WEBVTT

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>> Hi, and thank you for joining
us as we open up our discussion

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on the five-number
summary and boxplots.

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Heading into our first objective
then, we need to learn how

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to compute the five-number
summary.

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We're going to do this using
our calculator, but first,

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the five-number summary is
a summary of a set of data

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that includes the minimum, Q1,
median, Q3, and maximum values.

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That is, the five-number
summary gives us a good spectrum

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of what our data is doing and
how the overall picture looks.

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So, to find the five-number
summary, we're going to go ahead

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and jump into our example.

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We're told the following
values are final exam scores

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of the 12 students in a
statistics summer course.

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We have our data
here, and we're asked

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to determine the
five-number summary

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of the data using
our calculators.

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Well, we have our
instructions in front of us.

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Let's go ahead and jump over
to the calculator in order

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to demonstrate how we're going

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to find this five-number
summary step-by-step.

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Well, the first thing we're
going to do is hit Stat,

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and then hit Enter on Edit
in order to enter our lists.

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And then, I have my data in my
list already, but we're going

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to put our data into L1.

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And now we're just
inputting the data

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from the problem given to us.

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So, the first piece of
data we have was 71.

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We input 71, and then hit Enter.

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It would take us
to the next line.

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You can go ahead and do
that for each piece of data.

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Once you're all set with that,

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notice that down here it tells
us that we have our number

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that is number 76 as
the 12th piece of data.

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So, we know we didn't miss any.

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If that's okay, the
next part would be

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to find the five-number
summary now

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that we have our data
in there already.

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Go ahead and hit
Stat one more time.

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We're going to arrow to the
right to highlight Calc,

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and we want option number
one that is the 1-Var Stats.

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To go ahead and run
this program,

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we're going to hit Enter.

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And, now our calculator
is asking us, "Well,

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what list of data
do you want to use

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with this 1-Var Stats program?"

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We put our data into L1, so
I'm going to hit 2nd and then 1

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in order to pull that data.

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When you hit Enter now, it
provides us with a full output

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of a lot of information.

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From our 1-Var Stats program
we get the sample mean,

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we get the sample of
standard deviation,

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we get population
standard deviation.

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So we have the standard
deviation whether our data is a

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sample or a population, but what
we want right now would be the

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five-number summary.

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Well, in order to see the
five-number summary all we need

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to do is hit the down arrow.

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If you down-arrow a few times,

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notice that we get
some more information.

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We see that we have
our minX, our Q1,

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our median, our Q3,
and our maxX.

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Heading back to the notes just
to show it a little bit better,

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we're just highlighting
our five-number summary one

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more time.

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Note that minX represents
our minimum value,

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Med represents the
median of our 72,

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and maxX represents the
maximum value of our data.

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If that's okay, we have
a five-number summary

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that would be our minimum of 46,
Q1 is a 69, 72 is our median,

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Q3 is our 79, and our
maximum value being the 94.

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And that's all there is to
the five-number summary.

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Pause the video and
try these problems.

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Well, that takes us to our
next objective where we need

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to be able to draw and
interpret boxplots.

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Our calculator will
do a lot for this.

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We're going to present this on
how we do this by hand first

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to make sure we understand
each part of the boxplot.

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Well, first, a boxplot provides
us the five-number summary,

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along with any outliers
if there are any,

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while also giving us the
shape of the distribution

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and the general trend.

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In order to draw a
boxplot, first we want

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to find the five-number summary,

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because those are the key
values in the boxplot.

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Once we have our
five-number summary we want

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to determine the lower
and upper fences.

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Now, this is a big
deal, because the lower

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and upper fences are
not provided for us

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when we use our calculators
to draw this graph.

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So, make sure we
have a hand on that.

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You can see the formulas for
the lower and upper fence.

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The lower fence uses Q1,

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and then we subtract
the 1.5 times that IQR.

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Just as a refresher, remember
that IQR is the difference

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between our Q3 and our Q1.

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And then the upper
fence would be our Q3,

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but now we add the product
that is 1.5 times our IQR.

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We will, of course, show
this in the example, though.

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Step three would be to draw a
number line that's long enough

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to include our max and min
values, and then we're going

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to enter some vertical lines
at Q1, our median, and Q3.

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We enclose these lines to
form a box, and that's kind

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of our middle structure
of our boxplot.

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In step four, we will label
the lower and upper fences,

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and then an easy step to
forget, unfortunately,

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would be step five, where
we're drawing the whiskers

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of our boxplot.

