For the given frequency distribution, let denote the mean deviation about the mean, denote the mean deviation about the median, and denote the variance. [2025]

Question

Consider the following frequency distribution: [2025]

Value	4	5	8	9	6	12	11
Frequency	5	$f_{1}$	$f_{2}$	2	1	1	3

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6.

For the given frequency distribution, let $α$ denote the mean deviation about the mean, $β$ denote the mean deviation about the median, and $σ^{2}$ denote the variance.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

	List - I		List - II
(P)	$7 f_{1} + 9 f_{2}$ is equal to	(1)	146
(Q)	$19 α$ is equal to	(2)	47
(R)	$19 β$ is equal to	(3)	48
(S)	$19 α^{2}$ is equal to	(4)	145
			55

Smart Achievers · Accepted Answer

(3)

$Given sum of frequencies = 19$

$12 + f_{1} + f_{2} = 19$

$f_{1} + f_{2} = 7$

$∵$

$x_{i}$	$f_{i}$	$C . F .$
4	5	5
5	$f_{1}$	$5 + f_{1}$
6	1	$5 + f_{1} + 1$
8	$f_{2}$	$6 + f_{1} + f_{2}$
9	2	$8 + f_{1} + f_{2}$
11	3	$11 + f_{1} + f_{2}$
12	1	$12 + f_{1} + f_{2}$

$∵$ $Median = 10^{th} observation$

$∴$ $5 + f_{1} + 1 = 10$

$f_{1} = 4$ $∴$ $f_{2} = 3$

$∵$ $\bar{x} = \frac{\sum x_{i} f_{i}}{\sum f_{i}}$ $= \frac{20 + 20 + 6 + 24 + 18 + 33 + 12}{19}$ $= 7$

$x_{i}$	$f_{i}$	$f_{i} \| x_{i} - \bar{x} \|$
4	5	15
5	4	8
6	1	1
8	3	3
9	2	4
11	3	12
12	1	5

$∴$ $\sum f_{i}$ $| x_{i} - \bar{x} |$ $= 48$

$∴$ $α = \frac{\sum f_{i} | x_{i} - \bar{x} |}{\sum f_{i}} = \frac{48}{19}$

$x_{i}$	$f_{i}$	$f_{i}$ $\| x_{i} - M \|$
4	5	10
5	4	4
6	1	0
8	3	6
9	2	6
11	3	15
12	1	6

$∴$ $\sum f_{i}$ $| x_{i} - M |$ $= 47$

$∴$ $β = \frac{47}{19}$

$∵$ $σ^{2} = \frac{\sum f_{i} {(x_{i} - \bar{x})}^{2}}{\sum f_{i}}$

$= \frac{5 \times 9 + 4 \times 4 + 1 \times 1 + 3 \times 1 + 2 \times 4 + 3 \times 16 + 1 \times 25}{19}$

$σ^{2} = \frac{146}{19}$

Solution

1 (P)–(5), (Q)–(3), (R)–(2), (S)–(4)

2 (P)–(5), (Q)–(2), (R)–(3), (S)–(1)

3 (P)–(5), (Q)–(3), (R)–(2), (S)–(1)

4 (P)–(3), (Q)–(2), (R)–(5), (S)–(4)

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