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Trying to understand why Standard Deviation (SD) is widely accepted as a measure of dispersion instead of Mean Deviation (MD) $\frac {1}{N}\sum|X-\mu|$.

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 ...

which is from:

Fisher (1920) countered Eddington's empirical evidence with a mathematical argument that SD was more efficient than MD under ideal circumstances, and many commentators now accept that Fisher provided a complete defense of the use of SD (e.g. MacKenzie 1981, Aldrich 1997). Fisher had proposed that the quality of any statistic could be judged in terms of three characteristics.

  • The statistic, and the population parameter that it represents, should be consistent (i.e. calculated in the same way for both sample and population).

  • The statistic should be sufficient in the sense of summarizing all of the relevant information to be gleaned from the sample about the population parameter.

  • In addition, the statistic should be efficient in the sense of having the smallest probable error as an estimate of the population parameter.

Both SD and MD meet the first two criteria (to the same extent).

According to Fisher, it was in meeting the last criteria that SD proves superior. When drawing repeated large samples from a normally distributed population, 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 than its plausible alternatives as a way of estimating the standard deviation in a population using measurements from a sample (Hinton 1995, p.50). That is the main reason why SD has subsequently been preferred, and why much of subsequent statistical theory is based on it.

Question

Please explain in layman's words:

  1. Why using SD again to compare MD and SD and why it can be a criteria to compare?
  • standard deviation of their individual mean deviations
  • standard deviations of their individual standard deviations
  1. Why SD is less is more consistent as an estimate?

(SD of MD) is 14% higher ... thus SD of such a sample is a more consistent estimate

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kjetil b halvorsen
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  • Does this answer your question? [Why square the difference instead of taking the absolute value in standard deviation?](https://stats.stackexchange.com/questions/118/why-square-the-difference-instead-of-taking-the-absolute-value-in-standard-devia) – Pitouille Dec 13 '21 at 13:25
  • Do you mean why we care about standard deviation or why we use sample standard deviation as an estimator of population standard deviation as opposed to sample MD? Those are different questions. – Dave Dec 13 '21 at 13:25
  • The fact that the SD requires ideal conditions in order to be efficient makes it not an ideal measure of dispersion. Consider instead Gini's mean difference, mean absolute difference, median absolute difference, and other measures. Note that when normality holds, Gini's mean difference is 0.98 as efficient as SD. – Frank Harrell Dec 13 '21 at 13:45

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