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The whiskers tend to
cause a little confusion,

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as our lower whisker
would be from Q1

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to the smallest data value that
is larger than the lower fence;

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that is, it's the smallest piece

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of data we have that's
not an outlier.

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Another way to say that
is it's the smallest piece

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of data we have that's
within our fence.

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So, we draw this line, the
whisker, to the smallest piece

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of data that we have within
our boundaries created

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by the lower and upper fences.

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And then you have another
whisker going from Q3

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to the largest data
value, again,

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where that data value
is within our fences.

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As always, though, if that's
confusing to think about,

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we'll make sure to
elaborate on our example.

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And our last step would be

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to denote any outliers
that we have.

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An outlier would be any piece of
data that is outside our fences.

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You see how important it is to
get those fences correct, then,

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as any value that is less than
the lower fence or greater

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than the upper fence
ends up being an outlier.

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Well, let's go ahead and
jump into our example then.

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We're back to the final exam
scores of the 12 students

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in that stats summer class.

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We have our data
provided again below.

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This time we're told
to construct a boxplot.

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Well, remember, we previously
found our five-number summary.

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We had a minimum value
of 46, our Q1 was a 69,

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our median ended up being a
72, our Q3 ended up being a 79,

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and our maximum value was a 94.

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Well, now that we have our
five-number summary again,

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next we need to calculate
our lower and upper fences.

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And just as a refresher,
one more time,

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remember the IQR is the
difference between Q3 and Q1

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where we have a 79 minus
the 69 from our Q3 and Q1.

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And that makes our
lower fence 69, that Q1,

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minus 1.5 times the 10, our IQR,

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and that gives us a
lower fence value of 54.

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If that's okay, our upper fence,
making sure we're a bit careful,

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as now we're using our
value of Q3 rather than Q1.

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We have 79.

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Be sure to add that now to the
product that is 1.5 times 10,

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and we end up with
an upper fence of 94.

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So, now we have our
lower and upper fences,

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and that means we're all
set to sketch our boxplot.

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Well, sketching our boxplot
then and referring to our steps,

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we already found our five-number
summary like we were told

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to do so in our first step.

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We found our lower
and upper fences.

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And now we're on step three,

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where we have now our
number line long enough

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to include our maximum
and minimum values.

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That means we're ready to
enter our vertical lines

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at Q1, our median, and Q3.

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Well, Q1 we found to be
69, our median we had a 72,

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and our Q3 we found to be 79.

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Those three values coming from
our five-number summary again.

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We take those three values
and the vertical lines,

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and we enclose that box,

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and this would be the
box in our boxplot.

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Now step four tells us to label
the lower and upper fences.

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We had a lower fence of 54, an
upper fence of 94, putting those

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in the approximate ballpark

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of where they belong
in terms of values.

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And now, pausing for a minute,

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notice that I've highlighted
the key values that we're using.

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It's up to your instructor
as far as how

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to label your number line:
whether we're supposed

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to label every five values
or every integer, et cetera.

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In this demonstration, though,

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we're just showing those
key values approximating the

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location of each
of those numbers.

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If that's okay, we're
on step five now,

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where we need to
draw our whiskers.

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Well, the whisker on the right
side ends up being a bit tricky,

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because it's a line from
Q3 to our maximum value

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that is smaller than
the upper fence.

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Well, our maximum value is
94 and our upper fence is 94.

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So, that whisker goes right

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up to our maximum value
and our upper fence.

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Our whisker on the left side
a little more clear, though,

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would be the smallest
data value that's greater

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than our lower fence, and
that would be a value of 61.

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We're all set with
our whiskers then.

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In step six, we need to finish
up our boxplot by tossing

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in our outlier, the value
that is 46; 46 is an outlier

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because it's outside
those fences, or it's less

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than the lower fence, 46 being
less than our lower fence of 54.

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Just to be clear
with our labeling,

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we show the reader we have
our minimum value that ended

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up being an outlier of
46, our lower fence of 54,

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our Q1 being 69, our
median of 72, Q3 is 79,

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and our maximum value of 94,

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which was also the value
for our upper fence.

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And we've got ourselves
one beautiful boxplot.

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Well, now that we've drawn our
boxplot, let's go ahead and walk

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through how to construct this
thing using our calculator.

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We see the instructions
in front of us.

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So, let's go ahead and jump

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over to the calculator
again to demonstrate.

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Looking at our instructions,
we're told first

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to enter our data into
our list and just showing

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that we still have
our data in L1.

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Going to the new part
now, we want to hit 2nd,

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and that Y equals button shows
the option of a stat plot

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above it, meaning
with 2nd activated,

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we can get into our stat plots
by hitting that Y equals button.

