A pp (not me) asked AI a question, which replied that
"Privately educated individuals in the UK generally earn significantly more than their state-educated peers, with studies showing a substantial salary premium that widens over time, even when controlling for university, subject, and job"
You dismissed it, claiming that the AI question should be re-phrased to exclude 90-95% of state students, and compare only the outcomes for the most academic 5-10% of state students to the whole private cohort.
You mentioned cohort size (only 7% educated privately), but I assumed it was obvious that it's not necessary to have the same cohort size. That doesn't need explaining, does it? If everything else was equal, you'd expect the same spread of earnings outcomes - proportionally - even though there would more individuals at every income level who were educated in state.
I thought that was obvious and that your objection is to the difference in cohort spread of ability
The difficulty in comparing state and private outcomes is that private education is self-selecting, and requires material/career success in the parents... which correlates to parental ability... and IQ is heritable... so there's a different spread of abilities in the private cohort than in the state cohort. So when you compare outcomes, you need to be careful to make sure you're comparing the outcome for children coming from private against the outcome for equivalent ability children coming from state.
I pointed out that the AI answer controls for university, subject and job - so it's already comparing people with equivalent A levels (assuming that universities make offers equitably based on A level grades). That makes the comparison fair despite different ability across the respective cohorts
Ie when you group people people together who you would expect to have the same earnings based on their University degree and job (and by extension, equivalent A levels)... but then you split that out again by private/state... there is a big enough difference that you can conclude that education sector does change lifetime earnings outcomes.
Now I have no idea whether the data the AI based it's answers on is correct, whether the AI has hallucinated a wrong answer from correct underlying studies, or even whether the pp genuinely got such an AI answer or just made it up for fun.
My point is that based on that answer, your request to "compare private school pupils outcomes against the top 5 or 10% of state school pupils (as defined by GCSE/A level grades)." is unnecessary. In fact, it would give a less accurate result, since it assumes that the whole private school cohort matches the ability of the top 5-10% in state school. Which is an assumption, not based on anything.
You didn’t seem to understand my comment about the AI answer "controlling for university, subject, and job" so I gave more detail. You still don't seem to understand it, but I'm not sure how else to explain.
You need to pay attention to phrases like "controlling for" - and make sure you really understand how people use underlying data to reach meaningful conclusions - if you want to understand the conclusions you read about.