Question

If the median of the distribution given below is 28.5, find the values of x and y

$\begin{array}{cc}\text{Class Interval}& \text{Frequency}\\ 0-10& 5\\ 10-20& x\\ 20-30& 20\\ 30-40& 15\\ 40-50& y\\ 50-60& 5\\ Total& 60\end{array}$

Answer 1

$\begin{array}{ccc}\text{Class Interval}& \text{Frequency}& Cumulative\\ & & Frequency\\ 0-10& 5& 5\\ 10-20& x& 5+x\\ 20-30& 20& 25+x\\ 30-40& 15& 40+x\\ 40-50& y& 40+x+y\\ 50-60& 5& 45+x+y\\ Total& 60& \end{array}$

$n=60$

$\Rightarrow 45+x+y=60$

$\Rightarrow x+y=60-45$

$\Rightarrow x+y=15\dots \dots \left(1\right)$

The median is 28.5, which lies in the class 20 - 30

So, l = 20, f=20, cf=5+x, h=10

$\therefore Median=l+\left(\frac{\frac{n}{2}-cf}{f}\right)\times h$

$\Rightarrow 28.5=20+\left\{\frac{\frac{60}{2}-(5+x)}{20}\right\}\times 10\Rightarrow 28.5=20+\frac{25-x}{2}$

$\Rightarrow \frac{25-x}{2}=28.5-20\Rightarrow \frac{25-x}{2}=8.5$

$\Rightarrow 25-x=8.5\times 2\Rightarrow 25-x=17$

$\Rightarrow x=25-17=8\dots \dots \left(2\right)$

From (1) and (2), 8 + y = 15

$\Rightarrow y=15-8=7$

Hence, the values of x and y are 8 and 7 respectively