Question

Below Given Are The Daily Wages Of $110$ Workers, Obtained In A Survey. Compute The Mean Daily Wages And Modal Daily Wages Of These Workers.

$=18$

Answer 1

Given data:

Daily wages(in Rs)	100-120	120-140	140-160	160-180	180-200	200-220	220-240
Number of workers	10	15	20	18	18	12	13

Daily Wages (in Rs.)	$NumberOfWorkers\left(f_i\right)$	Class Marks (xi)	xi– 170	ui = $\frac{x_i-70}{20}$	fi ui
$100-120$	$10$	$\frac{100+120}{2}=110$	$110-170=-60$	$\frac{-60}{20}=-3$	$10\times (-3)=-30$
$120-140$	$15$	$\frac{120+140}{2}$$=130$	$130-170=-40$	$\frac{-40}{20}=-2$	$15\times (-2)=-30$
$140-160$	$20$	$\frac{140+160}{2}=150$	$150-170=-20$	$\frac{-20}{20}=-1$	$20\times (-1)=-20$
$160-180$	$22$	$\frac{160+180}{2}=170$	$170-170=0$	$\frac{0}{20}=0$	$22\times 0=0$
$180-200$	$18$	$\frac{180+200}{2}=190$	$190-170=20$	$\frac{20}{20}=1$	$18\times 1=18$
$200-220$	$12$	$\frac{200+220}{2}=210$	$210-170=40$	$\frac{40}{20}=2$	$12\times 2=24$
$220-240$	$13$	$\frac{220+240}{2}=230$	$230-170=60$	$\frac{60}{20}=3$	$13\times 3=39$
∑ fi = $110$	∑ fi ui= 1

Here, $u_i=\frac{x_i-a}{h}$

where $a$is assumed mean i.e., $70$.

Mean ($\overline{x}$) $=a+h\times \frac{\sum f_i{x}_i}{\sum f_i}$

where,

$a$ is assumed mean = $170$

$h$ is class interval = $20$

∑ f_i = $110$

∑ f_i u_i= 1

Now by substituting the values, we get:

$\Rightarrow \overline{x}=170+20\times \frac{1}{110}$

$\Rightarrow \overline{x}=170+0.18$

$\Rightarrow \overline{x}=170.18$

Therefore, the mean daily wages = $Rs170.18$

$Mode=l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$

Where:

The modal class=class interval with highest frequency $=160–180=2$

$l$ = lower limit of modal class$=160$

$h$ = class interval $=120–100=20$

$f_1$ = frequency of the modal class$=22$

$f_0$ = frequency of the class before the modal class$=20$

$f_2$ = frequency of the class after the modal class$=18$

Mode =$160+\frac{22-20}{2\left(22\right)-20-18}\times 20$

$=160+\frac{2}{44-38}\times 20$

$=160+\frac{2}{6}\times 20$

$=160+\frac{40}{6}$

$=160+6.67$

$=166.67$

Therefore, the mode daily wages =$Rs166.67$