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I know how variance is defined in statistics, but I do not know why variance is defined that way for describing how a set of numbers spread out.

Why can't it be defined other way around?

Can anyone provide the reason why it is defined like this? (Explanataions with examples like uncertainty principle in physics would be appreciated.)

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There are many useful notions of how "spread out" a collection of numbers is. Variance happens to have some nice technical properties, most importantly that the variance of the sum of two independent random variables is the sum of the variances. –  André Nicolas Aug 2 '12 at 5:59
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What exactly is the "other way around"? –  J. M. Aug 2 '12 at 6:01
    
Apropos to @André's comment, see this. –  J. M. Aug 2 '12 at 6:02
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Have you seen the previous question on the motivation behind standard deviation? Variance is just the same except without the square root on top. –  Rahul Aug 2 '12 at 6:11
    
The variance is also useful because the standard deviation (square root of the variance can tell you through the Chebyshev inequality what minimum percentage of the distribution is within k standard deviations of the mean. Also for the normal distribution and distributions close to the normal 68% of the distribution is within 1 standard deviation of the mean, 95.4% within 2 standard deviations and more than 99% within 3 standard deviations. –  Michael Chernick Aug 3 '12 at 19:11

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