Question

Let \(\bar x\), M and \(\sigma^2\) be respectively the mean, mode and variance of n observations \(x_1,x_2,\ldots,x_n\) and \(d_i=-x_i-a,\ i=1,2,\ldots,n\), where a is any number.
Statement I : Variance of \(d_1,d_2,\ldots,d_n\) is \(\sigma^2\).
Statement II : Mean and mode of \(d_1,d_2,\ldots,d_n\) are \(-\bar x-a\) and \(-M-a\), respectively.

• A. Statement I and Statement II are both false
• B. Statement I and Statement II are both true
• C. Statement I is true and Statement II is false
• D. Statement I is false and Statement II is true

Hint:

Each \(d_i\) is a linear (one-to-one) function of \(x_i\): mean and mode transform the same way; variance is unchanged by shift and sign.

Answer 1

The Correct Option is B

\(d_i=-x_i-a\). Mean: \(\bar d=-\bar x-a\).
Variance: \(\sigma_d^2=(-1)^2\sigma^2=\sigma^2\) (shift does not matter, scaling by \(-1\) squares to \(1\)). Statement I is true.
Mode: the most frequent value \(M\) of \(x\) maps to \(-M-a\), the most frequent value of \(d\), since the map is one-to-one. Statement II is true.
Both are true.
Answer: (2) Statement I and Statement II are both true