Hello again.
Mr. Tarrou.
We are going to continue our conversation today with what I started in the notes yesterday
in statistics.
That is describing or analyzing graphs.
We finished off with shape, center, spread, and outliers.
When you describe a distribution like histograms, stem plot, or dot plot you want to include
those four statistics in your discussion.
I want to continue with I what I said the other day which was we have measures of spread
that we use quite often.
Besides the ones that I have already talked about, another measure of spread besides range
and IQR is standard deviation.
This is by far the commonly used measure of spread in statistics.
But before I keep rambling on let me pause for just a second and get out of your way
so you can copy these down.
Ok, so hopefully you hit pause and copied that if you like an organized set of notes
if you don't already have some from your teacher.
Now, standard deviation...
What is it?
Standard deviation is the average distance each piece of data is from the mean or from
the mean... from the center of a bell shaped distribution or the mean.
uh...
The standard deviation is not a good measure of spread to use unless you are looking at
a bell curve distribution.
If your histogram is...
You don't know what Normal means yet.
If your histogram has one mound in the middle, one mode in the middle... it is unimodal and
is symmetric then we will be discussing later how that is Normal, or it could be Normal.
For right now we are getting the basics down.
So standard deviation is the average distance each piece of data is from the mean.
We are going to really dive into this in the next section of book or in a later video.
So we will be talking about doing an example with formulas and showing you exactly some
data points and calculating standard deviation and showing you really how they are measured
from the mean, which is one of the three measures of center of a distribution.
It is not a resistant measure.
You know I am throwing a lot of vocabulary out here and unfortunately there is a lot
of vocabulary to take care of in statistics.
I am kind of pulling concepts from the first two sections of my textbook together so a
lot of this vocabulary you are going to hear again in the following videos.
We will revisit this.
I am going to just say resistant.
I will verbally say it now, but it will not be defined formally with a written definition
and an example until a later video.
But a resistant measure if you have some values like 6, 7, 8, 10, 10, 6, 5... just some random
numbers... or they represent data... if you have an outlier that gets introduced it will
greatly effect the standard deviation.
It will change a lot.
So if I give you a group of numbers and maybe the standard deviation is 2 and to those numbers
I include a very large outlier.
That outlier, even though it maybe one number, if it is far away from the original group
of numbers will greatly increase the standard deviation to where it is not really representing
the spread of the data any more.
So standard deviation is not resistant and it is not resistant because the mean is used
in this calculation and the mean is not resistant.
Again all of these vocabulary words will be re-explained further in a later video.
Standard deviation.
Ok, so we have two symbols for standard deviation.
We have s of x and sigma of x. s of x is what we are going to calculate with our graphing
calculators.
So if you have a TI-83, TI-84, and TI-NSPIRE you are going to be using Sx out of that calculator
when you do 1 var stats.
We will be given and using this math symbol of sigma of x, but sigma of x is standard
deviation of a population.
When you are in a textbook and you are in Intro to Statistics we will never be using
the entire population.
So we will be given this value, the symbol, but we will never compute it.
We are never going to use sigma of x out of our graphing calculator.
So please be aware of that because it will give you both Sx and sigma of x when you do
your 1 var stats.
So again, we have our measures of spread which are range of max minus min, we have IQR which
is Q3-Q1... we are going to break that down a little bit more today and show you how to
find those Q1 and Q3 values, and then we have standard deviation.
Only a good measure of spread when your histogram or stem plot has a bell shaped curve to it.
Later will refer to that... or really define what that means to be Normal.
For right now unimodal symmetric distributions will be the only time you should be talking
about standard deviation later on after we really get into some real statistics problems
and not just building up the bones of the book.
Great.
When you look at a smoothed over distribution curve, this would be traced over some type
of histogram.
I don't want to mark up this diagram, but if there was a histogram behind here and I
kind of just squinted a drew a smooth line over it, that distribution curve...
If I were to draw that curve perfectly another representation of standard deviation, this
will be revisited later when we talk about normality, but just so you hear it more than
once...
When you look at a perfect bell shaped curve, the horizontal distance from the mean which
will be the center because it is a bell shaped symmetric curve...
The horizontal distance between the mean of that bell curve and the inflection point,
where the curve stops curving down and starts to curve back up, that horizontal distance
will represent if done with a computer or found in a textbook will represent the standard
deviation value.
So if this were on a number line, let's put one here quickly.
Let's say that this has a mean of 10 and a tick mark here, a tick mark here.
One, two, three.
