Median minimizes absolute deviation

1. Does the median minimizes the sum of absolute deviations?
2. Which of the following functions does the median of minimize?
3. What is the value that minimizes the sum of all absolute deviations from it?

Does the median minimizes the sum of absolute deviations?

It is widely known among statisticians that the median minimizes the sum of the absolute deviation about any point for a set of x's, xl, x2, x3, . . . , x,.

Which of the following functions does the median of minimize?

The median minimizes the sum of absolute deviations (the ℓ1 norm)

What is the value that minimizes the sum of all absolute deviations from it?

The Median minimizes the sum of absolute deviations. The Mean minimizes the sum of squared deviations (sum of squared errors). The Midpoint minimizes the biggest deviation.