Hello, Mr. Tarrou again.
We are going to now start learning how to describe distributions with numbers as opposed
to just describing their shape, center, spread, and outliers with a visual inspection.
We are going to put some concrete numbers on those, more thoroughly go over the 5 number
summary, and especially after this video talk about how to and what does standard deviation
mean and break it down into very very small parts.
So please if you would, if you would like an organized set of notes for your class,
if you don't already have some hit pause and copy these.
Ok, so Mr. Tarrou back.
I have written the next notes for the next section of the textbook.
Because of how you get my notes a lot of these vocabulary words, and even some of the formulas
and the descriptions, you would have seen in a previous video.
But, to rehash, we are going to describe distributions with numbers.
One vocabulary word I have not emphasized very much yet, even though I have said it,
is this idea of resistant.
Now, a resistant measure is a measurement that is not strongly effected by outliers.
And my exam is here.
Very simple example.
There are four numbers.
1, 2, 3, 4are one, two, two, and three.
If we look at these four numbers we have a mean of two and a median of two.
GREAT!
Two measures of center.
They are equal because the distribution is symmetric...of my four simple numbers.
Now if I bring in the dark evil outlier of 100.
Well, now my mean jumps up greatly.
When I take these five numbers and add them up and have to divide by 5, my mean varies
very, very much.
It goes from to 2 to 21.
That is a huge difference.
The reason is why because when you calculate mean, you have to use all of the data points.
Well, when you add in that 100, it gets thrown into the calculation.
Whereas the median still stays a value of 2.
Now to be resistant, the mean is not resistant, it changed greatly due to the outlier.
The median, excuse me, now is a resistant measure.
It does not have to stay the same.
The definition of resistant is that it is not strongly effected by outliers.
Now in intro to statistics we actually have just a very small number of measurements that
are resistant.
We have median, we have the IQR and it is calculated through the median process so the
IQR is resistant, and if you want to say the mode is resistant as well you can.
Though that is not something we normally worry too much about.
But really there is only two measurements, if you want to include mode there are 3 measurements
in statistics that are resistant... median, the IQR which uses the median in its calculation,
and maybe if you want to consider mode.
Everything else in the book, all of the other formulas you are going to learn this year
are going to be not resistant.
They are going to be heavily influenced and effected by outliers.
So the mean.
The mean again is closest to the tail in a skewed distribution.
Here is the formula for calculating mean.
You got this really scary looking Greek symbol in it, a summation symbol.
Don't fret.
You have learned and have known how to find mean for years for now.
You know to find mean, you just add all your numbers and divide by how many you have.
That is what this formula says.
Add all of your individual, with the x sub i, add all of your individual x's and then
when you are done, multiply that by 1/n or divide by how many numbers you have.
So statistics formulas are a lot scarier than they really are.
That Greek summation symbol (Sigma) just means add over, and over, and over, and over, and
over, again which is why love graphing calculators and computers now to help us with our statistics.
We just learn the concepts and learn how to analyze and let the computers do the grunt
work.
x bar is the sample of the mean... or the mean of a sample.
Mu is the mean of a population.
This will be give to use, basically never calculated by us in intro to statistics.
The measure of center for bell curves.
If you have bell curve distribution, a unimodal symmetric distribution you want to use the
mean.
That is the best measure of center for symmetric unimodal distributions.
Uh, means are not resistant.
I seem to have somebody at the door so I will be right back.
Ok, so he came for a textbook.
And we are back.
So the mean is closest to the tail if you have a skewed distribution, the formula, the
measure of center, the best choice if you have a bell curve, and again it is not resistant.
Median and IQR are the only resistant measures that we have and if you want to throw in mode,
that is fine.
Ok.
Median, a measure of center that we have talked about already.
Quite a bit.
It used for non-normal data.
It is the middle number or the average of the two middle numbers.
And the median is resistant like I just said twice now which is why it is the best measure
of center if you have non-normal data.
Let's key in on this idea that the median is the middle number or the middle two numbers.
We have had some problems in the textbook about... if you have a group of people, say
10 people and their average income is $40,000.
Well, if I could tell you out of those ten people the average income is $40,000 I could
just take the $40,000, multiply it by 10 and say ok everyone make 400,000 dollars total.
I can do that because when I say that the average is $40,000 I have added all ten of
those numbers up divided by 10.
The middle number or the median value only is calculated the middle number or the middle
two numbers.
So, if I told you that ten people had made a median of 40,000 dollars of income...
how much do the ten people make in total?
I don't know.
I only used one or two numbers to calculate the median.
I don't know how small the other numbers are, the other what...
4 or 5 numbers.
And I don't know how large the other 4 or 5 vales are.
I say 4 or 5 because you can vary a little bit in how you find the median and what you
do from there.
So, you need realize the difference in the calculation in means and medians and how that
can effect some of your questions from the textbook.
Again it is resistant.
Quartiles.
I gave you an example of that in a previous video.
If you want to find quartiles, take the low to high and find the median.
