Re-understanding Standard Deviation

Hongtao Hao / 2022-08-27

The other day I was thinking about the meaning of standard deviation:

$$\sigma = \frac{\sum_{i=1}^{N}(x_i - \bar{x})}{N-1}$$

What is it about? I think it is measuring the expectation of the deviation of a variable from its mean. Then why don’t we just measure the absolute deviation: $E|x_i - \bar{x}|$?

I saw this discussion on Stack Overflow and was taken aback by it. Reed’s answer was especially inspiring for me. Indeed, if we consider $x_i - \bar{x}$ as a vector in an $N$ dimensional space, then standard deviation is in fact measuring the length of this vector using Euclidean distance whereas the absolute deviation is measuring the length of this vector using Manhattan distance.


Last modified on 2022-10-01