Let $n \geq 3$. A list of numbers $x_1, x_2, \ldots, x_n$ has mean $\mu$ and standard deviation $\sigma$. A new list $y_1, y_2, \ldots, y_n$ is made: $y_1 = y_2 = (x_1+x_2)/2$ and $y_j = x_j$ for $j = 3,4,\ldots,n$. The mean and standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Then which is necessarily true?