Constructing a box-and-whisker plot

The owner of a restaurant
wants to find out more about

where his patrons
are coming from.

One day, he decided
to gather data

about the distance
in miles that people

commuted to get
to his restaurant.

People reported the
following distances traveled.

So here are all the
distances traveled.

He wants to create
a graph that helps

him understand the
spread of the distances--

this is a key word--
the spread of distances

and the median distance
that people traveled

or that people travel.

What kind of graph
should he create?

So the answer of what kind
of graph he should create,

that might be a little
bit more straightforward

than the actual creation of the
graph, which we will also do.

But he's trying to visualize
the spread of information.

And at the same time,
he wants the median.

So what a graph captures
both of that information?

Well, a box and whisker plot.

So let's actually try to
draw a box and whisker plot.

And to do that, we need to
come up with the median.

And we'll also see the median
of the two halves of the data

as well.

And whenever we're trying to
take the median of something,

it's really helpful
to order our data.

So let's start off by
attempting to order our data.

So what is the
smallest number here?

Well, let's see.

There's one 2.

So let me mark it off.

And then we have another two.

So we've got all the 2's.

And then we have this 3.

Then we have this 3.

I think we've got all the 3's.

Then we have that 4.

Then we have this 4.

Do we have any 5's?

No.

Do we have any 6's?

Yep.

We have that 6.

And that looks like the only 6.

Any 7's?

Yep.

We have this 7 right over here.

And I just realized
that I missed this 1.

So let me put the 1 at
the beginning of our set.

So I got that 1
right over there.

Actually, there was two 1's.

I missed both of them.

So both of those 1's
are right over there.

So I have the 1's,
2's, 3's, 4's, no 5's.

This is one 6.

There was one 7.

There's one 8 right over here.

And then, let's see, any 9's?

No 9's.

Any 10s?

Yep.

There's a 10.

Any 11s?

We have an 11 right over there.

Any 12s?

Nope.

13, 14?

Then we have a 15.

And then we have a
20 and then a 22.

So we've ordered all our data.

Now it should be relatively
straightforward to find

the middle of our
data, the median.

So how many data
points do we have?

1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17.

So the middle number
is going to be

a number that has 8
numbers larger than it

and 8 numbers smaller than it.

So let's think about it.

1, 2, 3, 4, 5, 6, 7, 8.

So the number 6 here is
larger than 8 of the values.

And if I did the
calculations right,

it should be smaller
than 8 of the values.

1, 2, 3, 4, 5, 6, 7, 8.

So it is, indeed, the median.

Now, when we're trying to
construct a box and whisker

plot, the convention is,
OK, we have our median.

And it's essentially dividing
our data into two sets.

Now, let's take the median
of each of those sets.

And the convention is to
take our median out and have

the sets that are left over.

Sometimes people leave it in.

But the standard convention,
take this median out.

And now, look
separately at this set

and look separately at this set.

So if we look at this first
bottom half of our numbers

essentially, what's the
median of these numbers?

Well, we have 1, 2, 3, 4,
5, 6, 7, 8 data points.

So we're actually going to
have two middle numbers.

So the two middle numbers
are this 2 and this 3,

three numbers less
than these two,

three numbers greater than it.

And so when we're
looking for a median,

you have two middle numbers.

We take the mean of
these two numbers.

So halfway in between
two and three is 2.5.

Or you can say 2 plus 3
is 5 divided by 2 is 2.5.

So here we have a median
of this bottom half of 2.5.

And then the middle
of the top half,

once again, we
have 8 data points.

So our middle two
numbers are going

to be this 11 and this 14.

And so if we want to take the
mean of these two numbers,

11 plus 14 is 25.

Halfway in between
the two is 12.5.

So 12.5 is exactly
halfway between 11 and 14.

And now, we've figured
out all of the information

we need to actually
plot or actually

create or actually draw
our box and whisker plot.

So let me draw a number line,
so my best attempt at a number

line.

So that's my number line.

And let's say that this
right over here is a 0.

I need to make sure I get all
the way up to 22 or beyond 22.

So let's say that's 0.

Let's say this is 5.

This is 10.

That could be 15.

And that could be 20.

This could be 25.

We could keep
going-- 30, maybe 35.

So the first thing we might
want to think about-- there's

several ways to draw it.

We want to think about
the box part of the box

and whisker
essentially represents

the middle half of our data.

So it's essentially trying to
represent this data right over

here, so the data between the
medians of the two halves.

So this is a part
that we would attempt

to represent with the box.

So we would start right
over here at this 2.5.

This is essentially
separating the first quartile

from the second quartile, the
first quarter of our numbers

from the second
quarter of our numbers.

So let's put it right over here.

So this is 2.5.

2.5 is halfway between 0 and 5.

So that's 2.5.

And then up here, we have 12.5.

And 12.5 is right
over-- let's see.

This is 10.

So this right over here would be
halfway between, well, halfway

between 10 and 15 is 12.5.

So let me do this.

So this is 12.5 right over here.

So that separates
the third quartile

from the fourth quartile.

And then our boxes,
everything in between,

so this is literally the
middle half of our numbers.

And we'd want to show
where the actual median is.

And that was actually
one of the things

we wanted to be able
to think about when

the owner of the
restaurant wanted

to think about how far
people are traveling from.

So the median is 6.

So we can plot it
right over here.

So this right here is about six.

Let me do that same pink color.

So this right over here is 6.

And then the whiskers of
the box and whisker plot

essentially show us
the range of our data.

And I can do this in a different
color that I haven't used yet.

I'll do this in orange.

So essentially, if
we want to see, look,

the numbers go all
the way up to 22.

So they go all the
way up to-- so let's

say that this is
22 right over here.

Our numbers go all
the way up to 22.

And they go as low as 1.

So 1 is right about here.

Let me label that.

So that's 1.

And they go as low as 1.

So there you have it.

We have our box
and whisker plot.

And you can see if you
have a plot like this,

just visually, you
can immediately

see, OK, what is the median?

It's the middle of
the box, essentially.

It shows you the middle half.

So it shows you how
far they're spread

or where the meat
of the spread is.

And then it shows, well, beyond
that, we have the range that

goes well beyond that or how
far the total spread of our data

is.

So this gives a pretty good
sense of both the median

and the spread of our data.