I will explain why I am doing three tick marks later when we learn something called the Empirical
Rule.
Let's say that this is 13, 16, 19, 7, 4, and 1.
The graph tapers off and could potentially go on forever.
But this horizontal distance between the center, the average of 10, and this inflection point
which I have now put over 13 would represent that for this bell curve the standard deviation...
the visual inspection...
would be about 3 units.
So that is a quick over view of standard deviation.
Again we will revisit this, use the formula and do some actual examples and talk about
other concepts and properties of standard deviation in a later video.
Let's move on to outliers.
We talked about describing shapes of distributions.
Again when you are describing shape it is shape, center, spread, and outliers.
Kind of pausing to count how many minutes I have left to keep talking for this little
video.
So we have outliers.
Now an outlier is just simply...
copy... ok.
An outlier is just simply a point that falls far outside the pattern.
I talked about, well, I actually did not write this.
A outlier is just simply a point that falls outside the overall pattern of data.
So if you are looking at a graph and it has a nice smooth curve whether it is left skewed,
right skewed, or symmetric and all of a sudden way over somewhere there is like a random
point.
It is just floating around out in the middle of nowhere by itself.
You know, it falls outside of the pattern that you are looking at.
So that is an outlier.
Another way of finding an outlier if you have univariate data like maybe we are only measuring
test scores like the other day, shoe sizes, or heart rate per minute.
We are only looking at one variable.
We have the IQR test.
The IQR test consists of three steps.
The first, find the IQR, the inter quartile range.
The spread of the middle 50 percent of the data.
Q3 is the 75th percentile of data.
You have gone to where 75 percent of the data is less than that value.
Q1 is the first quartile.
We will find these again when we look at box plots in a later video.
So Q3-Q1 is the IQR, the spread of the middle 50 percent.
This comes out to a number like 10, or 20, or 30.
We are going to take that IQR value and multiply it by 1.5 and add it to Q3.
That is going to be an upper bound.
For lower bound it is going to be Q1-1.5(IQR).
That is going to be the lower bound.
If we have data points that fall outside of this upper and lower boundary, it will be
marked as a possible outlier.
But a visual inspection if possible is your best of identifying outliers because you can
easily see if a number falls outside of an overall pattern.
And data outside of...
OH!
Yes, ok... data outside of these limits are marked as possible outliers.
You cannot toss out an outlier just because it looks funny or does not seem to make any
sense.
Let's come up with a very, very simple example to try and identify some outliers.
So we are going to make this extremely easy because of time allotment and I know we are
going to be looking at these medians, IQR, and Q1 and Q3's later more in-depth.
But let's make up a simple example.
So we have got some numbers of... let's see...
1, 31, 32, 33, 33.
Well ok these are too similar.
1, 31, 35, 36, 40, and 45, and 49.
Ok so we are going to do an IQR test on this data.
But for IQR test we have to know Q1 and Q3.
Let's go ahead and find those now with this really simple example.
We will revisit this later as we go through the book.
How do you find Q1, Q3, and just basically find the 5 number summary.
Well we have some numbers.
We are going to find the median first, the middle value.
They are already listed low to high, so the middle number is 36 and this would be the
my median.
Q1 then, I would take the first half of my data and find the median again.
Well the first half of my data is only 3 values because I did make a very simple example.
That is Q1.
Then the right side of my data, I will find the median again... again the middle number.
Since I made such a small example I can just see that without work that Q3 is 45.
Let's try and walk through quickly the IQR test before my 15 minutes are up.
Ok, so I have got Q3 which is 45 minus Q1 which is 31 gives me a difference of 14.
Now I am going to take Q3 which is 45, add it with 1.5 times my IQR of 14.
Well half of 14 is 7, so 45 and 21 is 45... 65... 66.
And then I am going to do Q1 which is 31 minus 1.5 times the IQR of 14.
Again that is 21, so 31 minus 21 is 10.
Ok, so now I have my upper limit and my lower limit.
Any data that falls outside of that domain is a possible outlier.
Well, clearly I have made an obvious outlier of 1 and any data outside those values will
be marked as a possible outlier.
So a very small simple example, hopefully not too simple, of finding Q1, Q3, and median.
You find median first and then the quartiles.
Then the actual steps of finding of doing the IQR test.
Because the value of 1 is outside that range... that lower and upper limit it is going to
be marked as an outlier, or as I like to emphasize to my students a possible outlier.
Visual inspection is always best.
So!
A couple more concepts about analyzing the shape of statistical graphs.
See you in a minute.
BAM!