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I understand from my lecturer that variance an standard deviation are central to statistics.

I do not understand the signifigance of both values, except that both measures the variability, and variance is the square of standard deviation. Why is there a need for two standards then?

Why must sd be squared to obtain variance? Why can't it be sd cubed, or even sd square rooted? Wouldnt sd cubed give us a more esaggerated value, which is better?

Also, how were these standards invented? They seem so non intuitive to me, then just takig a value then taking the difference wrt to the mean

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Well , s.d. can be cubed or squared to obtain " fractional order moments about mean " , though I have not seen such kind of a thing anywhere. – Souvik Dey Oct 19 '12 at 5:21
The variance has a number of nice properties, most importantly that the variance of a sum of independent random variavles is the sum of the variances. – André Nicolas Oct 19 '12 at 5:35
The nice thing about standard deviation, on the other hand, is that it scales like the random variable ($\sigma_{aX} = |a| \sigma_X$), so it has the same units as the random variable. – Robert Israel Oct 19 '12 at 5:55
up vote 2 down vote accepted

Variance is one of the so called moments and plays a very important role in Statistics. To be more intuitive just imagine how would be a measure of variance if you just take the normalized sum of differences between all values around some central value: those differences who are negative will influence in a mistaken way your dispersion measurement.

There are many applications of these concepts. To cite one example: with standard variation, engineers can guarantee if 99% of all production of an industry will lay within a specified interval of tolerance.

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If you need further information, see the Central Limit Theorem – Veeggon Oct 19 '12 at 6:01

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