Two-way frequency tables and Venn diagrams

- So what I have here
are 12 pieces of candy.

The ones that are colored
in brown have chocolate

on the outside and the ones that have a C

on them means that they have coconut

on the inside.

So for example, this one
over here in the top left,

it's made out of chocolate on the outside,

but it doesn't have coconut on the inside.

While this one right over
here is chocolate on the

outside and it has coconut on the inside.

While this one, whoops,
I didn't want to do that,

while this one right
over here does not have

chocolate on the outside, but it does have

coconut on the inside.

This one right over here has neither

chocolate nor coconut.

What I want to think
about is ways to represent

this information that we are looking at.

One way to do it is using a Venn diagram.

So let me draw a Venn diagram.

So Venn diagram is one
way to represent it.

The way it's typically
done, the convention,

is that you would make
a rectangle to represent

the universe that you
care about, in this case

it would be all the chocolates.

All the numbers inside
of this should add up

to the number of chocolates I have.

So it should add up to 12.

So that's our universe right over here.

Then I'll draw circles
to represent the sets

that I care about.

So say for this one I care about the set

of the things that have chocolate.

So I'll draw that with a circle.

You could draw them to scale, but I'm not

going to draw them to scale.

So that is my chocolate set, chocolate.

That is my chocolate set.

Then I want to have a coconut set.

So coconut.

Once again, not drawn to scale.

I drew them roughly the same size,

but you can see the
chocolate set is bigger

than the coconut set in reality.

Coconut set.

Now we can fill in the different sections.

So how many of these things have chocolate

but no coconut?

Let's see, we have one,
two, three, four, five,

six have chocolate but no coconut.

Let me do that in a different color

because I think the colors are important.

So let me do it in green.

So one, two, three, four, five, and six.

So this section right over here is six.

Once again, I'm not
talking about the whole

brown thing, I'm talking about just this

area that I've shaded in green.

Now how many have chocolate and coconut?

Chocolate and coconut.

Well that's going to be one, two, three.

So three of them have
chocolate and coconut.

Notice that's this section here that's

in the overlap between.

Three of them go into both
sets, both categories.

These three have coconut
and they have chocolate.

How many total have chocolate?

Well six plus three, nine.

How many total have coconut?

Well we're going to have to figure that

out in a second.

So how many have coconut but no chocolate?

Well there's only one with
coconut and no chocolate.

So that's that one right over there,

and that represents this are that I'm

shading in in white.

So how many total coconut are there?

Well one plus three, or
four, and you see that.

One, two, three, four.

The last thing we'd want to fill in,

because notice, six plus three plus one

only adds up to 10.

What about the other two?

Well the other two are
neither chocolate nor coconut.

Actually, let me color this.

So that's one, two, these are neither

chocolate nor coconut.

I could write these two right over here.

These are neither chocolate nor coconut.

So that's one way to
represent the information

of how many chocolates, how many coconuts,

and how many chocolate and coconuts,

and how many neither.

But there's other ways
that we could do it.

Another way to do it would
be with a two way table.

Two way table.

On one axis, say the vertical axis,

we could say, let me write this,

so has chocolate, I'll
write choc for short,

and then I'll write no
chocolate, choc for short.

Then over here I could write coconut.

I want to do that in white.

I got new tools and sometimes

the color changing isn't so easy.

So this is coconut and then over here

I'll write no coconut.

Then let me make a little table.

Make it clear what I'm doing here.

So a line there and a line there

and why not add a line over here as well.

Then I can just fill in
the different things.

This cell, this square,
this is going to represent

the number that has coconut and chocolate.

Well we are already looked into that,

that's one, two, three, that's these three

right over here.

So that's those three right over there.

This one right over
here, it has chocolate,

but it doesn't have coconut.

Well that's this six right over here.

It has chocolate, but
it doesn't have coconut.

So let me write this is
that six right over there.

Then this box would be it has coconut,

but no chocolate.

Well how many is that?

Well coconut no chocolate
that's that one there.

Then this one is going to be no coconut

and no chocolate.

We know what that's going to be.

No coconut and no chocolate
is going to be two.

If we wanted to, we could even throw

in totals over here.

We could write, actually
let me just do that

just for fun.

I could write total, I could write total,

and if I total it
vertically, so three plus

one this is four, six plus two is eight.

So this four is the total number

that have coconut that has chocolate

and doesn't have chocolate.

That's three plus one.

This eight is the total
that does not have coconut.

No coconut.

So the total of no
coconut, and that of course

is going to be the six plus this two.

We could total horizontally.

Three plus six is nine.

One plus two is three.

What's this nine?

That's the total amount of
chocolate, six plus three.

What's this three?

This is the total amount no chocolate.

That's this one plus two.

Anyway, hopefully you
found that interesting.

This is just different
ways of representing

the same information.