Then take the first half of your numbers and find the median, that is Q1.
Take the second half of your numbers and find the median, that is Q3.
Again if you want an example of that look at the previous video.
Alright.
So going over to some more summaries, we have got the 5 number summary which I have just
maybe mentioned in passing.
I kind of forgot what I have said by now.
And range and percentiles.
Some of these values again I have already mentioned.
We are moving into a graphical display of a box plot in a minute.
So here are some other concepts in section 1.2 of my textbook.
You will also maybe have seen these in other videos because I repeat myself a lot in the
beginning of the year trying to...
You can't give notes when no one has a clue about what you are talking about.
Five number summary.
That is min, Q1, median, Q3, and max.
The graphical display of the five number summary is the box plot.
The left whisker is the min.
Where the box starts is the Q1 value, median, Q3, and the max.
We are going to break that apart some more here in a minute.
Trying to make sure that I don't run out of time.
Your range, again I have said that a number of times, that is the max minus the min value.
The range is one single value.
So if your low is 20 and your highest is 100, your range is 80.
Don't say that your range is between 20 and 100.
I used to do that and got corrected eventually after a few years of teaching.
That was a little bit late, but anyway.
Percentiles.
You know if you take the SAT's and you score in the 95th percentile, then you scored better
than 95 percent of your fellow students.
Graphical display, we just covered that.
It is an Ogive graph or a Relative Cumulative Frequency Distribution.
So make sure you go and look those up.
Let's talk a little bit about box plots.
They are our graphical display of the five number summary.
They display it.
Ok, great.
And more time, your five number summary is what we use to describe the shape of non-normal
data.
It is not written up here because I have not talked about standard deviation a lot, but
if it is bell curve... a symmetric unimodal distribution your best description for shape
is not the 5 number summary but mean and standard deviation.
We will talk about that later again.
Box plots can show skewness, but not gaps.
So let me explain what I am talking about there.
uh...
Here we go.
So here is a box and whisker plot.
Ok.
We have talked about left skewed and how if it was left skewed the data would just sort
of taper off to the left and become more spread out.
Does this box plot look left skewed?
The answer is no.
Because these three quarters are all very evenly spaced with just one long whisker.
That is probably due to an outlier.
To show skewness, it would need to... and it will not be perfect.
We always deal with real life data so this will vary.
But you are looking for your segments to kind of increase in width as you either to the
left or the opposite would be right skewed.
This would be more like a left skewed distribution with a tight upper 25 percent and as I work
my way down towards the minimum my gaps get farther and farther apart.
This is skewness.
One thing that I have not written up here though is one of the very best uses of box
plots.
They are not to analyze the shape of the data.
That is what we have histograms for.
No.
If you go into your calculator you can put data in one, two, or three lists and you can
display up to three of those box plots at once.
You will also see examples in your textbook where there are multiple box plots together,
maybe up to four or five of them.
You want to use these box plots for comparing multiple distributions.
You put those on the same x scale, usually x, sometimes your book will put them vertically.
When they are side by side then you can talk about any comparisons or contrasts you see
between these.
Uh...
With these being the medians, maybe the data values that these two box plots now represent...
that I made up that don't have any numbers... but the median values are the same but this
grouping of data has a much small range or a much small spread.
Later when we are talking about standard deviation, it has a much smaller standard deviation.
All this box plot shows is that the data is more spread out.
This might actually have more numbers here than up in this graph.
Because cannot see how much data is displayed just how far it is spread out.
This minimum is smaller than this minimum.
The maximum is larger than this maximum value.
The medians are about the same.
Remember if you are describing distributions...
shape, center, spread, and outliers.
I am just doing a quick overview.
Well, I just repeated here what this says.
And I think I am running off screen.
Let me see if I have any time left.
Yes I do.
So one last concept before we call this video a wrap is modified box plots.
Now we are going to have a video... actually we have already done a video of how to find
histograms and box plots with our graphing calculators.
So we definitely want to make sure we know how to do that.
But um, I have not introduced it in an official set of notes yet.
Modified box plots.
Those are awesome because they take care of the IQR test for us which means that they
will show us the outliers with individual points on the side.
I just talked about how do you that IQR test.
It is the IQR range which is Q3-Q1 times 1.5.
You add that number to Q3, you subtract that number from Q1 and you get this boundary...
this imaginary boundary you don't want your data to fall outside of.
Well, your calculator will do that for you and when you see this box and whisker plot
or box plot your whiskers will start and end at the first or the last piece of data that
actually passes the IQR test.
So if the lower bound is maybe 40 but the first piece of data that is greater than that
lower bound is 50, then that whisker is going to be at 50 not 40... not that calculated
lower bound that we set up by doing the IQR test.
You definitely want to chose this from your calculator because why not get more information
from your graph automatically if you can.
Well, we are going to erase all of this and talk about standard deviation in a second
and talk about properties of standard deviation which is the most important measure of spread
that we are going to talk about this year in Intro to Statistics.
BAM!