Question

The table below shows the salaries of 280 persons:

Salary (in thousand Rs.)	No. of persons
5-10	49
10-15	133
15-20	63
20-25	15
25-30	6
30-35	7
35-40	4
40-45	2
45-50	1

Calculate the median salary of the data.

Answer 1

Hint: First check the total number of observations. If it is even then our median class is that class which includes the cumulative frequency \[\dfrac{N}{2}\] and \[\dfrac{N}{2}+1\] . Using this, decide the median class. Now use the formula, \[\text{Median= L +}\left( \dfrac{\dfrac{N}{2}-C.F}{f} \right)\times h\] and solve it further.

Complete step-by-step answer:

Salary (in thousand Rs.)	No. of persons(f)	Cumulative frequency
5-10	49	49
10-15	133	182
15-20	63	245
20-25	15	260
25-30	6	266
30-35	7	273
35-40	4	277
40-45	2	279
45-50	1	280

The value of N is the total number of persons.
N = 280 …………..(1)
We have to get the value of \[\dfrac{N}{2}\] . We have the value of N. So, using the value of N we can find the value of \[\dfrac{N}{2}\] .
\[\dfrac{N}{2}=\dfrac{280}{2}=140\] and \[\dfrac{N}{2}+1=141\] …………….(2)
As the total number of persons is 280 which is an even number. So, our median class is defined where the cumulative frequency value \[\dfrac{N}{2}\] and \[\dfrac{N}{2}+1\] lies.
Since the number 140 and 141 is less than 182, hence consider 182 cumulative frequency for the median class.
Now, the median class that we have to consider is 10-15.
We know the formula,
\[\text{Median= L +}\left( \dfrac{\dfrac{N}{2}-C.F}{f} \right)\times h\] ………………..(3)
Here, L is equal to the lower limit of the median class, N is equal to the number of observations, f is the frequency of the median class, C.F is the cumulative frequency of the class preceding the median class, and ‘h’ is the difference between the lower limit and upper limit of the median class.
From the table, we have to consider the class 10-15 as the median class.
L=10 …………(4)
C.F=49 …………..(5)
N = 280 ……………(6)
h=15-10=5 ……………..(7)
f=133 …………(8)
Now, from equation (3), equation (4), equation (5), equation (6), equation (7), and equation (8) we get
\[\text{Median= 10 +}\left( \dfrac{140-49}{133} \right)\times 5=13.42\] .
Hence, the median salary is Rs.13.42 thousand or Rs.13420.

Note: In this question, one might take C.F as the cumulative frequency of the median class which is wrong. Here, C.F is the cumulative frequency of the class preceding the median class.