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as a newbie,it keeps bugging me,why we need do square the difference rather than absolute value in root square mean as well ,if we need to calculate average size of data set say (distance from an arbitrary point like in one dimension with direction),and what is the exact difference between standard deviation and variation?

Erebius
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    When you search for this information on the internet, what isn't clear? – David G. Stork Sep 21 '24 at 16:47
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Sep 21 '24 at 16:48
  • @DavidG.Stork yes,it was "it is the best way to measure spread and squaring the differences would become more intuitive" – Erebius Sep 21 '24 at 16:55
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    It's a choice. It is often better to work with Absolute value for statistical reasons, but the squared version is analytically much, much better. See this question – lulu Sep 21 '24 at 17:00
  • @lulu why is that analytically better? – Erebius Sep 21 '24 at 17:04
  • Because squaring is analytic and absolute value isn't even differentiable. That's a huge problem. – lulu Sep 21 '24 at 17:07
  • @Erebius The absolute value function is very pointy at the origin. ;) It's nicer to work with a function that has a well-behaved derivative. – PM 2Ring Sep 21 '24 at 17:09
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    @lulu why weed to differentiate sd,in first place.since sd is constant – Erebius Sep 21 '24 at 17:12
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    It is easier to minimize the square instead of the absolute value. The reason we use the mean is that it minimizes the square of the errors. Another reason to care about squares is that Gaussian distributions naturally have squares and Gaussian show up a lot – whpowell96 Sep 21 '24 at 17:14
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    Please look up Mean Absolute Deviation. I told you it is a choice. It's not a question of right versus wrong. You can compute whatever you want. But usually, you want to manipulate the computation in some way or other, and these manipulations can run into problems. – lulu Sep 21 '24 at 17:17
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    In my opinion, the most important aspect of the variance is that, together with the mean, it completely characterizes the normal distribution. So, if you believe your distribution is close to normal, which is often the case due to the central limit theorem, then finding the variance can nail down your distribution of interest. Also, variance naturally yields covariance through polarization, a useful tool for measuring how two distributions are "entangled". In contrast, absolute value does not immediately give rise to a tool for measuring relatedness. – Sangchul Lee Sep 21 '24 at 17:24
  • Of course, as others have already stressed out, variance is merely a choice for measure of spread. There are many other choices, including $\mathbf{E}[|X-\mathbf{E}X|^p]^{1/p}$ for $p\geq 1$ (with $p=2$ corresponding to the variance), quantiles, entropy, to name a few. – Sangchul Lee Sep 21 '24 at 17:28
  • @SangchulLee For a normal distribution, the mean absolute deviation is $\sqrt{\frac2\pi}$ times the standard deviation, so knowing one tells you the other. This suggests that to me characterizing the normal distribution is not a particularly strong argument, though there are many others. – Henry Sep 22 '24 at 00:50
  • @Henry, You are right. Maybe I should have mentioned why variance is a natural choice for a parameter of the normal distribution. And unfortunately, this point becomes clear only when we consider multivariate normal distributions, so I guess my comment does not immediately address to OP's question. – Sangchul Lee Sep 22 '24 at 10:16
  • @SangchulLee - yes, meaningful and easily calculated correlation and covariance is indeed one of the many justifications for using variance. – Henry Sep 22 '24 at 16:14
  • Essentially the same question: https://math.stackexchange.com/questions/4787/motivation-behind-standard-deviation and https://stats.stackexchange.com/questions/118/why-square-the-difference-instead-of-taking-the-absolute-value-in-standard-devia – Henry Sep 22 '24 at 19:53
  • https://math.stackexchange.com/questions/3071367/whats-so-special-about-standard-deviation also shows some reasons for using the variance and standard deviation – Henry Sep 22 '24 at 20:28

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