>> Hi, and thank you for joining us as we open up our discussion
on the five-number summary and boxplots.
Heading into our first objective then, we need to learn how
to compute the five-number summary.
We're going to do this using our calculator, but first,
the five-number summary is a summary of a set of data
that includes the minimum, Q1, median, Q3, and maximum values.
That is, the five-number summary gives us a good spectrum
of what our data is doing and how the overall picture looks.
So, to find the five-number summary, we're going to go ahead
and jump into our example.
We're told the following values are final exam scores
of the 12 students in a statistics summer course.
We have our data here, and we're asked
to determine the five-number summary
of the data using our calculators.
Well, we have our instructions in front of us.
Let's go ahead and jump over to the calculator in order
to demonstrate how we're going
to find this five-number summary step-by-step.
Well, the first thing we're going to do is hit Stat,
and then hit Enter on Edit in order to enter our lists.
And then, I have my data in my list already, but we're going
to put our data into L1.
And now we're just inputting the data
from the problem given to us.
So, the first piece of data we have was 71.
We input 71, and then hit Enter.
It would take us to the next line.
You can go ahead and do that for each piece of data.
Once you're all set with that,
notice that down here it tells us that we have our number
that is number 76 as the 12th piece of data.
So, we know we didn't miss any.
If that's okay, the next part would be
to find the five-number summary now
that we have our data in there already.
Go ahead and hit Stat one more time.
We're going to arrow to the right to highlight Calc,
and we want option number one that is the 1-Var Stats.
To go ahead and run this program,
we're going to hit Enter.
And, now our calculator is asking us, "Well,
what list of data do you want to use
with this 1-Var Stats program?"
We put our data into L1, so I'm going to hit 2nd and then 1
in order to pull that data.
When you hit Enter now, it provides us with a full output
of a lot of information.
From our 1-Var Stats program we get the sample mean,
we get the sample of standard deviation,
we get population standard deviation.
So we have the standard deviation whether our data is a
sample or a population, but what we want right now would be the
five-number summary.
Well, in order to see the five-number summary all we need
to do is hit the down arrow.
If you down-arrow a few times,
notice that we get some more information.
We see that we have our minX, our Q1,
our median, our Q3, and our maxX.
Heading back to the notes just to show it a little bit better,
we're just highlighting our five-number summary one
more time.
Note that minX represents our minimum value,
Med represents the median of our 72,
and maxX represents the maximum value of our data.
If that's okay, we have a five-number summary
that would be our minimum of 46, Q1 is a 69, 72 is our median,
Q3 is our 79, and our maximum value being the 94.
And that's all there is to the five-number summary.
Pause the video and try these problems.
Well, that takes us to our next objective where we need
to be able to draw and interpret boxplots.
Our calculator will do a lot for this.
We're going to present this on how we do this by hand first
to make sure we understand each part of the boxplot.
Well, first, a boxplot provides us the five-number summary,
along with any outliers if there are any,
while also giving us the shape of the distribution
and the general trend.
In order to draw a boxplot, first we want
to find the five-number summary,
because those are the key values in the boxplot.
Once we have our five-number summary we want
to determine the lower and upper fences.
Now, this is a big deal, because the lower
and upper fences are not provided for us
when we use our calculators to draw this graph.
So, make sure we have a hand on that.
You can see the formulas for the lower and upper fence.
The lower fence uses Q1,
and then we subtract the 1.5 times that IQR.
Just as a refresher, remember that IQR is the difference
between our Q3 and our Q1.
And then the upper fence would be our Q3,
but now we add the product that is 1.5 times our IQR.
We will, of course, show this in the example, though.
Step three would be to draw a number line that's long enough
to include our max and min values, and then we're going
to enter some vertical lines at Q1, our median, and Q3.
We enclose these lines to form a box, and that's kind
of our middle structure of our boxplot.
In step four, we will label the lower and upper fences,
and then an easy step to forget, unfortunately,
would be step five, where we're drawing the whiskers
of our boxplot.
The whiskers tend to cause a little confusion,
as our lower whisker would be from Q1
to the smallest data value that is larger than the lower fence;
that is, it's the smallest piece
of data we have that's not an outlier.
Another way to say that is it's the smallest piece
of data we have that's within our fence.
So, we draw this line, the whisker, to the smallest piece
of data that we have within our boundaries created
by the lower and upper fences.
And then you have another whisker going from Q3
to the largest data value, again,
where that data value is within our fences.
As always, though, if that's confusing to think about,
we'll make sure to elaborate on our example.
And our last step would be
to denote any outliers that we have.
An outlier would be any piece of data that is outside our fences.
You see how important it is to get those fences correct, then,
as any value that is less than the lower fence or greater
than the upper fence ends up being an outlier.
Well, let's go ahead and jump into our example then.
We're back to the final exam scores of the 12 students
in that stats summer class.
We have our data provided again below.
This time we're told to construct a boxplot.
Well, remember, we previously found our five-number summary.
We had a minimum value of 46, our Q1 was a 69,
our median ended up being a 72, our Q3 ended up being a 79,
and our maximum value was a 94.
Well, now that we have our five-number summary again,
next we need to calculate our lower and upper fences.
