfrac{{{\left( \dfrac{n}{2} \right)}^{\text{th}}}\text{observation}+{{\left( \dfrac{n}{2}+1 \right)}^{\text{th}}}\text{observation}}{2} \]
\[ \text{median}=\dfrac{{{\left( \dfrac{100}{2} \right)}^{\text{th}}}\text{observation}+{{\left( \dfrac{100}{2}+1 \right)}^{\text{th}}}\text{observation}}{2} \] 
\[ \text{median}=\dfrac{\text{50th observation}+51\text{st observation}}{2} \]
\[ \text{median}=\dfrac{135+140}{2} \]
\[ \text{median}=\dfrac{275}{2} \] 
\[ \text{median}=137.5 \]
Therefore, the median wage of a worker is Rs. 137.50.

Note: The final cumulative frequency is always equal to the total frequency of the data. If the final cumulative frequency does not equal the total frequency, the solution will be incorrect. Always remember to take the observation corresponding to the cumulative frequency and not the given frequency. If the number of observations does not correspond to any cumulative frequency on the table, the observation corresponding to the next higher cumulative frequency is taken. For example, for the 51st observation, the observation corresponding to 78 was taken.