> the operational definition of no measurements outside of 2.66 times the mean consecutive difference between observations
Not even a simple Gaussian distribution can hold up to this standard of "stability" (unless I understood incorrectly what you mean here):
> data <- rnorm(1000) # i.i.d. normal data
> mcd <- 2.66*mean(abs(diff(data))) # mean consecutive difference * 2.66
> sum(as.numeric(abs(data) > mcd))/length(data) # fraction of bad points
[1] 0.002
Unless you are willing to add additional conditions (e.g., symmetry), I still don't see how criteria that pertain to variance and kurtosis (e.g., "the operational definition of no measurements outside of 2.66 times the mean consecutive difference between observations") can imply any strong relationship between the (sample) arithmetic mean (or any other mean) and the (population) median.
In fact, even distributions for which the "arithmetic mean is approximately equal to the median" claim is roughly correct will almost certainly not display the same property when you use some other mean (e.g., geometric or harmonic mean).
Either way, if you have some reference that supports the stated claim, I will be very happy to take a look at it (and educate myself in the process).
Not even a simple Gaussian distribution can hold up to this standard of "stability" (unless I understood incorrectly what you mean here):
> data <- rnorm(1000) # i.i.d. normal data
> mcd <- 2.66*mean(abs(diff(data))) # mean consecutive difference * 2.66
> sum(as.numeric(abs(data) > mcd))/length(data) # fraction of bad points
[1] 0.002
Unless you are willing to add additional conditions (e.g., symmetry), I still don't see how criteria that pertain to variance and kurtosis (e.g., "the operational definition of no measurements outside of 2.66 times the mean consecutive difference between observations") can imply any strong relationship between the (sample) arithmetic mean (or any other mean) and the (population) median.
In fact, even distributions for which the "arithmetic mean is approximately equal to the median" claim is roughly correct will almost certainly not display the same property when you use some other mean (e.g., geometric or harmonic mean).
Either way, if you have some reference that supports the stated claim, I will be very happy to take a look at it (and educate myself in the process).