From the NAIS web site -
Need-based Financial Aid
How much aid is available? For the academic year 2005-06, 972 National Association of Independent Schools (NAIS) members awarded $957.7 million in financial aid. These same schools reported that 18.7 percent of their students received some financial aid. The average award for boarding schools was $17,295 and for day schools, $9,232.
So lets just say 19% of the football team receive need-based financial aid.
BA roster of 100 players means 19 recieve need-based financial aid on average.
The school administrators and coaches do not determine which student gets the aid.
The parents application, not the students 40 yard dash time, goes to a "clearing house" where the student with the most need gets the aid.
Most aid is capped around 75% of the tutition.
Parents will still pay for books and meals.