Great point. We have been living abroad for the last three years so I do not have any AMC10 results. The UK uses the UKMT. On the intermediate challenge (Grades 8-10), I do not have his score on hand but remember it being 110-120 (out of 125). Award boundaries for 8-10th grade were:
Gold - 71+ points - 8% of entrants
Silver - 55-70 points - 17% of entrants
Bronze - 42-54 points - 25% of entrants
That implies a mean of ~42 points. I can’t work out the standard deviation since the distribution does not look normal.
Additionally, the cut-off for 8th graders to qualify for the next round was 95 points (Top 1500 nationally of ~100,000 test takers per year).
This test is optional for students so I’d assume the sample is already on the higher end of math aptitude.
CO_Dad