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I took the new data as b and the data removed as a and calculated the new mean and used that to find the new mean and deviation in terms of the old. But it gets too complicated and there is no way to get the relation looking at the terms.
Basically the question is, if after changing only one value of a data set, the mean absolute deviation increases, will the standard deviation always increase? Or is there any case where it can decrease too?
The standard deviation effectively weights the deviations from the mean by those very deviations, whereas all the deviations are equally weighted in the mean absolute deviation (mad). If we can arrange to change some of the most heavily weighted values, perhaps we can find a counterexample.
Consider, then, changing the dataset $(0,0,48,-40)$ to $(0,0,48,48),$ which changes one value $-40$ to $48$. The deviations are originally $(-2,-2,46,-42)$ and become $(-24,-24,24,24).$ The sd changes by $18\sqrt{3}-24 \approx -7.2$ while the mad changes by $+1,$ of opposite sign to the sd change, as intended.
Thus, changing even a single value in a dataset may effect (appreciable) changes in the sd and mad of opposite sign.
