Question

For the following information of wages of $30$ workers in a factory, the value of $Q_1+Q_2+Q_3$ is

S. No.	Wages (Rs.)	S. No.	Wages (Rs.)
1	330	16	240
2	320	17	330
3	550	18	420
4	470	19	380
5	210	20	450
6	500	21	260
7	270	22	330
8	120	23	440
9	680	24	480
10	490	25	520
11	400	26	300
12	170	27	580
13	440	28	370
14	480	29	380
15	620	30	350

• A. $1187.50$
• B. $797.50$
• C. $705$
• D. $872.50$

Answer 1

The correct option is A $1187.50$

Arranging the wages in ascending order, we obtain the following table:

S. No	Wages (Rs.)	S. No.	Wages (Rs.)
1	120	16	400
2	170	17	420
3	210	18	440
4	240	19	440
5	260	20	450
6	270	21	470
7	300	22	480
8	320	23	480
9	330	24	490
10	330	25	500
11	330	26	520
12	330	27	550
13	350	28	580
14	370	29	620
15	380	30	680

We have, $n=30$
Lower Quartile:
$Q_1=$ Value of ${\left(\frac{n+1}{4}\right)}^{th}$ observation
$\Rightarrow Q_1=$ Value of ${\left(\frac{30+1}{4}\right)}^{th}$ observation
$\Rightarrow Q_1=$ Value of ${\left(7.75\right)}^{th}$ observation
$\Rightarrow Q_1=$ Value of ${\left(7\right)}^{th}$ observation +
$\frac{3}{4}$(Value of $8^{th}$ observation - Value of
$7^{th}$ observation)
$\Rightarrow Q_1=300+\frac{3}{4}(320-300)=315$
Middle Quartile (Median):
$Q_2$
= A.M. of ${\left(\frac{n}{2}\right)}^{th}$ and
${\left(\frac{n}{2}+1\right)}^{th}$ observation
$\Rightarrow Q_2=\frac{\text{value of}{15}^{th}\text{observation}+\text{value of}{16}^{th}\text{observation}}{2}$
$\Rightarrow Q_2=\frac{380+400}{2}=390$
Upper Quartile:
$Q_3=$ Value of $3{\left(\frac{n+1}{4}\right)}^{th}$ observation
$\Rightarrow Q_3=$ Value of $3{\left(\frac{30+1}{4}\right)}^{th}$ observation
$\Rightarrow Q_3=$ Value of ${23.25}^{th}$ observation
$\Rightarrow Q_3=$ Value of ${23}^{rd}$ observation +
$\frac{1}{4}($ Value of ${24}^{th}$ Observation - Value
of ${23}^{rd}$ observation $)$
$\Rightarrow Q_3=480+\frac{1}{4}(490-480)=482.50$
$\therefore Q_1+Q_2+Q_3=315+390+482.50=1187.50$