Question

The following table gives the daily income of 50 workers of a factory:

Daily income (in Rs)	100-120	120-140	140-160	160-180	180-200
Number of workers	12	14	8	6	10

Find the mean, mode and median of the above data.

Answer 1

We have the following:

Daily income	Mid value $\left(x_i\right)$	Frequency $\left(f_i\right)$	Cumulative frequency	$f_i\times x_i$
100-120	110	12	12	1320
120-140	130	14	26	1820
140-160	150	8	34	1200
160-180	170	6	40	1020
180-200	190	10	50	1900
		∑$f_i=50$		∑$\left(f_i\times x_i\right)=$7260

Mean, $\overline{x}$=$\frac{\sum \left(f_i\times x_i\right)}{\sum f_i}$
=$\frac{7260}{50}$
=145.2

$$\begin{gathered} \mathrm{Here},N=50 \\ \Rightarrow \frac{N}{2}=25 \end{gathered}$$
The cumulative frequency just greater than 25 is 26 and the corresponding class is 120-140.
Thus, the median class is 120-140.
$Now,l=120,h=20,f=14,c=cf$of preceding class = 12 and$\frac{N}{2}=25$
Median, $M_e=l+\left\{h\times \frac{\left(\frac{N}{2}-c\right)}{f}\right\}$
$$\begin{gathered} =120+\left\{20\times \frac{\left(25-12\right)}{14}\right\} \\ =\left(120+20\times \frac{13}{14}\right) \end{gathered}$$
=138.57
Mode = 3(Median) $-$ 2(mean)
$$\begin{gathered} =3\times 138.57-2\times 145.2 \\ =125.31 \end{gathered}$$