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I'm currently working on exam review for an upcoming statistics exam and I've managed to dig myself to far into the theoretical background of basic statistical principles. I'm currently looking at the definition of variance and standard deviation.

I understand that when given a sample, lets say $S=\{s_1, s_2, s_3, s_4, ...., s_n\}$ from an observation I can calculate the following $$d(S,\bar{S})=\sqrt{(s_1 - \bar{S})^2 + (s_2 - \bar{S})^2 +\ ...\ + (s_n - \bar{S})^2} = \sqrt{\sum_{i=1}^n(s_i - \bar{S})^2}$$

which is the Euclidean distance or as I interpret it, the combined deviation of all my observations.

My question is then, is the standard deviation the same as the average deviation? Also, could someone explain to me the reasoning for dividing by $n - 1$ instead of $n$ without involving the use of moments? And why do we divide inside the squareroot?

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The standard deviation is the square root of the average squared distance from the mean. From that definition, the average (division by n or n - 1) is determined before taking the square root. Without the square root, the value is the variance. For smaller samples, the division by n-1 is a more accurate estimate of the population standard deviation.

Average deviation is simply the average distance from the mean.

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  • $\begingroup$ Is there reasoning behind the definition of standard deviation or did it simply fit with other models? I'm just trying to understand how one came up with standard deviation and variance as good ways to interpret distributions. Instead of using (for example; the average Euclidean distance or the range of the results). Thanks for taking your time :) $\endgroup$ – Gjert G May 17 '18 at 18:44
  • $\begingroup$ The standard deviation was developed as part of the statistical analysis process. With it. one can determine the probability of an outcome from a sample study. Example, a result outside of 2 standard deviations from the mean has only a 0.05 probability of occurring by chance. When we get this result, we usually say that it's such a low probability that something else must be going on to influence it. So the standard deviation gives us the ability to estimate probabilities of outcomes by partitioning the normal distribution curve into known probability regions. $\endgroup$ – Phil H May 17 '18 at 19:35
  • $\begingroup$ I understand why we use it as I've read up on hypothesis testing and regression by using this approach. I'm more interested in the reasoning, not the use. I get that it suits the models we use, however, they are constructed to suit the standard deviation, which I'm curious as of why one chose to define it as it is. $\endgroup$ – Gjert G May 17 '18 at 19:47
  • $\begingroup$ I'm currently reading this Wikipedia page about the derivation. However, dont understand the third and second last step of the derivation. Could you help me out here? $\endgroup$ – Gjert G May 17 '18 at 20:10
  • $\begingroup$ Standard deviation was invented more than 100 years ago. Its use is more to do with tradition than anything else. The arguments for its use in the past won out over other measures. I would say that was because standard deviation was more accurate in it's predictions than say mean absolute deviation. In my opinion what seemed to work best won out. If you want further explanation for why I think that, please ask. In the wiki article I pulled up, I don't see a derivation heading. Do you have a link? $\endgroup$ – Phil H May 17 '18 at 20:25

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