Marginal and conditional distributions

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 and so what we could do is we could set up some buckets of time studied in 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 sixty and seventy nine on percent 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 forty out of the two hundred got between eighty and a hundred sixty out of the two hundred got between sixty and seventy nine so on and so forth now a marginal distribution could be represented discounts or as percentages so if you represented as percentages you would divide each of these counts by the total which is two hundred so forty over two hundred that would be twenty percent sixty out of two hundred that'd be thirty percent seventy out of two hundred that would be thirty five percent twenty out of two hundred is ten percent and ten out of two hundred is five percent so this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets so you could say ten percent got between a twenty and eight thirty nine 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 and so then you would look at these counts right over here you would say a total of 14 students studied between zero and twenty 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 distribution of percent correct correct given that students students the 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 and so that would be this column right over here and then that column the information 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 160 to 79 40 to 59 continued over here 20 to 39 and 0 to 19 we'd want to 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 100 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 point six percent and then to get the full conditional distribution we would keep doing that we would figure out the percentage 62 79 that would be 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