Understanding Standard Deviation: Squaring Variances

[tumelo]

Can somebody explain to me why we have to square the variances when calculating the deviation and then finding the square root(whc is suppose to reverse the squaring) ,it doesn't make sense to me

 

[mathman]

It is simply a matter of definition. Let X be a random variable, and let A=E(X) (average).
Then the variance V is DEFINED by V=E((X-A)²) and the standard deviation is DEFINED as the square root of the variance.

 

[Mute]

Can somebody explain to me why we have to square the variances when calculating the deviation and then finding the square root(whc is suppose to reverse the squaring) ,it doesn't make sense to me

If you want to find the typical deviation from the average, you can't calculate the average of x -<x>, because that average is zero. You need a measure where the deviations from the average don't cancel out. Taking the average of |x-<x>| works, but that's not nice because the absolute value function isn't differentiable at zero. (x-<x>)^2 also works, and is differentiable. However, if x has units, you can't compare <(x-<x>)^2> directly to x, because the units don't match. So, you need take the square root of to get something with the same units of x that you can treat as a deviation from the mean.

 

[chiro]

Can somebody explain to me why we have to square the variances when calculating the deviation and then finding the square root(whc is suppose to reverse the squaring) ,it doesn't make sense to me

One other thing to keep in mind is that the definition provided for obtaining the variance is used directly in statistical theory like the normal distribution: ie the std deviation that is calculated with the variance that uses difference in squares is used in the normal pdf.

Also you'll find that this definition is useful when dealing with other properties of random variables.

Another thing to keep in mind is that you could picture the variance as a length in an n dimensional euclidean space where n is the number of elements in the random variable.

For example if we have a three dimensional vector where X(1), X(2), AND X(3) represent the difference between the average and the element of the random variable, then the "length" of this vector is basically found using the pythagorean theorem where length = SQRT(X(1)^2 + X(2)^2 + X(3)^2). This definition makes sense when you interpret this geometrically as the length of a vector in an n dimensional euclidean space.

 

[CellCoree]

Standard deviation is a measure of how spread out a set of data is from its mean. It is calculated by finding the square root of the variance, which is the average of the squared differences from the mean. The reason for squaring the differences before finding the average is to eliminate any negative values that may occur when calculating the differences. This ensures that all values contribute positively to the calculation and results in a more accurate measure of variability. Taking the square root at the end reverses the squaring process and gives us a value in the same units as the original data. This process is a standard mathematical practice and has been found to be the most effective way to measure variability in data.