The value of third quartile $Q_{3}$ for the following distribution is

Marks
Group $5 - 10$ $10 - 15$ $15 - 20$ $20 - 25$ 2 $5 - 30$ $30 - 35$ $35 - 40$ $40 - 45$

No. of
Students $5$ $6$ $15$ $10$ $5$ $4$ $2$ $1$

15

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21.5

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25

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25.5

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Solution

The correct option is C $25$
The cumulative frequency distribution is as given below

Marks
Groups (class) No. of Students Cumulative
frequency

5-10 5 5

10-15 6 11

15-20 15 26

20-25 10 36

25-30 5 41

30-35 4 45

35-40 2 47

40-45 1 48

$N = 48$

We have $N = 48 \Rightarrow \frac{3 N}{4} = 36$
The cumulative frequency just greater than $\frac{3 N}{4}$ is $41$ . The corresponding class is $25 - 30$ . It is the upper quartile class.
$∴ l = 25, f = 5, h = 5, F = 36$
$∴ Q_{3} = l + \frac{\frac{3 N}{4} - F}{f} \times h = 25 + \frac{36 - 36}{5} \times 5 = 25$

Suggest Corrections

Marks Group	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	2 $5 - 30$	$30 - 35$	$35 - 40$	$40 - 45$
No. of Students	$5$	$6$	$15$	$10$	$5$	$4$	$2$	$1$

Class:	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$
Frequency:	$5$	$6$	$15$	$10$	$5$	$4$	$2$	$2$

CI	$0 - 5$	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$	$45 - 50$
$f$	$9$	$13$	$6$	$12$	$9$	$6$	$10$	$15$	$6$	$4$

Size of item	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$	$45 - 50$
No. of items	$4$	$8$	$18$	$30$	$20$	$10$	$5$	$2$

Class	5-10	10-15	15-20	20-25	20-30	30-35	35-40	40-45
Frequency	5	6	15	10	5	4	2	2