Well, squaring is needed because it is "reasonalbe" to measure deviation in that way.

However, there is a simplified Variance formula. It is useful if you are doing it by hand:

"Squared-mean minus mean-squared".

Mean is the average squared. And squared-mean is the mean of the squares. Thus,

Where is the mean, and is the squared-mean. It is also easy to remember because "mean" means the average thus it is in the middle of the sentence. NOT! Mean-squared minus squared-mean. Now to find the standard deviation just take the square root of the variance.