In the following frequency distribution of $165$ observations, the mode is $34.5$ . Find the missing frequencies $a$ and $b$ .
Class $5 - 14$ $14 - 23$ $23 - 32$ $32 - 41$ $41 - 50$ $50 - 59$ $59 - 68$
Frequency $5$ $11$ $a$ $53$ $b$ $16$ $10$

Open in App

Solution

$\Rightarrow$ Here, modal class $= 32 - 41$
$\Rightarrow$ $l = 32, f_{0} = a, f_{1} = 53, f_{2} = b$ and $h = 9$
$\Rightarrow$ $M o d e = l + \frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}} \times h$

$\Rightarrow$ $34.5 = 32 + \frac{53 - a}{2 \times 53 - a - b} \times 9$

$\Rightarrow$ $\frac{2.5}{9} = \frac{53 - a}{106 - a - b}$

$\Rightarrow$ $265 - 2.5 a - 2.5 b = 477 - 9 a$
$\Rightarrow$ $7.5 a - 2.5 b = 212$
$\Rightarrow$ $75 a - 25 b = 2120$
$∴$ $13 a - 5 b = 424$ ------ ( 1 )
As total observation are $165$
$\Rightarrow$ So, $5 + 11 + a + 53 + b + 16 + 10 = 165$
$\Rightarrow$ $95 + a + b = 165$
$∴$ $a + b = 70$ -------- ( 2 )
$∴$ $b = 70 - a$
Substitute $b = 70 - a$ in equation ( 1 )
$\Rightarrow$ $13 a - 5 (70 - a) = 424$
$\Rightarrow$ $13 a - 350 + 5 a = 424$
$\Rightarrow$ $18 a = 424 + 350$
$\Rightarrow$ $18 a = 774$
$∴$ $a = 43$
$\Rightarrow$ $b = 70 - 43 = 27$
$∴$ Frequency $a = 43$ and $b = 27$

Suggest Corrections

Similar questions

30

1 - 10, 11 - 20 . . . . . . . . . . . . .41 - 50;

400

Marks	$1 - 10$	$11 - 21$	$21 - 30$	$31 - 40$	$41 - 50$	$51 - 60$	$61 - 70$	$71 - 80$	$81 - 90$	$91 - 100$
Frequency	$8$	$14$	$32$	$56$	$102$	$80$	$54$	$30$	$16$	$8$

16

Class	0-4	4-8	8-12	12-16	16-20	20-24	24-28	28-32	32-36
Frequency	6	8	17	23	16	15	-	4	3

Class Interval	0−9	10−19	20−29	30−39	40−49	50−59	60−69
Frequency	5	15	20	23	17	11	9

60

X	$30 - 34$	$40 - 44$	$45 - 49$	$50 - 54$	$55 - 59$	$60 - 64$
f	$3$	$5$	$12$	$18$	$14$	$62$

Join BYJU'S Learning Program

Class	$5 - 14$	$14 - 23$	$23 - 32$	$32 - 41$	$41 - 50$	$50 - 59$	$59 - 68$
Frequency	$5$	$11$	$a$	$53$	$b$	$16$	$10$