I am Mr. Tarrou.
So once again we are still working on evaluating the shape of distributions.
In the second part of this video I am going to roll it into something called a relative
cumulative frequency distribution where I want to spend most of my time.
Now I have been talking about describing the shape of distributions and saying the four
characteristics that you must include in your description of a graph, the last video is
kind of vague on that, I am talking about stem plots, dot plots, and histograms or univariate
quantitative graphs we need to talk about shape.
and I did put this down very quickly in a previous video, but I want to make sure you
understand we have three measures of center... mean, median, and mode.
When you are making just a random argument, you can use whatever measure of center you
would like that best suits your argument.
However, we need to know how these three measures of center are arranged when we have some different
shapes of distributions.
Now these are all sample distributions or distribution graphs.
They would be a smoothed over line that you put on top of a histogram, or a dot plot,
or a stem plot that you made to kind of like... squint and see that general shape.
If you have a right skewed distribution, which as I mentioned in the last video as an example
is all of your money questions, if you have a right skewed distribution your mode is going
to be the smallest measure of center even though it shows up the most often.
A little bit larger will be the median.
And a little bit larger still will be the mean.
So if you notice that your mode, median, and mean... you can find the mean and median out
of your calculator when you do 1 Var Stats... if you notice the mean is greater than the
median, especially because that comes out of 1 Var Stats, that is an indication the
data is right skewed or that you have a large outlier.
If you have a left skewed distribution, the means are always closest to the tail...
I mentioned that before in previous video in passing, but I wanted you to have a really
nice set of sketches you could copy and use as reference.
If you have a left skewed distribution, your mean is the smallest measure of center, median,
and then mode.
Let's talk about what I just said a second ago about when you are just using Statistics
to make an argument, what do you want to use?
Well, you know, you always about teachers complaining about how much we make.
And you know, I don't want to go into that... whether I agree or disagree or how I feel
about it.
But if I have a money question and I do want to argue that maybe I don't make enough.
If my argument is I don't make enough, then since it is a money question and all money
questions are right skewed I am going to argue about what most teachers make or the median
salary.
Those will best suite my argument because they are lower.
However, if the county wants to argue that we get paid fine, or the state, then in their
argument they are going to talk about average teacher income.
Because, that is the largest measure of center.
We are all using valid numbers, but a lot of people may not understand why they are
different.
Why is someone saying that it is x number of thousands of dollars and somebody else
is saying that it is this number of thousands of dollars.
It is like somebody is lying, but not necessarily.
They are just using the number that fits their argument the best.
If you have a symmetric distribution, now these are not bell shaped or what we will
call later hopefully prove as Normal, but they are symmetric.
And if you have a symmetric graph then when you do 1 Var Stats with your calculator you
will notice that the mean and median will not be exactly equal, we are talking about
real data, but they will be approximately equal.
Then finally with the bell shaped curve (isosceles triangle too), that is when you are looking
at the mean, the median, and the mode...
I abbreviated that one... mean median and mode are all approximately the same number
right in the center of that bell curve.
Hopefully later with something called the Empirical Rule we will prove that data is
Normal.
Or, we are not going to show it today, but you will be learning about a graph called
a Normal Probability Plot which will help to verify the data is Normal as well.
But that is coming up.
Ok.
Over to the right, I do have a lot of chalk board space which I love very much, we have
other types of graphs.
We are going to cover two types of graphs, but within this video I believe only going
to fit in something called a relative cumulative frequency distribution.
Such a long drawn out name.
Well, your book might also call this an Ogive graph.
So you get ogive... you get jivey with it.
It displays percentiles along the y axis.
So if you have a certain value, a statistic like maybe a test score like your SAT's and
ACT's are always talking about...
They come back and hopefully you are in the 90 or 95th percentile or even higher... and
maybe lower.
But eh, you know...
We all have our different strengths and weaknesses.
If you want to display your percentile, if you want to know how you stand against all
of the other people that took this test... or we measured this number.. and it is nice
to do.
What we are going to do to explain a relative cumulative frequency distribution is...
We are going to turn this histogram into a Ogive graph.
Well...
When you make a histogram you can make a frequency histogram or a relative frequency histogram.
One more time relative frequency is fancy language for percent.
That is why or how I have labelled my y axis to keep this as simple as possible.
I am going to go from this graph to a distribution table which may or may not be necessary.
But my kids have been asking about exactly that means in class, and then convert that
into an Ogive graph.
So we have data points of 5 to 8, remember this is x is greater than or equal to 5 and
less than 8.
So we have data between 5 and 8, 8 and 11, 11 and 14, 14 and 17, and 17 to 20.
Notice these numbers diagonally are repeated.
In a distribution table but not repeated on the bottom of a histogram.
Our relative frequencies that I am pulling out of my histogram are 20, 40, 20, 10, and
10 percent making this a right skewed distribution.
So my relative frequencies between each of these classes is 20%, 40%, 20%, 10%, and 10%.
Great!
So a abbreviated version of a distribution chart.
There are better ones in your book with more detail on how to set these up if you have
counts instead of percents which is division, but anyway.
