Let ¯ x , M and σ ^ 2 be respectively the mean, mode and variance...

Let $\overset{¯}{x}$ , $M$ and ${\sigma}^{2}$ 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 I: Variance of ${d}_{1}, {d}_{2}, . . . , {d}_{n}$ is ${\sigma}^{2}$ .

Statement II: Mean and mode of ${d}_{1}, {d}_{2}, . . . . , {d}_{n}$ are $- \overset{¯}{x} - a$ and $- M - a$ , respectively.

Options

AStatement I and Statement II are both true
BStatement I and Statement II are both false
CStatement I is true and Statement II is false
DStatement I is false and Statement II is true

Correct answer

A. Statement I and Statement II are both true

Step-by-step solution

Mean of ${d}_{1}, {d}_{2}, {d}_{3} . . . {d}_{n}$

$= \mathop{∑}_{i = 1}^{n}\left(\dfrac{- {x}_{i} - a}{n}\right)$

$= - \mathop{∑}_{i = 1}^{n}\dfrac{{x}_{i}}{n} + \dfrac{- a}{n}\mathop{∑}_{i = 1}^{n}1$

$= - x + \dfrac{- a}{n}n$

$= - \overset{\to}{x} - a$

And, the mode of ${d}_{1}, {d}_{2}, {d}_{3} . . . {d}_{n}$ will be $- M - a$

Also, we know that the variance ${\sigma}^{2}$ is unaffected by change of origin, hence it remains same.

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