# How to account for reduced student capacity when calculating program retention?

I work at a nonprofit youth center that has 2 distinctive programs throughout the year. Our afterschool program runs during the school year, and our summer camp program runs during the summer.

I am working on calculating our student retention between Afterschool 2019-20 and SummerCamp 2020 to see which students we retained, however, Between AS2019-20 and SC2020, we reduced our maximum student capacity by 10. I figure that this definitely impacts the retention rate calculation, but I have no idea how to account for this reduction in capacity. Does this impact anything, or am I overthinking?

I have to assume it would be some sort of calculation like returned students / eligible students = retention and then maybe dividing that percentage by the quotient of SC2020 capacity / AS2019-20 capacity = proportion to account for the proportion. Is this correct?

### Rationale

Some of the terms are a little vague, particularly what you refer to as eligible students and returned students. I'll set some variables for clarity, but tell me if I defined them incorrectly.

I assume them to mean:
eligible students $$= A$$ being the set of all students in the after-school program 2019-2020
returning students $$= A\cap S$$ where $$S$$ is the set of all of summer camp 2020 students

Now we can define retention rate to be $$\frac{|A\cap S|}{|A|}$$. I base this on the US government definition of university student retention rate, which I think is pretty similar, but please correct me if it's not.

The reasons as to why you might have $$|A\cap S|<|A|$$ are irrelevant.

### Example

Let's say you have $$|A|=100$$ so that, in your situation, $$|S|=90$$. For simplicity, let's also say that there are no newcomers to the mix.

The retention is, quite literally, how many students you retained. If you retained $$90$$ students, then the retention rate is $$90/100$$. Even though $$|S|=|A|-10$$, the number of students you retained $$|A\cap S|$$ as a percentage of the number of students in the original program $$|A|$$ is still $$90\%$$.

It wouldn't make sense to normalize this to $$100\%$$.

• Some of the terms are a little vague, particularly what you refer to as eligible students and returned students. By "eligible" I am referring to students who attended last year's program for more than 2 months and did not graduate. This also includes any other reason they could not have returned the next year like moving away. "returned" students are the students that have "returned" from the last year/session and were eligible. So essentially I am just getting the percentage of students that returned out of the students that could've possibly returned. Sep 8, 2020 at 13:34
• So, if we had 33 eligible afterschool students, and 12 total enrolled summer camp students, it would be 33 ∩ 12 / 12. I believe we are on the same page. But, looking at the formula, wouldn't it be |A ∩ S| / |A| not |A ∩ S| / |S| if you are calculating the number of retained students from the first program to the second, wouldn't you divide by the first program, not the second? I have been using the formula, |A ∩ S| / |A|. How is it calculating the retention for AS19-20 if I'm not dividing by AS19-20? Sep 8, 2020 at 14:28
• Yes! You're right, sorry for my mistake. The example still holds, but the denominator in the rationale was certainly wrong. Thanks for catching that
– Ben
Sep 8, 2020 at 14:38
• No worries. I appreciate the effort you put into the initial response, nonetheless. Just making sure I'm grasping the concepts correctly. The way you framed it made sense to me and I see why normalizing it doesn't actually add anything to the formula. Thank you, Benji! Sep 9, 2020 at 18:14