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Regarding the difference between mean absolute deviation & standard deviation the both involve the deviation of ALL the points from the mean. One involves the sum of the absolute deviations from the mean while the is the square root if the sum of the squared deviation.. – Michael R. Chernick. Sep 18, 2019 at 21:14.
The mean absolute deviation is about .8 times (actually $\sqrt{2/\pi}$) the size of the standard deviation for a normally distributed dataset. Regardless of the distribution, the mean absolute deviation is less than or equal to the standard deviation. MAD understates the dispersion of a data set with extreme values, relative to standard deviation.
$$ Mean\;Absolute\;Deviation = \frac{1}{n}\sum_{i=1}^n |x_i-mean(X)| $$ Almost all textbooks and papers are using Standard Deviation as a measurement of dispersion. And of course, almost all the models are built based on Standard Deviation. But I don't understand how Standard Deviation has gained such popularity.
7. Many of the comments in posts about using variance rather than mean absolute deviation from the mean (e.g. here) apply also to median absolute deviation from the median. Then on top of that, generally speaking properties of medians are not as nice as those of means. For example in general med(X + Y) ≠ med (X + Y) ≠ med(X) + med(Y) med (X ...
In the case of the scaled mean deviation vs the standard deviation as an estimate of $\sigma$ in the normal, Fisher derived the ARE to be $\frac{1/2}{(\pi/2)-1} = \frac{1}{\pi-2} \approx 0.87597$. This will be the result Tukey refers to. Some details of the derivation (and other references) are given in Pham-Gia and Hung (2001)[1].
For instance, the mean absolute deviation is not. Second, it's one of the central moments: $$\mu_k=\sum_ip_i(x_i-\mu_1)^k$$ Here $\mu_2$ is a variance. Being a moment is important, for it defines the distribution when combined with all other moments.
Revisiting a 90-year-old debate: the advantages of the mean deviation says: the standard deviation of their individual mean deviations is 14% higher than the standard deviations of their individual standard deviations (Stigler 1973). Thus, the SD of such a sample is a more consistent estimate of the SD for a population, and is considered better ...
MAD and standard deviation will not change if the mean shifts. Yes wiki gives a reference where essentially there is simply the conclusion, which is why I am putting the question here. $\endgroup$ – user34829
The reason that the standard deviation is more commonly used as a measure of spread is that it has better properties than the mean absolute deviation in most contexts. One of the desirable properties of the sample variance (the square of the sample standard deviation) is that it is an unbiased estimator of the true variance for any sample of ...
The standard deviation is the square root of the variance. The standard deviation is expressed in the same units as the mean is, whereas the variance is expressed in squared units, but for looking at a distribution, you can use either just so long as you are clear about what you are using.