Question

(CBSE 2006C]
1. The mean of the following data is 42. Find the missing frequencies x
and y if the sum of frequencies is 100.
Class interval
0-10 10-20 20-30 | 30-40 40-50 50-60 | 60-70 70-80
Frequency
7
10
X
13
Y
10
9
[CBSE 2014]​

Answer 1

Answer:

Given that mean of data is 42

sum of frequencies (N) or ∑f=100

Class interval 0−10 10−20 20−30 30−40 40−50 50−60 60−70 70−80

Frequency 7 10 x 13 y 10 14 9

Class interval frequency(f

i

) mid value of (x

i

) f

i

x

i

0−10 7 5 35

10−20 10 15 150

20−30 x 25 25x

30−40 13 35 455

40−50 y 45 45y

50−60 10 55 550

60−70 14 65 910

70−80 9 75 675

___________ ______________

63+x+y 2775+25x+45y

Given N=100

⇒63+x+y=100

x+y=100−63=37

x+y=37 ,,,,,,,,,(1)

we know mean =∑f

i

x

i

/∑f

i

42=

100

2775+25x+45y

4200=2775+25x+145y

⇒25x+45y=1425

5(5x+9y)=5×285

5x+9y=285 .............(2)

(1)×9=9x+9y=333

(2)=5x+9y=285

___________________

4x=48

⇒x=12

(1) ⇒x+y=37

⇒y=37−x=37+12=25

y=25.

Answer 2

$\begin{gathered}\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\cline{1-9} \tt Class & \tt 0-10 & \tt 10-20 & \tt 20-30 & \tt 30-40 & \tt 40-50 & \tt 50-60 & \tt 60-70 & \tt 70-80 \\\cline{1-9}\tt Frequency &\tt 7 & \tt 10& \tt x & \tt 13 & \tt y & \tt 10 & \tt 14 & \tt 9 \\\cline{1-9}\end{tabular}\end{gathered}$

We have to find, value of x and y.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

$\boxed{\begin{array}{cccc}\sf Class\: interval&\sf Frequency\: (f_i) &\sf x_i &\sf f_i x_i\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 0 - 10&\sf 7&\sf 5 &\sf 35\\\\\sf 10 - 20 &\sf 10&\sf 15 &\sf 150\\\\\sf 20 - 30 &\sf x&\sf 25&\sf 25x\\\\\sf 30 - 40&\sf 13&\sf35&\sf445\\\\\sf 40 - 50&\sf y&\sf 45&\sf45y\\\\\sf 50-60&\sf 10 &\sf 55 & \sf 550\\\\\sf 60-70&\sf 14 &\sf 65 & \sf 910\\\\\sf 70-80&\sf 9&\sf 75 & \sf 675\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf Total& \sf63+x+y&& \sf 2775+25x+45y\end{array}}$

Sum of frequencies = 100

$\quad:\implies\sf \sum f_i = 100$

$:\implies\sf 63 + x + y = 100$

$:\implies\sf x + y = 100 - 63$

$\quad\quad:\implies\sf x + y = 37$

$\quad\quad:\implies\sf y = 37 - x\qquad\qquad\bigg\lgroup\bf eq\;(1) \bigg\rgroup$

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━

$\dag\;{\underline{\frak{Formula\;to\:find\;Mean,}}}$

$\star\;{\boxed{\sf{\purple{\bar{x} = \dfrac{ \sum f_ix_i}{f_i}}}}}$

$\: \: :\implies\sf 42 = \dfrac{2775 + 25x + 45y}{100}$

$:\implies\sf 4200 = 2775 + 25x + 45y$

$:\implies\sf 4200 - 2775 = 25x + 45y$

$\qquad:\implies\sf 1425 = 25x + 45y$

$\: \: :\implies\sf 1425 = 25x + 45(37-y)\qquad\qquad\bigg\lgroup\bf from\;eq\;(1) \bigg\rgroup$

$:\implies\sf 1425 = 25x + 1665 - 45x$

$:\implies\sf 45x - 25x = 1665 - 1425$

$\qquad \: \quad:\implies\sf 20x = 240$

$\qquad \: \quad \: \: :\implies\sf x = \cancel{ \dfrac{240}{20}}$

$\qquad \: \quad \: :\implies{\underline{\boxed{\frak{\pink{x = 12}}}}}\;\bigstar$

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━

Now, Putting value of x in eq (1),

$\qquad\quad:\implies\sf y = 37 - 12$

$\qquad\qquad:\implies{\underline{\boxed{\frak{\pink{y = 25}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;the\;value\;of\;x\;is\;12\;and\;y\;is\;25.}}}$