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You see your Plot1, 2 and 3.

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We want to make sure whichever
plot we're using is on.

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Right now it shows that
it is on, but just in case

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as we hit Enter, if it was off,

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the Off word would
be highlighted.

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We would just arrow
to On and hit Enter

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to make sure that's
highlighted back to On.

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Next, we're told to
arrow down to Type.

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The type of boxplot we want
would be here with the outliers.

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Make sure you're a
bit careful with this.

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This option tells us that we
want to illustrate the outliers.

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If we accidentally
chose the other option,

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it would treat all our
data as quality data rather

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than calculating
outliers for us.

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So, make sure we choose
the correct option

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that is the boxplot
with outliers.

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Our Xlist needs to be L1,
since our data is in L1.

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Frequency stays at 1, since
we're working with raw data,

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and any mark that you
want to do will work.

00:11:13.296 --> 00:11:16.556 A:middle
So, if you are just loving
these boxplots, feel free to use

00:11:16.556 --> 00:11:18.926 A:middle
that plus sign to
show your positivity.

00:11:18.926 --> 00:11:21.176 A:middle
If you've got an empty
heart, we'll go ahead

00:11:21.176 --> 00:11:22.516 A:middle
and use this open square.

00:11:23.196 --> 00:11:26.106 A:middle
Once you're all set there,
as you see in the notes,

00:11:26.106 --> 00:11:27.676 A:middle
we need to head over to Zoom.

00:11:28.216 --> 00:11:31.146 A:middle
And we want option number
9: that would be ZoomStat.

00:11:31.646 --> 00:11:33.506 A:middle
You can arrow down to ZoomStat,

00:11:33.936 --> 00:11:35.716 A:middle
or you could just
hit the number 9.

00:11:36.146 --> 00:11:37.106 A:middle
Either way will work.

00:11:37.496 --> 00:11:40.136 A:middle
If I hit 9 or hit Enter
since I'm highlighted,

00:11:40.136 --> 00:11:42.236 A:middle
it will generate my graph,

00:11:42.236 --> 00:11:44.236 A:middle
and we've got ourselves
our lovely boxplot.

00:11:45.676 --> 00:11:47.816 A:middle
Making sure we're a bit
careful to notice, though,

00:11:48.016 --> 00:11:50.046 A:middle
it does not provide
us the fences.

00:11:50.046 --> 00:11:52.426 A:middle
Remember, we had a fence
here when we drew it by hand,

00:11:52.426 --> 00:11:55.296 A:middle
and we had another fence
matching our maximum value.

00:11:55.296 --> 00:11:58.456 A:middle
So, our calculator does not give
us those values of the lower

00:11:58.456 --> 00:12:01.246 A:middle
and upper fence; we have
to make sure we remember

00:12:01.246 --> 00:12:02.946 A:middle
to do those ourselves.

00:12:02.946 --> 00:12:05.636 A:middle
And, one last piece, though,
we could come over here

00:12:05.636 --> 00:12:07.236 A:middle
and hit the Trace button.

00:12:07.236 --> 00:12:10.816 A:middle
If you hit Trace and then use
the right and left arrows,

00:12:10.956 --> 00:12:13.866 A:middle
it will jump to your key points.

00:12:13.866 --> 00:12:16.246 A:middle
That is, if we need to find
the five-number summary,

00:12:16.326 --> 00:12:17.926 A:middle
we could always draw a boxplot,

00:12:17.926 --> 00:12:19.846 A:middle
and it will show
us our key points

00:12:19.846 --> 00:12:21.626 A:middle
that would be the
five-number summary.

00:12:22.316 --> 00:12:25.696 A:middle
Over here our outlier was the
minimum, like we had by hand,

00:12:26.196 --> 00:12:32.166 A:middle
a value of 46; the whisker
was set at 61; our Q1 was 69;

00:12:32.606 --> 00:12:36.916 A:middle
our median was 72;
Q3 we found to be 79;

00:12:37.236 --> 00:12:39.226 A:middle
and our maximum was that 94.

00:12:39.226 --> 00:12:41.936 A:middle
If that all makes sense,

00:12:41.936 --> 00:12:44.976 A:middle
you're all set drawing a
boxplot with your calculator.

00:12:45.636 --> 00:12:47.496 A:middle
Pause the video and
try these problems.

00:12:54.366 --> 00:12:56.106 A:middle
And, that wraps up
our conversation

00:12:56.236 --> 00:12:59.186 A:middle
on the five-number
summary and boxplots.

00:12:59.186 --> 00:13:01.016 A:middle
And thank you one more
time for joining us.