Method of collecting and comparing outliers from sets of sets of populations Background
I am a PhD student co-supervising a Master's student in our lab. I am mostly familiar with discrete mathematics, signal processing, and programming simulations. My statistics background exists, but it is not formal or comprehensive. Without disclosing too much detail, the student's project is exploring an old biological conjecture. The idea being:
Say you have a set of populations called the long-beaks, and each population's beak length $\sim N(\mu, \sigma)$. Now say you look at the distribution of all           population $\mu$'s, the statistical interpretation of the conjecture is something like, the more a $\mu$ is extreme in value (e.g. an outlier, but not necessarily an outlier) w.r.t. to the distribution of all $\mu$'s, then its $\sigma$ will also be extreme w.r.t. to the distribution of all population $\sigma$'s. In other words, if a set of populations share a common defining trait, those populations which are further from their shared central tendency are more likely to be highly variable in that trait.
Problem
There are two questions we have been continually re-visiting, and have still not settled on.

*

*What is the most appropriate method of identifying those extreme values within a set of populations.

*What is the best method of determining whether this rule holds across multiple sets of populations.

What We've Considered So Far
As for 1):
We have played with idea of using the coefficient of variation, the MADM, quartile coeffecient of dispersion, and the variation of log-transformed data, as a function of the mean or median trait values. We've experimented with arbitrary cut-offs of extremeness, and have also used methods like z-scores or the Mahalanobis distance to identify outliers (assuming $\mu \: \& \: \sigma$ are multivariate normal).
Is there anyone best method -- a most robust method? Or is perhaps a suite of methods more appropriate?
As for 2):
We've a hypothesis of how the data might look, should the data evidence our hypothesis. Say we min-maxed the $\mu$'s and $\sigma$'s of each set of populations, collected the 'extreme' cases from each set, and superimposed them all on the same plot. There are three plots we think that would support the hypothesis:

*

*A cluster of points in the upper-left of the 0-1 space.

*A cluster of points in the upper-right.

*A cluster of points in the upper-right and left.

These cases all depend on the datasets collected; whether extremeness tends to be to the left and/or right of the distributions.
A plot that would not support the hypothesis is any of the last 3 cases, but where the cluster spans the y-axis and/or x-axis, i.e. there is no relationship between extreme $\mu$ and $\sigma$.
Are we perhaps overthinking this problem? Or has a common statistical test completely gone over our heads?
Thank you very much for your advice.
 A: Comment in Answer formal to show figures:
One common method of identifying 'outliers' is to use a boxplot. Typically, boxplot outliers are greater then 1.5(IQR) above the upper quartile or smaller than 1.5(IQR) below the lower quartile (where IQR is the distance between lower and upper quartile). Ends of boxes in boxplots are distance IQR apart.
It is not unusual for truly normal samples of as many as 100 observations to show several outliers. More than half of the normal samples plotted below show boxplot outliers. Outliers are shown as black dots in the boxplots below. [It would help if you could say what importance 'outliers' may have in your particular situation.]
set.seed(2021)
n=100; m = 20
hdr = "n=100: Boxplot Outliers in 20 Samples from NORM(100,1)"
x = rnorm(m*n, 100, 1)
g = rep(1:m, each=n)
boxplot(x ~ g, col="skyblue2", pch=20, main=hdr) 


If the the normal $\sigma$ increases (say from $\sigma=1$ above to $\sigma=15$ below), that does not increase the frequency of boxplot outliers. (Everything scales proportionately).  [If you mean something different by increasing the normal standard deviation, please give details.]

set.seed(2021)
n=100; m = 20
hdr = "n=100: Boxplot Outliers in 20 Samples from NORM(100,15)"
x = rnorm(m*n, 100, 15) # change in this line only
g = rep(1:m, each=n)
boxplot(x ~ g, col="skyblue2", pch=20, main=hdr) 

