Question

For the following data, the value of $Q_1+Q_3-Q_2$ is :

Age in years:	20	30	40	50	60	70	80
No. of members:	3	61	132	153	140	51	3

• A. $Q_1$
• B. $Q_2$
• C. $Q_3$
• D. $2Q_2$

Answer 1

The correct option is B $Q_2$

The cumulative frequency table is as given below

Age in years	No. of students	Cumulative frequency
20	3	3
30	61	64
40	132	196
50	153	349
60	140	489
70	51	540
81	3	543
N $=$543

Lower Quartile: We have, $\frac{N}{4}=\frac{543}{4}=135.75$
Cumulative frequency just greater than $N/4$ is $196$ and the corresponding value of the variable is $40$.
$\therefore Q_1=$ Lower Quartile = $40$ years
Middle Quartile (Medium):
We have, $\frac{N}{2}=\frac{543}{2}=271.5$
Cumulative frequency just greater than $N/2$ is $349$ and the corresponding value of the variable is $50$.
$\therefore Q_2=$ Middle quartile = $50$ years
Upper Quartile:
We have $\frac{3N}{4}=\frac{3\times 543}{4}=407.25$
Cumulative frequency just greater than $407.25$ is $489$ and the corresponding value of the variable is $60$.
$\therefore Q_3=$ Upper quartile $=60$ years
$\therefore Q_1+Q_3-Q_2=40+60-50=50=Q_2$