Question

The median of the following data is 50. Find the values of p and q , if the sum of all the frequencies is 90 .

Marks: 20-30 30-40 40-50 50-60 60-70 70-80 80-90

Frequency: p 15 25 20 q 8 10

Answer 1

The given series is in inclusive form. Converting it to exclusive form and preparing the cumulative frequency table, we have

Class interval	Frequency(fi )	Cumulative Frequency (c.f.)
20–30	p	p
30–40	15	p + 15
40–50	25	p + 40
50–60	20	p + 60
60–70	q	p + q + 60
70–80	8	p + q + 68
80–90	10	p + q + 78
	78 + p + q = 90

Median = 50
It lies in the interval 50–60, so the median class is 50–60.
Now, we have
$l=50,h=10,f=20,F=p+40,N=90$
We know that

$$\begin{gathered} 50=50+\frac{45-\left(p+40\right)}{20}\times 10 \\ \Rightarrow 0=\frac{5-p}{2} \\ \Rightarrow p=5 \\ \mathrm{And}, \\ p+q+78=90 \\ \Rightarrow p+q=12 \\ \Rightarrow q=12-5=7 \end{gathered}$$