The overall average hides 2 different numbers

arm splits clinical-trial into 2 groups. Here is what baseline_score looks like inside each.

• treatment — mean 61.28, median 61.35 (450 rows)
• placebo — mean 61.00, median 61.30 (450 rows)

Top to bottom that is 61.28 against 61.00, a spread of 0.5%. The pooled average is 61.14.

select arm,
  count(*)                                                    as rows,
  round(avg(baseline_score)::numeric, 2)                      as mean,
  percentile_cont(0.5) within group (order by baseline_score) as median
from clinical_trial
group by 1
order by mean desc;

The groups are close enough that the pooled average is a fair summary. That is worth confirming rather than assuming: the check costs one query and the failure mode is invisible.

Notice the mean and median columns disagree only slightly here. Always compute both in the group-by. The comparison between them per segment is free and tells you whether you are looking at a level difference or a tail difference.

This is the setup for Simpson's paradox — the case where every segment moves one way and the total moves the other.