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