So if I had failure rate data, which in theory was independant (for each failure) and normally distributed, but in practice I knew it was probably clustered due to the various users number of items, service, duty and treatment of the parts, I could just submit this as multimodal data and get an idea of the likely distribution?
How would I aplpy a confidence test, or determine an interval for a given confidence, so I could define a likely reliability from the distribution, or would I end up with multiple piecewise intervals maybe?
I have a specific need in mind, but with this I am sort of a the edge of my detailed statistical understanding as it currently is.
I would tend to say granularity, which I take to be defined by surface area to volume ratio, but with the context of scale. It's sort of fractal, in a non-linear way.
So for this example, the number of connections you can get into a certain volume is limited by the the processing and receptor ability inside the volume, and the surface area to provide entry points, so this is potentially the ultimate metric of capability for a region/space/functional sub-system that is specialised to one function.
I make up a dimensionless unit for this, m2/m3 or metres squared per metre cubed, but it's not 1/m.
Surface area to volume ratio figures in all sorts of areas in life - lead is flammable in air if it is ground finely enough, do it under glycol, then rub it between your fingers until dry and then throw a flash of fire. (Clowns used to do this, not any more, obviously). But the reason powders are an explosion risk is surface area to volume ratio (eg flour, or the dust generated when machining grooves in buttons etc). All sort of chemistry and cooking depend on this factor, even if you don't realise when making your emulsion of tea or mayonnaise. And after all what is nano technology, but materials behaving differently at very small scale, and some of that is due to volume decreasing much faster than surface area when you get very small.
Take a 1m cube. 6m2/1m3 = ratio of 6.
Take a 1 cm cube 0.0006m2/0.0000001m3 = ratio of 600.
And that's still well into human scales, at a mm it's a ratio of 6000.
Stuff that needs to get in, like heat, chemicals or light, can get in to the entire volume much more easily and quickly, impacting the way almost everything works - reactions, heating and cooling.
Yet the flip side is if you want to abrade, coat or polish the surfaces, a huge amount more material or energy is now required because the surface area is going up exponentially, but the hardness, as one example does not lessen any (generally). All of a sudden the same volume of material needs a lot more paint, or grinding/polishing energy.
But also the contact areas change and fluidisation or non-newtonian behaviour can emerge. For a seemingly simple concept that doesn't really have an SI unit, it can have complex impacts.
For the brain, there is the wrinkling of the surface, meaning that for the volume of the head there is vastly increased surface area - this enables more connections and communication between "sub-systems" differing in specialisation, allowing specialisation within small regions, that can feed up in a hierarchy to consciousness, or unconscious thought, or whatever.
I suspect the surface area to volume ratio of brains compared between species could be a measure of "intelligence", because size alone seems to not matter as much as you might think it would.
"Smooth brain" is considered an insult of low intelligence, so somehow this concept is known in some way.
So, I have lots more of examples etc, but I have no qualifications or even formal learning in this area - I am an electrical engineer and I just made all this up (well I have been thinking about writing a book about granularity for while - the dimensionless dimension showing that size does matter, just the very small sizes more than the very large, because the nonlinearity really starts to kick in as you get smaller for all effects).
But, it is a fairly basic concept once you see it, and it would seem to potentially explain so much, so tell me where I am wrong?