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I wanted to measure the "average distance from the average value", but I can't understand the "standard deviation". Many text books say the standard deviation is used to "summarize" the deviation, but I have trouble understanding why it uses squares to do it.

If I have many x numbers the formula is:

$$\sigma = (\frac{1}{N} \sum_i \lvert x_i - avg \rvert^\color{red}{2.0})^{1/\color{red}{2.0}}$$

So, there it is... that magic number 2.0. It shows up twice. What is so magic about it? Can't I use 1, 3, 4, or 6? or even 1.9, or 2.1?

Why 2.0?

Edit: I just learned that 1 is called "mean deviation".

Joe DiNottra
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  • If we didn't square the numbers the positive deviations would cancel the negative deviations which would not give an accurate picture of the dispersion. – John Douma Aug 31 '20 at 02:15
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    Have you tried to search for similar questions? My first results via Google give https://math.stackexchange.com/q/717339/96384, https://math.stackexchange.com/q/684257/96384, https://math.stackexchange.com/q/1200211/96384, https://stats.stackexchange.com/q/118 and in there I found https://math.stackexchange.com/q/4787/96384 and https://mathoverflow.net/q/1048. All this goes to show it's a very good question, but it has been discussed many times. – Torsten Schoeneberg Aug 31 '20 at 02:20
  • The math works out nicely when the exponent is 2. – littleO Aug 31 '20 at 02:27
  • @littleO Just trying to understand. So that means that in the age before computers it was easier to compute. In other words, there's no real meaning behind the number "2.0" but it's more of a practicality, to save time/money and go home early. Is that what you mean? – Joe DiNottra Aug 31 '20 at 02:29
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    It is not just about "time/money". It has many practical reasons. A bit of research on your behalf would clarify that. –  Aug 31 '20 at 02:33
  • Not only in the age before computers. Calculations that we want computers to do might also be much easier or faster if the exponent is 2. One might also say that the math works out more beautifully if the exponent is 2. There are nice formulas and theorems that involve the variance, as we define it. – littleO Aug 31 '20 at 02:39
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    @TorstenSchoeneberg Thank you, thank you, thank you. I didn't know how to express the quesiton until I asked it. It seems there's a lot of recent debate about it. Very interesting. This has bothered me for a long time. I'm so happy I was not going crazy. At least now I know I'm not wrong in questioning this... and I'm not alone either. – Joe DiNottra Aug 31 '20 at 02:54

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