HELLO!
Mr. Tarrou Ok, so thanks for joining me again.
I hope these videos are helping.
In AP Statistics right now we are talking about...
We are going to just start tinkering into looking at relationships between categorical
variables.
Now actually testing whether or not two categorical variables are related or has a strong relationship,
that requires something called the Chi-Square test and the Chi Square Goodness of Fit Test.
We are very far away from that.
So we are just going to be looking at some two way tables.
Finding some row and column percents and just discussing what patterns we see in the numbers.
So what I have done is I have pulled some information off the internet which I will
summarize at the end of the video.
And I used it to make up this two way table studying the relationship between education
levels and how much people are getting paid.
So... um...
We have no high school, high school, and college.
I have got some pay scales broken down by category.
Now pay is money.
Money can be averaged.
You might think that is a quantitative variable.
But when you break it into categories of income, now it is no longer quantitative but is categorical.
Ok.
Now on the outside of this 4 by 3 table of income, four rows and three columns, we have
what is called a marginal distribution.
That is where I have added up all the totals.
So if you add up all of these people that supposed have been interviewed that did not
have a high school diploma there are 97 of them.
You add all up the high schools, that is 315.
There are totals going in the rows as well.
And ultimately we have 591 people in this survey that we are studying.
Alright.
What we are going to do is ask, Is there a relationship between those two.
Hopefully if we are in College and taking honors classes we are hoping that of course
there is.
While you can analyze the counts, percents are easier.
You may have a choice on how to set up those percents, but usually it will be dictated
to you in the problem based on the variables or the language of the question.
Now why would not want to just look at this table of counts and go, well clearly the more
school you have the more you make.
And we probably can, but these totals are not equal.
So we might feel like there is something going on and that there that really is not simply
because we are looking at these numbers but we are missing the fact that oh... this is
153 out of 315.
And while this is a lower number, it is also out of a much smaller group.
So we always want to use relative frequencies or percents when doing comparisons.
Much like we prefer relative frequency histograms to just regular frequency histograms when
comparing two distributions.
[nanananana] Ok.
Now on the camera we cannot show you the table and these questions at the same time.
So refer back to it, or maybe go back and copy that table for a minute as I talk about
these questions.
Ok.
So we have a question of simply out of all the people I questioned, what percent have
a college education?
That turned out, that would be a margin percent....
those were the totals at the bottom, and it was 97 out of 591.
That is going to be approximately 16.4 percent.
Ok.
Well, now all of these questions, the two white ones and the two yellow ones, are conditional
probabilities.
Conditional probabilities will come back in symbol format when we talk about probability
rules in a later chapter.
But, I would bet that reading these, you probably would have set these up correctly already.
What percent of college grads make 40 to 65 thousand dollars?
Well... um...
That is the number...
First of all, the condition is what the percent of college graduates.
So we are not using everybody in the study.
That was 591 people.
We are only considering the people that graduated from college and that was a total of 179.
Sorry about that.
And then... and this is the same question, "Given that you are a college educated person,
what is the probability you make between 40 to 65 thousand?"
Again that is 71 over 179.
So these two questions are both giving you... they are looking for the same fractional answer..
the same percent.
71/179.
But the second one is blatantly telling you what the condition is.
I don't want to know all the people, I don't want to talk about all the people that make
between 40 and 65 thousand dollars, I want to just talk about those that graduated from
college.
So the 179 as opposed to the 519 people.
You need to read these questions, you need to identify the condition, especially when
you are talking about two variables in the same question.
There are probably giving you a condition.
And that is going to dictate whether you do a row or column percent.
The college graduates, given that you are a college grad, what percent of college grads....
Well if you go back and you look at the previous part of the video, the college was on the
top of the column so again this 71 divided by 179 are column percents because this variable
is the top of the column.
So, what about...
What percent of those making 40 to 65 thousand went to college?
Or more obviously, given that you make 40 to 65 thousand dollars, what is the probability
you got a college diploma.
