Question

The given table shows the salaries of 81 people in thousands. Calculate the Median from the data given.

$\begin{array}{cc}\text{Salary(in thousands)}& \text{No. of people}\\ 5-10& 7\\ 10-15& 12\\ 15-20& 15\\ 20-25& 8\\ 25-30& 11\\ 30-35& 18\\ 35-40& 10\end{array}$

Answer 1

The Cumulative frequency table is given as :

$\begin{array}{cc}\text{Salary(in thousands)}& \text{Cumulative frequency}\\ <10& 7\\ <15& 19\\ <20& 34\\ <25& 42\\ <30& 53\\ <35& 71\\ <40& 81\end{array}$

N = 81 (An odd number)

Median = $\frac{(81+1)}{2}th$ observation = 41st observation.

So, the Median class is 20000 - 25000.

We know that there are 8 people from the 34th to the 42nd person whose salary is between ₹20000 and ₹25000.

However, we do not know their individual salaries.

We will divide ₹5000 from ₹20000 to ₹25000 into 8 equal parts and assume that one person is in each subdivision.

We will assume that the salary of each person in a subdivision is the mid - value of the subdivision.

So, the salary of the 35th person is the mid value of ₹20000 and ₹$20000\frac{5000}{8}$, that is ₹$20000\frac{5000}{16}$

So, from the 35th person to the 41st person, there are 6 people.

If the 35th term of the AP is $20000\frac{5000}{16}$ and the common difference is $\frac{5000}{8}$,

Then the 41st term = $20000\frac{5000}{16}$ + $6\times \frac{5000}{8}$

= $20000+\frac{5000+60000}{16}$

= $20000+\frac{65000}{16}$

= 20000 + 4062.5

= ₹24062.5