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Why to take square in the formula of standard deviation?
The sum of deviations from the mean will always be 0 and so does not provide any useful information. The absolute deviation is one solution to tat, the other is to take the square - and then take a square root.
When computing the sample variance the sum of squared deviations about the mean is used for what reason?
You want some measure of how the observations are spread about the mean. If you used the deviations their sum would be zero which would provide no useful information. You could use absolute deviations instead. The sum of squared deviations turns out to have some useful statistical properties including a relatively simple way of calculating it. For example, the Gaussian (or Normal) distribution is completely defined by its mean and variance.
How do you find the average absolute deviation?
Add all the absolute deviations together and divide by their number.
Why do you square the deviations and then turn around and find the square root of the sum of the squares?
If you simply added the deviations, their sum would always be zero. The derived statistic would not add any information. Essentially, the choice was between summing the absolute values or taking the square root of the squares. The latter has some very useful statistical properties.
How do you calculate the sum of the deviations from the mean?
multiply the mean by the amount of numbers
