Question

Let $n^3$. A list of numbers $x_1,x_2.....x_n$ has mean $m$ and standard deviation $s$. A new list of numbers $y_1,y_2....y_n$ is made as follows : $y_1=\frac{x_1+x_2}{2},y_2=\frac{x_1+x_2}{2}$ and $y_j=x$ for $j=3,4,.......$. The mean and the standard deviation of the list are $\hat{\mu }$ and $\hat{\sigma }$ . Then which of the following is necessarily true?

• A. $\mu =\hat{\mu }$ and $\sigma \le \hat{\sigma }$
• B. $\mu =\hat{\mu }$ and $\sigma \ge \hat{\sigma }$
• C. $\sigma =\hat{\sigma }$
• D. $\mu \ne \hat{\mu }$

Answer 1

The correct option is B

$\mu =\hat{\mu }$ and $\sigma \ge \hat{\sigma }$