How to calculate a random measurement error?

[Lotto]

TL;DR

How to calculate a random measurement error? If we did a measurement and want to calculate a random error, what formula are we to use?

I have seen this formula

$$\sigma=\sqrt{\frac {\sum_{i=1}^{N}{(X_i- \bar{X})^2}}{N(N-1)}}$$

but also this formula $$\sigma =\frac{\sum_{i=1}^{N}{|X_i- \bar {X}|}}{N}.$$ Which of them is correct?

 

[BvU]

The first one

$\$

 

[Dale]

The first one is the standard deviation. That is the usual one. The second one is the mean absolute deviation and it is rarely used at all.

 

[Ibix]

The second one is the mean absolute deviation and it is rarely used at all.

Some machine learning applications seem to use it - I think because it's cheaper to calculate, so it gives you a time saving if you can live with the less mathematically nice behaviour. But in experimental physics I agree it's a no, you want the standard deviation.