Absolute Deviation - Minimization

Minimization

The measures of statistical dispersion derived from absolute deviation characterize various measures of central tendency as minimizing dispersion: The median is the measure of central tendency most associated with the absolute deviation, in that

L2 norm statistics

just as the mean minimizes the standard deviation,

L1 norm statistics

the median minimizes average absolute deviation,

L∞ norm statistics

the mid-range minimizes the maximum absolute deviation, and

trimmed L∞ norm statistics

for example, the midhinge (average of first and third quartiles) which minimizes the median absolute deviation of the whole distribution, also minimizes the maximum absolute deviation of the distribution after the top and bottom 25% have been trimmed off.