OK!!! I am Mr. Tarrou. Now that we have talked about some of the quantitative graphs, you
might want to read about something about the dot plot. We have covered stem plots and histograms.
We want to discuss or talk about how we describe distributions. The four things you want to
remember when describing a distribution is Shape, Center, Spread, and Outliers. Now it
is very very important to be able to describe distributions well. You need to use number,
use your data, make sure you mention all four of these variables. The first free response
question of every AP Stats test I have seen so far, they release the questions and you
can look them up on APCentral.CollegeBoard.com, the very first question is always about data...describing
graphical situations like this. Shape, Center, Spread, and Outliers. So shape, is it symmetric,
or left skewed, or right skewed? Is there a big gap in the middle so it is bi-modal
or is there only one peak...Is it unimodal? Center, well you have three choices. Mean,
median, and mode. Mean is when you add up all the numbers and divide by the number of
numbers. If you have 100 numbers, you add them up and divide by 100. The median is when
you take your numbers and you list them from low to high and you just go...tick, tick,
tick...until you get to the middle one value or the middle 2 values. If you have the middle
2 values average those out. Or if you are going to describe a distribution(density curve),
the easiest measure of center to identify is the mode. We will be doing that in a minute.
So for measures of center you have three choices mean, median, and mode. If you just want to
look at a graph and describe what you see, the mode is the easiest measure of center
to talk about. Spread, well there is something range which is probably the most common measure
of spread. You just take the max value, that would be to my right, the max value and the
min value and subtract them. Remember that the Range is just one single value. So here
we have a graph displaying a min of 20 and a max of around a 100, that would give you
a range of 80. You don't ever want to say the range is between 20 and a 100. You can
say the min is 20 and the max is 100, but your range is that 1 value that one difference
which is 80. At least it will be in this example. The IQR, just write this down. There is a
lot to talk about in the beginning of statistics. I am not sure if I am going sure if I am going
to get two Q1 and Q3 type stuff in this little video. But there is another measure of spread
and that is the spread of the middle fifty percent of the data. You might want to just
write that down because you don't really understand how to find the quartiles yet. And then finally
your outliers. So here we have a histogram that displays again some grades. We are just
going back to that old example that I keep doing in these videos so far. And we have
one or two kids doing poorly, a lot of kids getting in the B range, and then 90 up to
a 100. 90 up to a 100 would be your A's. Why is this a histogram to begin with. Why is
this not a Bar Graph. Because I am letting my numerical values represent a continuous
random variable. I am not using the scores to group them into letter grades like A, B,
C, D, and F. I am keeping the grades as the raw score...so histogram. I have my values
in the middle of my classes here. Again if you have not watched my other video, you can
also have this displayed as 20, 30, 40, 50. Now I common mistake I see my students make
especially if they are pulling numbers out of their textbook that is in a table like
20-30, 1 kid: 30-40, 0 kids: 40-50, 4 kids, where they make the x axis look like this.
20 to 30, 30 to 40, they are attempting to label the x axis, but really what they are
using is the table labelling to put into their graph and this is not correct. If you are
going to put numbers on each of these tic marks, just do it once. 20, 30, 40, 50, 60,
70, 80, 90, 100, and for the one kid that scored a perfect 100 I have to go to 110.
This is because again this last interval represents x's that are greater than or equal to 100
and less than 110. So this interval is for x's that are greater than or equal to 90 and
less than 100. So if some kid scored a perfect 100 you have to make a new class interval
for them and put them over here. Ok, so how would you describe this distribution that
is skewed to the left. See I have not even talk about that with you guys yet. So if you
have a peak on the right side of your graph and it tapers down to the left, that is called
a left skewed distribution. I am going to break these shapes down for you and talk about
these three measures of center would show a left skewed distribution, and if it is right
skewed their relationship will be the opposite. I will highlight that before this 15 minute
period is up. Now I don't have time to write this but please pause and reply the video
and write it down or read your textbook for examples about how to describe a distribution.
This is a left skewed distribution, the pointy side is to the left. This is a left skewed
distribution...SHAPE. It has a min of 20 and a max in the 100's giving my a RANGE of approximately
80. I just talked about spread. So shape and spread are done. Most scores were in the the
80's. I just talked about the CENTER, that was mode. There appears to be no OUTLIERS.
