It's common for QC purposes, when there's a suite of measurements in different units and with different distributions, to normalise their display such that each value is shown as a deviation from a chosen central tendency (mean, etc), scaled according to its dispersion (sd, interquartile, etc). [I believe this is generally known as standardization, but am not a statistician]. Particularly when displayed alongside the same for a large number of samples, this can be an effective way to spot "defects" worth investigation.
How is this process best refined when it comes to skewed distributions, particularly when related to dispersion? I understand the common options for central tendency choice when it comes to skewness (and outliers, etc), but don't know anything similar to "x standard deviations" when a distribution is very skewed.
The skew on the data I'm concerned about is not pathological in any mathematical sense: usually it is just something which "looks" more like $xe^{-x^2}$ than $e^{-x^2}$, and restricted to positive $x$.
The best I've come up with so far is to scale each side of (whatever) chosen central tendency individually, but the discontinuity at the centre is troubling me. (For example, if above a mean to show a line of length scaled by the 2-3 quartile range but if below scaled to the 1-2 quartile range).
Are there better ways? Does going any further absolutely require a thorough knowledge of the nature of the underlying distribution in a situation like this? (Because we don't have it!) The data is only used for initial, rapid visualisation of a large number of samples side-by-side, for investigative purposes.
