How to properly calculate average need-based aid met by an institution?

Question

As an institution we give out a certain amount of financial aid every year (need-based aid) to each student. This need-based aid does not always meet the students estimated need. My task is to compare this need-based aid vs. estimated need for all students. I am trying to calculate an average of percent need-based aid met by my institution and I want to use the most precise and statistically-appropriate way to get this average (bear with me for I have never taken statistics before, and just testing my knowledge out with this simple task).

The following is a sample of data that I have come up with and will use it to show how I have been calculating the average:

The first way I have done this is by summing up column C and diving it by the sum of column B, this gives me an average percentage of 23.7%.

The second way I have done this is by getting a percentage for each individual student and then getting an average of all the percentages so: take C1/B1 + C2/B2 + C3/B3 + C4/B4 and then divide this by the number of students, which is four. This calculation gives me an average of 39.5%.

As you can see, both my averages are different (as they should be since I used different methods to calculate them). My question is which method of calculating average is better in this case? And if you know of any better way to calculate an average please do tell.

Answer 1

You need to make three decisions about what you are trying to accomplish. One is whether you want to look at this from the perspective of the institution or of the individual students. Second is whether percentages are really the best way to present the data. Third is whether the average alone provides enough information.

Your first way of doing the average represents the situation from the perspective of the institution. If you wanted to go to a donor you could say: "We only have enough resources to meet 24% of our students' needs. Please give us more money." The second way you did the average looks at it from the students' perspectives: what is the average percentage of need that an individual student can expect to receive? That is nearly 40% in your data.

The difference in these values is because you are working in percentages. That might not be a good way to proceed, the second decision you have to make. When I was the father of a college student receiving financial aid, I didn't care much about the percentage of need that would be met. I cared about the number of dollars we would have to come up with on our own. For example, say that a student had only \$500 need and received no aid. That would only be 0% need met. But is that student really worse off than the first student in your table, who received 2.3% of need but still had to come up with \$21,000? That's a good example of how misleading percentage values can be.

Third, your examples show a wide range of need met among the students. If that is the case in practice, then you need to include some measure of the differences among students, not just the average. If your institution advertised that they met 40% of need for the average student, the student receiving 60% might not complain, but the one receiving 2.3% would. And the differences among the students might be something the institution would want to deal with even without increased financial resources.