There is a gap in this class interval. No one scored between 30 and 40, but the grades
slowly taper down to the left so I would not call this data an outlier. So this is a left
skewed distribution, with a min in the 20's and a max in the 100's giving a range of approximately
80. The most common score is in the 80's and there is no apparent outliers. Within those
very short simple sentences I talked about the four things you need to discuss when you
are describing a distribution. Shape, Center, Spread, and Outliers. Here we have a left
skewed distribution. Now, I have not talked about these concepts yet, but you can hear
them and we will discuss them later as we continue on through the notes. When you have
a left skewed distribution, and we are talking about these three measures of spread... When
you have a skewed distribution the mean is always closest to the tail. Now I am not really
sure where the mean is in this data, but I do know it is going to be toward the left
side because that is where the tail tapers off. Probably not this far, so don't think
I am saying the mean if 50, I really don't know what it is. When you have a left skewed
distribution, a little bit larger than the mean is the median. Ok, I just eye balled
that. I have know idea exactly where the median is. We will maybe actually stop this and recalculate
where the median is in a minute. And the mode is always where you have the peak of your
graph, so the mode is in the 80's. When you have a left skewed distribution, the mean
is smaller than the median, and the median is smaller than the mode in numerical value.
If it is a right skewed distribution, let me put this here, if it is a right skewed
distribution then the mode is the smallest measure of center, the median is a little
bit larger, and the mean is always closest to the tail is going to be larger. So when
you have a right skewed distribution, this is right skewed...I am running out of room
to write. You have your three measures of center. When it is a right skewed distribution
the mode is the smallest measure of the center, then comes the median, and then always closest
to the tail is the mean. I jump around a lot when i talk about AP Stats in the beginning
because there is so little that you know, so I jump around a bit. But you will see this
in notes later as well. So if you don't have the written now, you will have it written
later as you keep watching my videos as we got through the year. If you have a symmetrical
distribution, whether it is a bell curve like this, or whether it is was bimodal such as
this, some of this might not make sense unless you also read your textbook, but if you have
a symmetric distribution your mean and your median will be roughly equal. So if your mean
and median are approximately equal that is your sign for symmetry. If your mean, median,
and mode are equal, this will get repeated later because I am jumping ahead a little
bit, if your mean, median, and mode are all the same that is a sign that you have a unimodal
symmetric distribution. Let's see here. Let's try and find where the median actually is
in this histogram. I have 1 number, 2 numbers, I have got 2 people in this category. That
is 1, 2 and 2 make 4...another 2 people make 6...another 3 people makes 9...let's call
that 4 so a total of 13...now 15 and 16. This graph represents, the way it is made as a
frequency histogram, represents 16 pieces of data. So if finding the median goes from
low to high, and the median is the middle number. So if there is 16 pieces of data,
that means that there is 8 over here in the low end and 8 over here in the upper end.
I want to find where is the 8th or 9th number to estimate the median from this histogram
that I have now scribbled all over. The eighth number right...low to high...counting the
tops I have 16 pieces of data... so I want the eight ninth number. 1, 2, 4, ok so I guess
I was wrong, 6 because I am not to the eighth number yet. Actually I wanted to draw this
correctly then this comes out and 1,2,4,6, and another 3 makes 9. The median is actually,
if you want to go back to your notes and change that, somewhere between 70 and 80. So, left
skewed distribution. Mean is the smallest measure of center, then median, and then mode.
If you have a right skewed distribution, then it is just the opposite. By the way, this
will come up in your textbook as well... All money questions are right skewed. You always
have a lot of people doing the grunt work and then a few people at the top of the company
making the big decisions and the CEO's of the company...the stars of the football team
or whatever. You always have that handful of people making a ton of money. All money
questions in our textbook are going to be right skewed distributions. That is a quick
introduction on how you describe the shape of a distribution. Don't forget...Shape, Center,
Spread, and Outliers. Don't forget to read your book. Read the examples and do your homework
because you are going to want to have those class discussions and get help from your teacher
because they will get a little bit tricky. There you go....BAM!!!