You might want to look that up.
A cumulative relative frequency just simply means add up the percents as you go down.
So 20 percent of the data is between 5 and 8.
But when we get to the 8 to 11, 8 to 11 is 40% of the data, but if you are adding up
the numbers as you go then that is 20+40 is 60%.
Another 20% gives you a total of 80% as you add as you go down.
Another 10% makes 90%.
Finally another 10% gives 100%.
So when you do a cumulative relative frequency distribution, those percents are going to
add up to 100%.
That is what allows you to observe on the y axis your percentiles.
If you are all the way to the right and you are the largest piece of data, then you are
larger than 100% of all the other data points.
So we are going to make a graph of this chart.
And I am going to run into my note there.
But I have got some kind of data down here.
I have a cumulative rel.
frequency and we are going to say... let's count by 20's.
20, 40, 60, 80, 100.
I am trying to... unfortunately trying to cram this together.
I am not sure if I am going to be off the screen of the camera here soon.
So hopefully it is ok.
The same x axis scale that we had in this histogram.
I just marked that off.
We have got 5, 8, 11, forgot what I counted by 14, 17, and 20.
So between 5 and 8 we had 20 percent of the data.
Between...
now notice that I put that dot on the right side of the interval because we are moving
through all the values from 5 up until the value of 8 when we get that 20% of all the
data.
Between 8 and 11, that is another 40 percent getting us to 60%... taking us to 60.
We have another increase from 20, 60, and now 80%.
So above 14 we are looking 80.
Then another increase of 10% and another increase of 10%.
So I made some extra space that I don't necessarily need.
And this is what our Ogive graph looks like.
So we have got this line segment, a slight steeper line segment, and then one... it flattens
out at 10 a piece.
So what can we learn from this Ogive graph?
Or why do we care about them?
What are we looking for as information?
Well, if you say picked a value of 10.
If your data was 10, what percentile would that be?
Well, we go to 10 and go up the graph and over and this is only a sketch, but it seems
like if your data is a 10 that your....
Can you see that?
If your data value was a 10, that would be about the 52 percentile.
What if I wanted to know what the 75th percentile is?
I want to read this backwards.
What is the 75th percentile?
Let's do 70.
Since I counted by 20 and that is kind of vague.
So go to 70 on the y axis, 70%, and go over to the line graph... go down... and it would
appear that if your data value is say maybe 13 that would be the 70th percentile.
By they way, that happens to be the third quartile if that happens to be drawn perfectly
correct (for 75th percentile) because when you talk about Q1 and Q3 Q1 means 25th percentile.
Q1 is bigger than 25 percent of the date.
Q3, well it is the third quartile. 25 and 25 and 25 percentile of the data, 75 percent
of the data is to the left or lower than your Q3 value.
So Q1 and Q3 are not just random values.
You can actually think of that subscript as Q1 is one 25% like a quarter.
Q3 is 1, 2, 3 25 percents or 75th percentile.
Well there we have it.
We have gone from a histogram through a distribution chart to a relative cumulative frequency distribution,
or an Ogive graph.
Sweet!
Let's see what else we can look at.
If we have time, and it looks like I just have a few minutes.
That might be enough time to talk about stem plots.
That is not a stem plot.. a time plot.
Ok.
Time plots we are all very very familiar with these, or most people are.
A time plot is when you track your data over time.
So if you want to look, say the sale of citrus over the years or months of the year, then
the months would be the x axis and it would go up and down.
Up in the winter and maybe down in the summer.
If you have data collected over time, or even sequence, if your doing an experiment... and
first experiment, second experiment, third experiment, and so on.
That is time as well.
We can use time plots to help analyze the cause of outliers and time is always the x
axis.
Over here I have some values that I have collected.
Maybe it was an experiment and it is 2,6 25, 12, 29, 28, 28, 27, and 25.
We have this outlier of 12.
Now this is a very small example, but if I were to say the sequence was 1, 2, 3, 4, 5,
6, 7, and 8 I could make a time plot out of this data.
And show that my outlier happened early in the experiment.
So the time is going to be, the sequence is going to be, the x axis.
The data is on the y axis.
And 1, 2, 3, 4, 5, 6, 7, 8 and we stopped at 8.
The data is ranged between...
let's sat this is 10 and this is 20.
My first measurement is 26.
My second measure was 25.
This is just a quick a quick sketch.
My third measurement is 12.
Then all my other measurement are around 28, 27, and 25.
They would all be up here somewhere.
And with a visual inspection of my time plot I can see that I have a nice horizontal band
of points.
I have a nice consistent score through all my sequence of experiment, but early on I
had an outlier.
Then I could say hey.. you know what... that outlier is probably due to user error or mis-calibration
on the equipment so I could throw out this outlier because I have shown that it was caused
due to user or equipment error or something to that effect.
So really super quick idea of time plots.
There are more examples in the book.
Make sure that you read about relative cumulative relative frequency distributions or Ogive
graphs in your book.
Read those distribution tables.
See the relationship between histograms and Ogive graphs.
And that is it.
I am out of time.
WHOOO!!!