>The choice of a curve isn't a random variable. The choice of a curve is a human selected, non-random variable.

You're misunderstanding statistics. If you have 5000 humans and let them each choose a choice for a curve. Then you select a random human to see what choice he or he/she chose, that choice might as well be a random variable. If you try to find the mode of which curving methodology the person chose you'll find one that's the majority. That's a good value to freeze the choice around.

https://www.wikiwand.com/en/Probability_distribution

For the purposes of this argument ^^ the above is the definition you should center your research around. Deeper meanings into "random variable" become too pedantic.

>IDK how you could possibly assert the curve bias is random.

Random variables don't technically exist unless you study quantum mechanics. It's not even a provable notion whether they actually do exist. But typically in statistics you can treat all measurements as taking a sample of a random variable. When you shuffle a deck of cards, when you draw from a raffle... or when a random school chooses what curve methodology it uses. The language I'm using here "random variable" might have thrown you off, but it's common parlance in science and statistics. Try re-reading my previous response with this knowledge in mind.

>You haven't proven this. It's a totally unsupported claim. There's zero evidence to indicate mere selectivity confers difficulty.

Again, please understand the concept of a random variable, then re-read the response again. Then you will see, the evidence is quite strong.

I'll restate it here for clarity. Read carefully. We can test if selectivity influences difficulty in a mathematical model of schools. First treat all parameters that are part of a model of a school as random. Because those parameters are random we can just freeze all those values at some average and make it the same for everything, because it's not something we're trying to measure... and likely these parameters cluster around some average anyway.

SO, as a result, we have a bunch of mathematical models of schools that are all EXACTLY the same. Each school model takes in an ass load of parameters as inputs, and outputs some number that measures "difficulty" as an output. Since all models are identical all input parameters yield the same Output.

To test "selectivity" we just take two identical models and fiddle with the "selectivity" input parameter. If the "difficulty" output changes when you adjust "selectivity" then you have proven selectivity is causal to difficulty for that model.

I don't have to know exactly what this mathematical model is. Just pieces of it. I know grading policy influences difficulty, I know the curve influences grading policy, and I know the amount of smart people in the school influences the curve, and I know selectivity influences the amount of smart people in the school.

Thus I know when you fiddle with that selectivity parameter, difficulty increases.

Your argument is saying that you can't ignore the other input parameters that the choice of the curve matters. I agree. It matters, but we're not trying to measure that. We want to throw it out of the measurement equation and see if selectivity influences difficulty. So we freeze the curve methodology at some arbitrary option and ignore it. We measure selectivity so that's the only parameter we fiddle with.

Think of it like this. You have 20 switches labeled with numbers 1 - 20 and a single lightbulb. Each switch may or may not be connected to that lightbulb. I don't know, but I only care about switch 5 and I want to know if switch 5 turns on that light bulb. I don't care about switch 3, 4, 6, or 7. Only switch 5.

I test switch 5 by freezing all other switches at ON then flipping switch 5 on and off to test if it has a causal effect on the light bulb. Switch 5 in this analogy would be "selectivity" and switch 7 can be "the choice for the curve methodology"

This is the exact SAME proof I'm using to show that selectivity influences difficulty. Same concept but instead switches are the input parameters and the output parameter is the lightbulb.