Investigation into the standard deviation has given me more insight into why it is defined as it is. It seemed a sensible definition when I was introduced to it, but I didn’t question the motivation for defining it in that particular way, when there may have been other possibilities which, without any further explanation, seemed just as valid. For example, the absolute deviation. Recently, however, I learned of a geometric relationship between the standard deviation and the mean which justifies its usage as a measure of deviation from the mean (it has to do with how distance is measured). A similar relationship exists between the median and the absolute deviation. In either case, the choice of how to measure deviation is the natural choice.
Does this geometric approach lend itself to analysis of higher order moments, or parameters or statistics based on them, such as the skewness and kurtosis? Is there a geometrical relationship between the third moment, the mean, and the standard deviation that makes the definition of skewness natural? Likewise for the 4th moment and the skewness, or higher moments and parameters defined in terms of them.
Can one define an “absolute skewness” or an “absolute kurtosis” in a natural way that describes the asymmetry or the peakedness about the median?
