Genesis Innovation
9811 West Charleston Boulevard
Suite 2556
Las Vegas, Nevada 89117 USA
(702) 524-2034
Genesis Innovation
9811 West Charleston Boulevard
Suite 2556
Las Vegas, Nevada 89117 USA
(702) 524-2034
- why the square, square root, and n-1?-
Why do we square the differences between our samples and the calculated mean? The normal distribution tends to be homogeneous around the mean, therefore differences between samples and the mean tend to be equal (or close enough). If, say the mean is 5 and there is a 2 to the left of the mean and another 2 to the right, these would end up summing to 0, which would be of no value to the calculation of Sigma. With the squares, however, the sum is 4: something we can work with.
What about the n-1? Why is it not simply n? When we calculate the mean before calculating Sigma, we in mathematical terms have "used up" one degree of freedom. But what does this mean (in a more practical way)? Good question. We use n-1 because of the former calculation of the mean. By knowing the mean and n-1 samples the nth (or last) sample could be easily calculated, so we would in essence be submitting "too much information" into the equation and biasing the result. As an example, if the mean of say five numbers is 4, and the first four terms are 4, 3, 5, 3, the last term must be 5; we cannot force the last term into the Sigma calculation.
Why the square root? The explanation is parallel to that of why we use the square, as noted above. Additionally, Sigma then "fits nicely" into the function that describes the normal distribution (shown below).
