Confidence in determining proportion of population with a given characteristic

Question

Say I have discipline incident data from a total population (not a sample) of students in a school district. If I split the students into groups by some demographic marker (such as race), then I will get very different numbers of students in each group and very different number of students with an incident in each group.

For example, if I had groups A,B,C, and for a given year,

Group A had 500 students with 50 of those students having discipline incidents

Group B had 200 students with 15 having incidents

Group C had 40 students with 8 having incidents

I can of course say that:

50/500 = 10% of Group A had incidents

15/200 = 7.5% of Group B had incidents

8/40 = 20% of Group C had incidents

How do I determine how much confidence I can have that the size of the group and the number of incidents is large enough for each group over the given year?

Most of the things I am reading related to confidence intervals and margins of error are related to samples from a population, which this is not. This is an entire population. They also often refer to finding a mean, such as if you were using data from a survey of student heights or something.

In other words, how does my confidence change if I keep growing the size of the group and number of incidents proportionally? For example if Group C, rather than being 40 students with 8 having incidents, we had 400 students with 80 incidents or 4000 students with 800 incidents?

And how does the number of students with incidents specifically affect my confidence? So for example, if I had 40 students with 20 having incidents or 40 students with only 1 having incidents?

What I would love to have would be a cutoff point for the minimum number of students needed and the minimum number of students with incidents needed to say that the data is valid. As well as a confidence level or margin of error for each proportion.

Thanks very much.

Answer 1

If you truly have data on the entire population of interest, you do not need to estimate anything. You know precisely the proportion in each class because you have measured every individual.

However, if what you are truly interested in is the underlying data generating process, you can think of your "population" as simply a sample. Take a look at the answers here Statistical inference when the sample "is" the population for a more in depth discussion.

In short, you can treat your "population" as a sample from an underlying generating process, or else you have perfect information and no need for statistics.