And just as a refresher, one more time,
remember the IQR is the difference between Q3 and Q1
where we have a 79 minus the 69 from our Q3 and Q1.
And that makes our lower fence 69, that Q1,
minus 1.5 times the 10, our IQR,
and that gives us a lower fence value of 54.
If that's okay, our upper fence, making sure we're a bit careful,
as now we're using our value of Q3 rather than Q1.
We have 79.
Be sure to add that now to the product that is 1.5 times 10,
and we end up with an upper fence of 94.
So, now we have our lower and upper fences,
and that means we're all set to sketch our boxplot.
Well, sketching our boxplot then and referring to our steps,
we already found our five-number summary like we were told
to do so in our first step.
We found our lower and upper fences.
And now we're on step three,
where we have now our number line long enough
to include our maximum and minimum values.
That means we're ready to enter our vertical lines
at Q1, our median, and Q3.
Well, Q1 we found to be 69, our median we had a 72,
and our Q3 we found to be 79.
Those three values coming from our five-number summary again.
We take those three values and the vertical lines,
and we enclose that box,
and this would be the box in our boxplot.
Now step four tells us to label the lower and upper fences.
We had a lower fence of 54, an upper fence of 94, putting those
in the approximate ballpark
of where they belong in terms of values.
And now, pausing for a minute,
notice that I've highlighted the key values that we're using.
It's up to your instructor as far as how
to label your number line: whether we're supposed
to label every five values or every integer, et cetera.
In this demonstration, though,
we're just showing those key values approximating the
location of each of those numbers.
If that's okay, we're on step five now,
where we need to draw our whiskers.
Well, the whisker on the right side ends up being a bit tricky,
because it's a line from Q3 to our maximum value
that is smaller than the upper fence.
Well, our maximum value is 94 and our upper fence is 94.
So, that whisker goes right
up to our maximum value and our upper fence.
Our whisker on the left side a little more clear, though,
would be the smallest data value that's greater
than our lower fence, and that would be a value of 61.
We're all set with our whiskers then.
In step six, we need to finish up our boxplot by tossing
in our outlier, the value that is 46; 46 is an outlier
because it's outside those fences, or it's less
than the lower fence, 46 being less than our lower fence of 54.
Just to be clear with our labeling,
we show the reader we have our minimum value that ended
up being an outlier of 46, our lower fence of 54,
our Q1 being 69, our median of 72, Q3 is 79,
and our maximum value of 94,
which was also the value for our upper fence.
And we've got ourselves one beautiful boxplot.
Well, now that we've drawn our boxplot, let's go ahead and walk
through how to construct this thing using our calculator.
We see the instructions in front of us.
So, let's go ahead and jump
over to the calculator again to demonstrate.
Looking at our instructions, we're told first
to enter our data into our list and just showing
that we still have our data in L1.
Going to the new part now, we want to hit 2nd,
and that Y equals button shows the option of a stat plot
above it, meaning with 2nd activated,
we can get into our stat plots by hitting that Y equals button.
You see your Plot1, 2 and 3.
We want to make sure whichever plot we're using is on.
Right now it shows that it is on, but just in case
as we hit Enter, if it was off,
the Off word would be highlighted.
We would just arrow to On and hit Enter
to make sure that's highlighted back to On.
Next, we're told to arrow down to Type.
The type of boxplot we want would be here with the outliers.
Make sure you're a bit careful with this.
This option tells us that we want to illustrate the outliers.
If we accidentally chose the other option,
it would treat all our data as quality data rather
than calculating outliers for us.
So, make sure we choose the correct option
that is the boxplot with outliers.
Our Xlist needs to be L1, since our data is in L1.
Frequency stays at 1, since we're working with raw data,
and any mark that you want to do will work.
So, if you are just loving these boxplots, feel free to use
that plus sign to show your positivity.
If you've got an empty heart, we'll go ahead
and use this open square.
Once you're all set there, as you see in the notes,
we need to head over to Zoom.
And we want option number 9: that would be ZoomStat.
You can arrow down to ZoomStat,
or you could just hit the number 9.
Either way will work.
If I hit 9 or hit Enter since I'm highlighted,
it will generate my graph,
and we've got ourselves our lovely boxplot.
Making sure we're a bit careful to notice, though,
it does not provide us the fences.
Remember, we had a fence here when we drew it by hand,
and we had another fence matching our maximum value.
So, our calculator does not give us those values of the lower
and upper fence; we have to make sure we remember
to do those ourselves.
And, one last piece, though, we could come over here
and hit the Trace button.
If you hit Trace and then use the right and left arrows,
it will jump to your key points.
That is, if we need to find the five-number summary,
we could always draw a boxplot,
and it will show us our key points
that would be the five-number summary.
Over here our outlier was the minimum, like we had by hand,
a value of 46; the whisker was set at 61; our Q1 was 69;
our median was 72; Q3 we found to be 79;
and our maximum was that 94.
If that all makes sense,
you're all set drawing a boxplot with your calculator.
Pause the video and try these problems.
And, that wraps up our conversation
on the five-number summary and boxplots.
And thank you one more time for joining us.