I'm still waiting for your citations of the statistics literature as to why that can't be done.
I'm also waiting for you to actually look at and think about that joint-probability distribution I've posted before of karyotypes & heights. You might reasonably quibble about the order,
You answered your own question. The "assigned" order is EXACTLY why you can't get a bimodal distribution from categorical data. It only superficially looks bimodal, as an artifact of an arbitrary and artificial order.
Imagine a fruit basket, containing:
30% Apples
20% Oranges
10% Bananas
15% Grapes
25% Mangoes
If you were to "order" the fruit as I've listed them (A, O, B, G, M), you'll have something kind of like a bathtub distribution.
If, on the other hand, you order them (B, G, O, M, A) you get something more resembling an exponential distribution
If you decide to plot them (B, O, A, M, G) you get a slightly skewed but relatively normal-looking distribution.
And if you plot them (B, A, O, M, G) you'll get something that looks like a bimodal distribution.
And THAT'S entirely the point. Because the data is categorical, it has no inherent order. You cannot say that apples are > bananas. The sentence has no meaning, because there is no ordinal criteria under discussion. You could potentially say that the weight of apples > the weight of bananas - because weight is an ordinal variable. You could say that the number of apples is > the number of bananas, because number is an ordinal variable.
But apple has no ordinal value. What a plot looks like is 100% entirely dependent on an
artificially imposed order that has nothing at all to do with the elements on the x-axis.
The same is true with your multidimensional speculation - it does not work when you mix ordinal and categorical data. It makes one of your n-dimensions meaningless.
There are methods for integrating categorical data into statistical analysis, so that you don't **** up the whole thing. But it is NOT done by assigning some random "order" to them. That biases the analysis beyond redemption.
but don't see how you reasonably deny that there IS a family of probability distributions for heights for EACH of the karyotypes listed. All of which might reasonably be put into the joint probability distribution shown.
I 100% deny that there is a family of probability distributions for the height of each karyotype. If you wish to argue about that, I will need you to tell me WHAT you are MEASURING that shows VARIANCE AROUND the karyotype. Let's make it simple: How would you calculate the mean of karyotype 46XXY? How would you determine the standard deviation?