Those are both the same question worded a little bit differently.
You know, with english there is all kinds of ways to move these words around a little
bit.
It can occasionally be a little bit tricky to determine which one is the condition and
which one is the... to set up the denominator of the fraction.
So what percent of those making 40 to 65 thousand?
Well 40 to 65 thousand was on the side of the table we just had up there a second ago
on that blackboard.
So that is going to be a row percent.
How many people make 40 thousand to 65 thousand that were in our survey?
It was 177.
So then out of that group how many went to college?
71.
So 71 divided by 177 will give the percent of people making 40 to 65 thousand that went
to college.
If you want to just a percent of people that made 40 to 65 thousand period with no condition,
it would be 177 divided by 519.
But these are all examples of conditional probabilities.
You must be a college grade to be considered.
You must make 40 to 65 thousand to be considered in these percents.
The marginal distributions do not have any conditions given.
So to describe a relationship among categorical variables, calculate the appropriate percent
based on the condition...
That is what I am talking about with column percent or in this second example row percent...
from the counts.
Ok.
That is how you set up conditional probabilities or marginal distributions (non-conditional
probabilities) out of a two way table.
[nanananana] Alright!
What you would want to do then for comparison purposes is...
I have gone through, "What is the distribution of income among the three levels of education?"
I have worded this, and we would think, education would help to explain how much you make.
So if your education is helping to explain how much you make, then that condition...
that variable...
needs to be how you set up the percents.
These are all going to be column percents. 62 with no high school making 0 to 25 thousand
out of the total of 97 non-high school grad students.
This 48.6% comes from 153 divided by 315, the number of people making 25 to 40 thousand
and having a high school diploma... given that they have a high school diploma.
So these are all column percents and I can now look at it and go, for non high school
students of course you have a much lower income bracket.
For college students, or for college grads, your higher percents are in your higher income
brackets of 25 to 40 thousand , 40 to 65 thousand, and 65 or more.
And if you want more than just looking at a table of percents, and again those column
percents will make it nice and easy for us to analyze... or at least hypothesize that
there is a relationship between education and income level.
To really prove there is a relationship will require again a Chi Square test.
We are not going to do that for many many months.
You can take these percents and create some bar graphs.
Remember bar graphs and pie charts are our two categorical graphs that we will be dealing
with.
Please do not talk about things like left and right skewed, or Normal, or symmetric.
Those are all language or vocabulary words for quantitative data.
This all set up in bar graph form.
This is all categorical data, so you do not use language you used for dot plots, stem
plots, histograms.
You can just discuss, look... non-high school graduates making 0 to 25 thousand dollars
is much higher than the high school and college graduate individuals.
For incomes of 40 to 65 thousand, I can compare non-high school, high school, and college
grads.
See what I have made here is a bar graph out of this first row of data and a bar graph
of the third row of data.
I could not really display all of this in a single bar graph, unless it was a bunch
of side by side bars.
Even then it would be confusing.
I think it is better to have it set up like this.
And to conclude the video before I do Simpon's Paradox, that is next.
That will be a second video.
I have a link on the top of this video for you to look at, or just easily go to.
I made up these numbers, but I did to the internet and I did find a website that talked
about degrees and how much people made.
And so you can clearly see that the higher education, the higher the median income.
Now why do you think on the internet they reported the median income for those education
levels?
Well income, if not broken down by category like 0 to 25 thousand, income is a quantitative
variable and is going to be right skewed.
A lot of people make very little money and a few at the top, the stars of the football
team or the CEO's of the companies make a tremendous amount of money.
So money is always going to be right skewed.
So they would not want to report the...
I guess you could.. but it would be a little misleading...
The average incomes, because all of these variables if the income is used as a quantitative
value is going to be right skewed, and the mean is always pulled closest to the tail
because it is not resistant, the median.
WHOOO!
Alright:D
Thank you very much!