JEE Advanced · Mathematics · 20. Statistics

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, and \(\sigma^2\) denote the variance.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
\(\begin{array}{|l|l|l|l|} \hline & \text{LIST - I} & & \text{LIST - II} \\ \hline (P) & 7 f_1+9 f_2 \text{ is equal to} & (1) & 146 \\ \hline (Q) & 19 \alpha \text{ is equal to} & (2) & 47 \\ \hline (R) & 19 \beta \text{ is equal to} & (3) & 48 \\ \hline (S) & 19 \sigma^2 \text{ is equal to} & (4) & 145 \\ \hline & & (5) & 55 \\ \hline \end{array}\)

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

A \((\mathrm{P}) \rightarrow(5),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(2),(\mathrm{S}) \rightarrow(4)\)
B \((\mathrm{P}) \rightarrow(5),(\mathrm{Q}) \rightarrow(2),(\mathrm{R}) \rightarrow(3),(\mathrm{S}) \rightarrow(1)\)
C \((\mathrm{P}) \rightarrow(5),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(2),(\mathrm{S}) \rightarrow(1)\)
D \((\mathrm{P}) \rightarrow(3),(\mathrm{Q}) \rightarrow(2),(\mathrm{R}) \rightarrow(5),(\mathrm{S}) \rightarrow(4)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \((\mathrm{P}) \rightarrow(5),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(2),(\mathrm{S}) \rightarrow(1)\)

Step-by-step Solution

Detailed explanation

\(\begin{array}{|c|c|c|c|c|c|c|} \hline X_i & f_{\mathrm{i}} & \begin{array}{l} \bar{x}=7 \\ d_i = \left|x_i-\bar{x}\right| \end{array} & \begin{array}{l} \mathrm{M}=6 \\ \mathrm{ei} = |\mathrm{xi}-\mathrm{M}| \end{array} & \sum f_{\mathrm{i}} \mathrm{d}_{\mathrm{i}} & \sum f_{\mathrm{i}} \mathrm{e}_{\mathrm{i}} & \Sigma f_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}^2 \\ \hline 4 & 5 & 3 & 2 & 15 & 10 & 45 \\ \hline 5 & 4 & 2 & 1 & 8 & 4 & 16 \\ \hline 6 & 1 & 1 & 0 & 1 & 0 & 1 \\ \hline 8 & 3 & 1 & 2 & 3 & 6 & 3 \\ \hline 9 & 2 & 2 & 3 & 4 & 6 & 8 \\ \hline 11 & 3 & 4 & 5 & 12 & 15 & 48 \\ \hline 12 & 1 & 5 & 6 & 5 & 6 & 25 \\ \hline & & & & 48 & 47 & 146 \\ \hline \end{array}\)
\(\begin{aligned} & f_1=4 \\ & f_2=3 \\ & \alpha=\frac{48}{19}, \beta=\frac{47}{19}, \sigma^2=\frac{146}{19}\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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	List-I		List-II
(i)	$\frac{1}{\sqrt{2}} (\sin ω t \hat{j} + \cos ω t \hat{k})$	(p)	$0$
(ii)	$\frac{1}{\sqrt{2}} (\sin ω t \hat{i} + \cos ω t \hat{j})$	(q)	$- \frac{α}{4} \hat{i}$
(iii)	$\frac{1}{\sqrt{2}} (\sin ω t \hat{i} + \cos ω t \hat{k})$	(r)	$\frac{3 α}{4} \hat{i}$
(iv)	$\frac{1}{\sqrt{2}} (\cos ω t \hat{j} + \sin ω t \hat{k})$	(s)	$\frac{α}{4} \hat{j}$
		(t)	$- \frac{3 α}{4} \hat{i}$