Why Square Of Difference In Standard Deviation Formula? Why Not Just Square

If we just add up the differences from the mean ... the negatives cancel the positives:

4 + 4 − 4 − 44 = 0

So that won't work. How about we use absolute values?

|4| + |4| + |−4| + |−4|4 = 4 + 4 + 4 + 4 4 = 4

That looks good (and is the Mean Deviation), but what about this case:

|7| + |1| + |−6| + |−2|4 = 7 + 1 + 6 + 2 4 = 4

Oh No! It also gives a value of 4, Even though the differences are more spread out.
So let us try squaring each difference (and taking the square root at the end):

		√( 42 + 42 + 42 + 424) = √( 64 4 ) = 4
		√( 72 + 12 + 62 + 22 4) = √( 90 4 ) = 4.74...

That is nice! The Standard Deviation is bigger when the differences are more spread out ... just what we want.
In fact this method is a similar idea to distance between points, just applied in a different way.
And it is easier to use algebra on squares and square roots than absolute values, which makes the standard deviation easy to use in other areas of mathematics.