The list is in no way arbitrary. 10 is the answer you would find when you think it's a trick question. 12 is the correct answer assuming your logic is sounds. 15 is the correct answer when you have unsound logic.
Which are absolutely unmeasurable properties! Whether someone thinks something is a trick question or not is not a quantifiable thing, you're taking a system of booleans and trying to say they have some continuous basis. You're putting data through a flow diagram and trying to map a function to the end results.
If you were asking two questions at once where the questions had the same structure (apparently trick answer, correct answer, stupid answer, irrelevant answers), you could show people who pick the logical answer are more likely to pick the logical answer both times, or you could run this alongside an IQ test and map whether they get the correct answer against their intelligence, but you're not. You're collecting completely useless options. If I go round asking people a simple maths question and tick a box at the end that says "right" or "wrong", I can't map a function to that, I just have a rough estimate of what proportion of the population are stupid. I can't then fit a dirac delta function to that and go OH SNAP IT FITS, IT MUST BE SOME FUNDAMENTAL CONSTANT OF NATURE. Correlation implies you're taking one value and mapping it to another. You're just taking a value.
Correlation can be talked about. Does someone else's vote have any influence on another person's vote? That's correlation; What I was talking about, is seeing the correct answer someone is more influenced to guess the correct answer. There are correct answers and there are not correct answers, there is nothing arbitrary about it.. In other words, this can be mapped out in a time domain. What are you talking about no metric... Are you suggesting the central limit theorem does not hold for a binomial distribution (or a sum of bernoulli distributions)? Because it's just frequency right?
Of course it holds for a continuous distribution, but that's completely removed from arbitary values.
You're trying to spot a pattern where there is none. You think just because you've offered numerical values that that makes it possible to fit your data.
If the true value is 12, someone might get 15 because he did the maths wrong. He's not going to accidentally get 9 though unless he makes a royal mess up, in which case he could equally get any other of those options, despite it being exactly the same displacement from the true value.
It's like you're asking people to add large numbers in their head on a tiny time limit, in which case you'd expect some people to overestimate and some to underestimate, but it to generally centre around the true value. We're talking about a fundamental lack of understanding in how they process the numbers. If they understand the numbers so poorly that they would end up with an answer like 18, they're no more likely to mess up and end up with 7. There's no feasible pattern, it's just a case of pure misjudgement, so you're essentially trying to map a plot to random noise. And that's what it is: completely unreasoned, illogical choice is essentially background noise. You could get a random number generator on the results that aren't 12 and 15 and expect the same kinds of values.
You have proven to me that you know how to use statistics but you lack in the most basic foundations; you don't have a clue what assumptions you are making when using your statistics, you just know what people tell you.
And you seem to think that everything that has a number associated with it therefore has some statistical meaning. I don't doubt you'd try and fit background radiation.
Here's the essence of the situation:
For correlation, you need one variable to be dependent on another, or both to be dependent on another property.
You have data you're collecting (frequency).
You're saying that your data is dependent on how logical the people are. Which is true. But you aren't measuring how logical the people are, because you have no metric with which to do so! You're just measuring whether people get the answer right, but you aren't determining whether or not those people are logical or not, because you're comparing it to nothing!
The only way you could correlate this is to assign a "logicalness" number to each option (which is horrendously flawed in itself), and see what level of logicalness the data centres around, but at that point you're completely deciding what you think is logical yourself. You could argue that all of the incorrect answers are illogical as each other, because they're nonsense.
