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I'm learning statistics and I can imagine pretty well what the standard deviation looks like (image here).

But, knowing that the standard deviation is the square root of the variance, I just can't figure out what that looks like.

Can anybody provide me with an illustration or a plot to help me understand that?

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    $\begingroup$ You are approaching this the wrong way. A graphical representation is not always the best way to look at things. Variance is just the square of the standard deviation, which you already understand. A better question is: why is the square interesting enough that it has its own name? The answer to that is that variances are additive (while standard deviation is not). $\endgroup$ – Szabolcs Nov 20 '13 at 17:31
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    $\begingroup$ Thanks for your replies. I do understand that the variance represents the dispersion of the values, and that the standard deviation includes 68.2% of the values in normally distributed, nominal number sets. Therefore, it must be interesting enough beecause it should kind of represent the area of the Gaußian distribution, but I can not calculate this area precisely. That's why I would like to understand that visually. Do you refer to 'additive' because the formula you provided (Bienaymé) doesn't contain the sum of(each value minus the difference to the average)^2 divided by (n-1) part? Thanks $\endgroup$ – nic Nov 21 '13 at 0:43
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I found this paper to be helpful. Hopefully you will find it useful too:

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9639.2010.00426.x

Paraphrasing what's explained in the article, if you understand SD, you can imagine that the squared deviation (x - mean of x)^2 can be represented graphically as a square (area = length of side ^2), where each side is the deviation from the mean for that observation. The numerator in the formula for variance is the sum of all n squared deviations, also known as SS, which graphically is the sum of the areas of all these squares. The sum of all these squares itself can also be represented graphically as a square, made up by smaller rectangles, all with the same width (square root of SS), but different heights (given by the squared deviation for that observation divided by the square root of SS). Again, the area of this bigger square is also SS. If you then convert this bigger square into (n-1) equal squares all with the same area, you get variance, which is the area of each one of these smaller square (aka SS/(n-1), or the formula for variance).

Interestingly, the length of each side of these variance squares will be equal to the standard deviation!

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