Statistics · JEE Advanced previous year questions

Consider the following frequency distribution: $$\begin{array}{cccccccc}\text{Value}& 4& 5& 8& 9& 6& 12& 11\\ \text{Frequency}& 5& f_1& f_2& 2& 1& 1& 3\end{array}$$ 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. $$\begin{array}{ll}\text{List-I}& \text{List-II}\\ \text{(P) }7f_1+9f_2\text{ is equal to}& \text{(1) }146\\ \text{(Q) }19\alpha \text{ is equal to}& \text{(2) }47\\ \text{(R) }19\beta \text{ is equal to}& \text{(3) }48\\ \text{(S) }19{\sigma }^2\text{ is equal to}& \text{(4) }145\\ & \text{(5) }55\end{array}$$

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

Let $X$ be a random variable, and let $P(X=x)$ denote the probability that $X$ takes the value $x$. Suppose that the points $(x,P(X=x))$, $x=0,1,2,3,4$, lie on a fixed straight line in the $xy$-plane, and $P(X=x)=0$ for all $x\in \mathbb{R}-\{0,1,2,3,4\}$. If the mean of $X$ is $\frac{5}{2}$, and the variance of $X$ is $\alpha$, then the value of $24\alpha$ is _____.

Consider the given data with frequency distribution $$\begin{array}{ccccccc}{x}_i& 3& 8& 11& 10& 5& 4\\ f_i& 5& 2& 3& 2& 4& 4\end{array}$$ Match each entry in List-I to the correct entries in List-II. $$\begin{array}{ll}\text{List-I}& \text{List-II}\\ \text{(P) The mean of the above data is}& \text{(1) }2.5\\ \text{(Q) The median of the above data is}& \text{(2) }5\\ \text{(R) The mean deviation about the mean of the above data is}& \text{(3) }6\\ \text{(S) The mean deviation about the median of the above data is}& \text{(4) }2.7\\ & \text{(5) }2.4\end{array}$$ The correct option is:

1. A.$(P)\to (3)(Q)\to (2)(R)\to (4)(S)\to (5)$
2. B.$(P)\to (3)(Q)\to (2)(R)\to (1)(S)\to (5)$
3. C.$(P)\to (2)(Q)\to (3)(R)\to (4)(S)\to (1)$
4. D.$(P)\to (3)(Q)\to (3)(R)\to (5)(S)\to (5)$