Consider $X = \{x_1, x_2, x_3\}$. Then $\bar{x} = \frac{1}{3} (x_1 + x_2 + x_3)$ with degrees of freedom, $df(\bar{x}) = n = 3$.
Now consider the total variation in $x$:
$$S_{xx} = (x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + (x_3 - \bar{x})^2$$
If we express $x_1$ in terms of $\bar{x}$, $x_2$, and $x_3$, then
$$S_{xx} = (3\bar{x} - (x_2 + x_3) - \bar{x})^2 + (x_2 - \bar{x})^2 + (x_3 - \bar{x})^2$$
Why does $S_{xx}$ have $2$ degrees of freedom?
My professor says that our knowledge of $\bar{x}$ causes $S_{xx}$ to lose one degree of freedom, but I have no idea why.
