Question

The mode of the following data is 67. Find the missing frequency x.

Class 40-50 50-60 60-70 70-80 80-90

Frequency 5 x 15 12 7​

Answer 1

Given:- Mode = 67

To Find:- Find missing frequency x

Step-by-step explanation:

Class           40-50   50-60   60-70   70-80   80-90

Frequency     5           x            15         12           7

Click on the image for solution.

Answer 2

☯ The mode of following data is 67.

⠀⠀⠀⠀

$\begin{tabular}{|c|c|c|c|c|c|c|}\cline{1-6} \sf Class & 40-50 & 50-60 & 60-70 & 70-80 &80-90\\\cline{1-6} \sf Frequency&5&x&15&12&7\\\cline{1-6}\end{tabular}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

$\boxed{\begin{array}{c|cc}\sf Class\: interval&\sf Frequency\: (f)\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}\\\sf 40 - 50&\sf 5\\\\\sf 50 - 60 &\sf x\\\\\sf 60-70 &\sf 15\\\\\sf 70 - 80&\sf 12\\\\\sf 80-90&\sf 7\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}\\\sf & \sf \sum f = 67& \end{array}}$

$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

$\sf {\underline{\sf{\bigstar\;Mode\;of\;the\;data\;is\;givEn\;by,}}}$

$\maltese\;{\underline{\boxed{\pink{\sf{Mode = L + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h}}}}}$

Given that,

• Mode of data is 67 which lies between 60 - 70.

⠀⠀⠀⠀

Therefore,

• Lower limit, L = 60
• Class interval, h = 50 - 40 = 10
• Frequency of modal class, $\sf f_1$ = 15
• Frequency of class before modal class, $\sf f_0$ = x
• Frequency of class after modal class, $\sf f_2$ = 12

⠀⠀⠀⠀

$\dag\;{\underline{\frak{Now,\; Putting\; values\;in\;formula,}}}$

$:\implies\sf 60 + \dfrac{15 - x}{(2 \times 15) - x - 12} \times 10= 67$

$:\implies\sf 60 + \dfrac{15 - x}{30 - x - 12} \times 10 = 67$

$:\implies\sf 60 + \dfrac{15 - x}{18 - x} \times 10 = 67$

$:\implies\sf \dfrac{15 - x}{18 - x} \times 10 = 67 - 60$

$:\implies\sf \dfrac{15 - x}{18 - x} \times 10 = 7$

$:\implies\sf 10(15 - x) = 7(18 - x)$

$:\implies\sf 150 - 10x = 126 - 7x$

$:\implies\sf - 10x + 7x = 126 - 150$

$:\implies\sf - 3x = - 24$

$:\implies\sf x = \cancel{ \dfrac{24}{3}}$

$:\implies{\underline{\boxed{\frak{\purple{x = 8}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{The\;missing\; frequency\;is\; {\textsf{\textbf{8}}}.}}}$