$250k is low average. When you look at average debt for med students, you have to leave out the students who aren't taking on debt, such as parents are paying their way. That's about 35% of the pie. (I lost my link to a recent report showing this - it was between 30% and 40%.) I'd guess out of a class of 150 students, you'll have maybe 20 who get a free ride scholarship. Others want to take a guess at this?
Scholarships totally depend on the school and your CV. If you're at an Ivy, there are more scholarships offered. If you're from a rural area or you're URM, there are more scholarships. The vast majority of scholarships are small.
At my school there is a long list of merit-based and need-based scholarships, most of which are for around $3k per year. That's against $60k instate and $80k out of state cost of attendance.
Make sure you really want to be a doctor, and if you do, then you have to just learn how to not think about this too much. Maybe 2x/yr look through what's new with the federal repayment programs. And good lord don't consider private loans with their shiny short-term rates, because these are not subject to the repayment programs for federal loans.
Best of luck to you.