Question

The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers, (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers.

$\begin{array}{ccccccc}\text{Daily Income (in Rs)}& \text{10 - 14}& 15-19& 20-24& 25-29& 30-34& 35-39\\ \text{Number of Workers}& 5& 10& 15& 20& 10& 5\end{array}$

Answer 1

$\begin{array}{ccc}\text{Daily income (in Rs)}& \text{Number of Workers}& \text{Cumulative Frequency}\\ \text{Class Interval}& \text{(f)}& \text{(cf)}\\ 9.5-14.5& 5& 5\\ 14.5-19.5& 10& 15\\ 19.5-24.5& 15& 30\\ 24.5-29.5& 20& 50\\ 29.5-34.5& 10& 60\\ 34.5-39.5& 5& 65\\ \text{Total}& \sum f=65& \end{array}$

(a) Highest income of lowest $50\%$ workers will be given by the median.

$\sum f=N=65$

Median class = 32.5th item.

32.5th item lies in the 50th cumulative frequency and the corresponding class interval is 24.5 - 29.5.

Median = $L+\frac{\frac{N}{2}-cf}{f}{X}_4$

$=24.5+\frac{32.5-30}{20}\times 5$

$=24.5+\frac{2.5}{20}\times 5$

=$Rs25.13$

(b) Minimum income earned by top $25\%$workers will be given by the lower quartile $Q_1.$

Class interval of $Q_1=\left(\frac{N}{4}\right)$ th item

$=\left(\frac{65}{4}\right)$ th item = 16.25 th item

16.25th item lies in yhe 30th cumulative frequency and the corresponding class interval is 19.5 - 24.5

$Q_1=L+\frac{\frac{N}{4}-cf}{f}{X}_4$

$=19.5+\frac{16.25-15}{15}\times 5$

$=19.5+\frac{1.25}{15}\times 5$

= Rs 19.92

(c) Maximum income earned by lowest 25% workers will be given by the upper quartile $Q_3$

Class interval of $Q_3=\left(\frac{N}{2}\right)$ th item

$=3\left(\frac{65}{4}\right)$ th item

$=3\times 16.25$ th item

= 48.75 th item

48.75th item lines in 50th item and the corresponding class interval is 24.5 - 29.5.

$Q_3=L+\frac{\frac{3N}{4}-cf}{f}{X}_i$

$=24.5+\frac{\frac{3\times 65}{4}-30}{20}\times 5$

=$24.5+\frac{\frac{195}{4}-30}{20}\times 5$

$=24.5+\frac{48.75-30}{20}\times 5$

$=Rs29.19$