Let $n \geq 3$. A list of numbers $0 < x_1 < x_2 < \ldots < x_n$ has mean $\mu$ and standard deviation $\sigma$. A new list of numbers is made as follows: $y_1=0, y_2=x_2, \ldots, x_{n-1}$ $=x_n-1, y_n=x_1+x_n$. The mean and the standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Which of the following is necessarily true?

  • [KVPY 2013]
  • A

    $\mu=\hat{\mu}, \sigma \leq \hat{\sigma}$

  • B

    $\mu=\hat{\mu}, \sigma \geq \hat{\sigma}$

  • C

    $\sigma=\hat{\sigma}$

  • D

    $\mu$ may or may not be equal to $\hat{\mu}$

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