The mean deviation gives better resuls when deviations are taken from the median instead from the mean, because the sum of the deviations from the median is less than the sum of the deviations from the mean
$$ \operatorname{M.D.}(M)=\frac{\sum|x_i-M|}{N}=\frac{\sum f_i|x_i-M|}{\sum f_i} $$
This is a widely used statement when defining mean deviation. I understand the proof by which we can say the mean deviation is minimized when deviations are taken from the median. But how come that somehow leads to the statement "mean deviation gives better resuls when deviations are taken from the median instead from mean as the former minimizes the sum of absolute deviations" ?
Screen shot of sample reference
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Whether item 3 under "merits" is a merit depends on what the purpose is.
Item 3 under "limitations" is nonsense. It does in fact not take into account whether an observation is more than the mean or less, at least after the point where the mean is computed, but why would that be a reason to consider anything mathematically incorrect?
Item 2 under "limitations" also depends on what the purpose is.
Item 1 under "limitations" may be subtle.
Now imagine a normally distributed population has standard deviation $\sigma,$ and $\sigma$ is not observable but a finite random sample of $n$ observations is available. The expect value of the mean absolute deviation of the sample is a known scalar multiple of $\sigma.$ Multiplying the mean absolute deviation of the sample by the reciprocal of that known scalar yields an unbiased estimator of of $\sigma.$ Similarly, multiplying the sample standard deviation by another known scalar yields an unbiased estimator of $\sigma.$
Theorem: The one based on the standard deviation has a smaller variance than the one based on the mean absolute deviation.
So that's one thing that could be called a demerit of the mean absolute deviation.
However if the population distribution is only slightly different from normal, that is no longer true.
These lists of "merits" and "limitations" look like something written by a person with little understanding of the subject matter.