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# Marginal and conditional distributions

AP Stats: UNC‑1 (EU), UNC‑1.Q (LO), UNC‑1.Q.1 (EK), UNC‑1.Q.2 (EK)

## Video transcript

- [Instructor] Let's say that we are trying to understand a relationship in a classroom of 200 students between the amount of time studied and the percent correct. What we could do is we could set up some buckets of time studied and some buckets of percent correct and then we could survey the students and/or look at the data from the scores on the test. And then we can place students in these buckets. So what you see right over here, this is a two-way table. And you can also view this as a joint distribution along these two dimensions. So one way to read this is that 20 out of the 200 total students got between a 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So, for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and a hundred. 60 out of the 200 got between 60 and 79, so on and so forth. Now, a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that would be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you can say 10% got between a 20 and a 39. Now, you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. Then you would look at these counts right over here. You would say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. And this would be 35% right over there. Now, another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And this is the distribution of one variable given something true about the other variable. So, for example, an example of a conditional distribution would be the distribution of percent correct given that students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. That would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now, an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets given that we're studying between 41 and 60 minutes. So this first one, 80 to a hundred, it would be 16 out of the 86 students. So we would write 16 out of 86, which is equal to 16 divided by 86 is equal to, I'll just round to one decimal place. It's roughly 18.6%. 18.6. Approximately equal to 18.6%. And then to get the full conditional distribution, we would keep doing that. We would figure out the percentage. 60 to 79, that would 30 out of 86. 30 out of 86, whatever percentage that is, and so on and so forth in order to get that entire distribution.