Question

Consider the following frequency distribution:

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:

• $ \alpha $ denote the mean deviation about the mean
• $ \beta $ denote the mean deviation about the median
• $ \sigma^2 $ denote the variance

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

List-I

• (P) $ 7f_1 + 9f_2 $ is equal to
• (Q) $ 19\alpha $ is equal to
• (R) $ 19\beta $ is equal to
• (S) $ 19\sigma^2 $ is equal to

List-II

• (1) 146
• (2) 47
• (3) 48
• (4) 145
• (5) 55

• A. (P) → (5), (Q) → (3), (R) → (2), (S) → (4)
• B. (P) → (5), (Q) → (2), (R) → (3), (S) → (1)
• C. (P) → (5), (Q) → (3), (R) → (2), (S) → (1)
• D. (P) → (3), (Q) → (2), (R) → (5), (S) → (4)

Hint:

For grouped data, apply formulas for mean, mean deviation and variance carefully, and use cumulative frequencies for median-based conditions.

Answer 1

The Correct Option is C

Step 1: Use total frequency = 19

\[ 5 + f_1 + f_2 + 2 + 1 + 1 + 3 = 19 \Rightarrow f_1 + f_2 = 7 \quad \cdots (1) \]

Step 2: Median = 6 ⇒ Cumulative frequency up to 6 is ≥ 9.5

Frequency table (partial):

Value	4	5	6	8	9	11	12
Freq	5	\( f_1 \)	1	\( f_2 \)	2	3	1

Cumulative frequency up to 6: \[ 5 + f_1 + 1 = 6 + f_1 \geq 9.5 \Rightarrow f_1 \geq 4 \] From (1): \( f_2 = 7 - f_1 \)

Try \( f_1 = 4 \Rightarrow f_2 = 3 \)

Step 3: Calculate (P)

\[ 7f_1 + 9f_2 = 7 \cdot 4 + 9 \cdot 3 = 28 + 27 = 55 \Rightarrow (P) \to (5) \]

Step 4: Construct full distribution

Value (x)	4	5	6	8	9	11	12	Total
Freq (f)	5	4	1	3	2	3	1	19
\( xf \)	20	20	6	24	18	33	12	133

Mean: \[ \bar{x} = \frac{133}{19} = 7 \]

(Q): Mean deviation about mean (\( \alpha \))

\[ \alpha = \frac{1}{19} \sum f_i |x_i - 7| = \frac{48}{19} \Rightarrow 19\alpha = 48 \Rightarrow (Q) \to (3) \]

(R): Mean deviation about median (\( \beta \))

\[ \beta = \frac{1}{19} \sum f_i |x_i - 6| = \frac{47}{19} \Rightarrow 19\beta = 47 \Rightarrow (R) \to (2) \]

(S): Variance \( \sigma^2 \)

Given: \[ \sum f(x - 7)^2 = 146 \Rightarrow \sigma^2 = \frac{146}{19} \Rightarrow 19\sigma^2 = 146 \Rightarrow (S) \to (1) \]

Final Matching:

• (P) → (5)
• (Q) → (3)
• (R) → (2)
• (S) → (1)