Does anyone know what is the name of this formula?
$$M_i = \displaystyle\frac{0.6745(x_i – \hat{x})}{\mathrm{MAD}}$$
where $\textrm{MAD}$ is the median absolute deviation and $\hat{x}$ is the median of $x$.
Does it appear in some scientific publication? I also wonder where the constant comes from (0.6745 is roughly 29/43). I am using it for outlier detection.
Best Answer
Suppose $x$ follows a standard normal distribution.
The $\mathbf{MAD}$ will converge to the median of the half normal distribution, which is the 75% percentile of a normal distribution, and $\mathbf{N}(0.75) \simeq 0.6745$
Since you are multiplying by $(x-\hat{x})$, this means that, for any normal distribution, your formula will converge to 1 for a large enough sample size.