Question

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

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

Answer 1

We may find the cumulative frequency for the given data as following:

$\begin{array}{ccc}Classinterval& Frequency& Cumulativefrequency\\ 0-10& 5& 5\\ 10-20& x& 5+x\\ 20-30& 20& 25+x\\ 30-40& 15& 40+x\\ 40-50& y& 40+x+y\\ 50-60& 5& 45+x+y\\ Total\left(n\right)& 60& \end{array}$

It is clear that, n = 60

45 + x + y = 60

x + y = 15 (1)

Median of data is given as 28.5 which lies in interval 20 - 30.

So, median class = 20 - 30

Lower limit $l$ of median class = 20

Cumulative frequency $c_f$ of class preceding the median class = 5 + x

Frequency $f$ of median class = 20

Class size $h$ = 10

$Now,median=l+\left(\frac{\frac{n}{2}-cf}{f}\right)\times h28.5=20+\left[\frac{\frac{60}{2}-(5+x)}{20}\right]\times 108.5=\left(\frac{25-x}{2}\right)$

17 = 25 - x

x = 8

From equation (1)

8 + y = 15

y = 7

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