No σ for you!

[carey]

(11-25-2024 12:54 PM)dm319 Wrote:  I never really got the whole squaring then square root thing. I'm my mind, the mean absolute variation seemed a more intuitive measure.

Since the mean of the deviations is 0, there are only two ways to avoid cancellation of the deviations when taking their mean. One way is to use absolute values of deviations (i.e., the mean absolute variation you mention). It works, but working with absolute values in subsequent equations becomes unwieldly.

The other way is to square the deviations, then take the mean and the square root as done in the standard deviation. In fact, the standard deviation is the root-mean-square (RMS) deviation or RMSD. If we read its name backwards (from right to left), applying one word at a time (like an RPL or FORTH program :), the SD algorithm is generated.

Step 1: Deviation \[x_{i} - \bar{x}\]
Step 2: Square \[(x_{i} - \bar{x})^{2}\]
Step 3: Mean \[ \frac{\sum{x_{i} - \bar{x})^{2}}}{N} \]
Step 4: Root \[ SD = \sqrt{\frac{\sum{x_{i} - \bar{x})^{2}}}{N}} \]