- [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.