If the median of the following frequency distribution is 46, find the missing frequencies.
Variable: 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Total
Frequency: 12 30 ? 65 ? 25 18 229

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Solution

Let the frequency of the class $30 - 40$ be $f_{1}$ and that the class of the class $50 - 60$ be $f_{2}$ . The total frequency is 229.
$12 + 30 + f_{1} + 65 + f_{2} + 25 + 18 = 229 \Rightarrow f_{1} + f_{2} = 79$
It is given that the median is $46.$
Clearly, 46 lies in the class $40 - 50$ . So, $40 - 50$ is the median class.
$∴ l = 40, h = 10, f = 65$ and $F = 12 + 30 \div f_{1} = 42 + f_{1}, N = 229$
Now, $M e d i a n = l + \frac{\frac{N}{2} - F}{f} \times h$
$\Rightarrow 46 = 40 + \frac{\frac{229}{2} - (42 + f_{1})}{65} \times 10$
$\Rightarrow 46 = 40 + \frac{145 - 2 f_{1}}{13}$
$\Rightarrow 6 = \frac{145 - 2 f_{1}}{13} \Rightarrow 2 f_{1} = 67 \Rightarrow f_{1} = 33.5$ or $34$ (say)
Since $f_{1} + f_{2} = 79$ . Therefore, $f_{2} = 45$ .
Hence, $f_{1} = 34$ and $f_{2} = 45.$

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Variable	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80	Total
Frequency	12	30	P	65	Q	25	18	230

46

p

q

class interval	10-20	20-30	30-40	40-50	50-60	60-70	70-80	Total
Frequency	12	30	p	65	q	25	18	230

230

46

a

b

Class	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	12	30	a	65	b	25	18

Class	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90
Frequency	9	11	15	24	19	9	8	5

\begin{matrix} Class & 10 - 20 & 20 - 30 & 30 - 40 & 40 - 50 & 50 - 60 & 60 - 70 & 70 - 80 Frequency & 180 & 34 & 180 & 136 & 50 \end{matrix}

685

42.6

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Variable:	10-20	20-30	30-40	40-50	50-60	60-70	70-80	Total
Frequency:	12	30	?	65	?	25	18	229