Let x , M and 2 be respectively the mean, mode and variance of n observations x 1 , x 2 , … , x n and d i = - x i - a , i = 1 , 2 , … , n where a is any number. Statement 1: Variance of d 1 , d 2 , … , d n is 2 . Statement 2: Mean and m

Question

Let $\bar{x}$ , M and $σ^{2}$ be respectively the mean, mode and variance of $n$ observations $x_{1}, x_{2}, \dots, x_{n}$ and $d_{i} = - x_{i} - a, i = 1, 2, \dots, n$ where $a$ is any number.

Statement 1: Variance of $d_{1}, d_{2}, \dots, d_{n}$ is $σ^{2}$ .

Statement 2: Mean and mode of $d_{1}, d_{2}, \dots, d_{n}$ are $- \bar{x} - a$ and $- M - a$ respectively.

Smart Achievers · Accepted Answer

(2)

$S_{1} \to V (a x + b) = a^{2} V (x)$

$S_{2} \to mean (a x \pm b) = a mean (X) \pm b$

$Mode (x_{1}, x_{2}, \dots, x_{n}) = K$

$Mode (a x_{1} \pm α, a x_{2} \pm α, \dots, a x_{n} \pm α) = a k \pm α$

$Central tendencies depend on shift of origin & scaling$

$Variance depends on scaling but not shift of origin$

Solution

1 Statement 1 and Statement 2 are both false

2 Statement 1 and Statement 2 are both true

3 Statement 1 is true and Statement 2 is false

4 Statement 1 is false and Statement 2 is true

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