Question

Calculate the median from the following data:

Rent (In Rs.)	15 – 25	25 – 35	35 – 45	45 – 55	55 – 65	65 – 75	75 – 85	85 – 95
No. of houses	8	10	15	25	40	20	15	7

Answer 1

Hint: First of all we will re – create the table using the data given in the question and adding a row or column of C.F. (cumulative frequency) in it. After that, we just have to put in the required values in the formula of the median and thus, we have the answer.

Complete step-by-step solution:
Let us first of all re – create the table with the required cumulative frequency and the data already given to us in the table in the question.

Rent (In Rs.)	No. of houses	C.F. (Cumulative Frequency)
15 – 25	8	8
25 – 35	10	10 + 8 = 18
35 – 45	15	18 + 15 = 33
45 – 55	25	33 + 25 = 58
55 – 65	40	58 + 40 = 98
65 – 75	20	98 + 20 = 118
75 – 85	15	118 + 15 = 133
85 – 95	7	133 + 7 = 140
Total	140

The students must note that cumulative frequency of some data gives the information of frequency up to it. Therefore, when we wrote cumulative frequency in row with 25 – 35 as 18, it means up to the rent 35, we total have 18 houses because it also includes the houses which have rent 15 – 25. Similarly going on, we keep adding the new frequency in previous C. F.
Now, let us first write the formula of median of grouped data.
The median is given by the following formula:-
$ \Rightarrow Median = l + \dfrac{{\dfrac{N}{2} - C.F.}}{f} \times h$, where l is the lower limit of the class where the media lies, N is the total number of objects encountered, C.F. is the cumulative frequency of the data previous to the interval where the median lies, f is the frequency of the interval where median lies and h is the the height of the class intervals throughout the table.
Here, we have N = 140.
$\therefore \dfrac{N}{2} = \dfrac{{140}}{2} = 70$
We see that 98 is just greater than 70. Therefore, the median lies in the class 55 – 65.
Therefore, looking at it, we see the lower limit of the class = l = 55, height of the class = h = 65 – 55 = 10, f is the frequency of this class = f = 40.
Now, let us put in all these values in the formula mentioned above.
$ \Rightarrow Median = 55 + \dfrac{{70 - 58}}{{40}} \times 10$
Simplifying the fractional part on the RHS, we will then get:-
$ \Rightarrow Median = 55 + \dfrac{{28}}{4}$
Simplifying the fractional part further on the RHS, we will then get:-
$ \Rightarrow Median = 55 + 7 = 62$
Hence, the median of the data given is 62.

$\therefore $ The required median of the data is 62.

Note: The median is basically the middle of the set of the numbers. The students might get confused between the terms “mean” and “median”. Mean refers to the average and median refers to the middle value of the set.
Now, let us understand how to calculate where the median lies. Like in the solution when we calculated $\dfrac{N}{2}$ to be 70 and then decided. We first calculate $\dfrac{N}{2}$, then we check its value in the table above, the class with Cumulative frequency just above it will be the class where the median lies.