The sum of the deviations from the mean is zero: $\sum_{i=1}^n (x_i - \bar{x}) = 0$.

The sum of the squared deviations $\sum_{i=1}^n (x_i - c)^2$ is strictly minimized when $c = \bar{x}$.

The sum of the absolute deviations $\sum_{i=1}^n |x_i - c|$ is minimized when $c$ is the median of the dataset.

If we define $d_i = x_i - \bar{x}$, then the mean of the deviations $\bar{d}$ is equal to $\bar{x}$.