3.3 Symmetric, Skewed Left or Right?

So we left off where we
should think our mode should be 49,

because that's the one
that occurs the most, right?

It's the only one that occurs twice.

But remember our other definition.

It's the location of the center
of the highest bar

on the histogram.

So I'm going to go back
to our histogram that we created

in the beginning of these notes.

And I went ahead,
I'm going to redo it.

I'm going to put our median.

We talked about 74,
so I'm eyeballing it.

There's our median.

And then we're going to put our mean,
which was 72.6.

So it's about right there;
there's our mean.

And if we're following that definition

of our mode being the highest
pretty much the peak,

the middle of our peak,
it would have to be there,

which is not 49, right?

Now let's make sense of this.

How we're graphing this data

is we're doing it in bin sizes of 10.

So that's very different
than just plotting this data

as just a single test 49,
a single test 68, and so forth.

They're in bins.

Why this is more informative
is because this

is where the majority of the scores
are lying, right?

Is right here,

not down here, all right?

So it's really important
that we understand the difference.

I'm really interested in
where does the majority

of our data lie, right?

Where did--
how did the majority of my students do?

Of course, these 49 students matter.

But overall, are students getting it?
Or overall, you know,

am I providing them
with the material they need

to do well on the exam?

That is what that information gives me
versus looking at, oh,

just two people got these 49s.

So that is why our...

mode is: 80 up points

is the mode.

Hopefully, that makes sense to you.

It's where the majority
of the data is lying.

Also, visually, it's the highest peak.

So, knowing this--

So we just learned:
mean, median, mode.

We're going to use
these three terms visually.

So we talked about--

and I lost my graph, so I'm just
going to give them to us again.

We talked about something
looking symmetric, okay.

Something looks skewed to the left,

which means you have a tail on the end.

And then something skewed to the right,

you have a tail on the right.

Well, what we just learned is
our mode

is our peak...

our peak.

Now remember, because this scale
right here is frequency.

That's how often it's happening.

So that's why that would be our mode,

because it's happening the most.

Okay, the other definition we
talked about is median.

And that cuts your data
right in half, okay?

So it cuts your histogram
in the middle.

So we have...

Here is the middle, right,
when something is symmetric.

Well, what do you know,
that means our median and our mode

will be the same,
when it's symmetric.

Whereas if I go over here,
skewed to the left,

I have to cut this data in half,

saying half the data is below it,
half the data is above it.

I'm completely eyeballing this,
but this looks pretty accurate.

This would be my median here.

Whereas on this one,

it'd be somewhere around there.

All right.
So the last piece of information is

we have to talk about the mean.

Remember, the mean is the average.

Well, think about if you
were adding up all these values,

your average is actually going to be...

the same as your mode and your median
when it's symmetric.

So that is the beauty
of having symmetrical data.

It doesn't always happen,
but it does happen.

Um, so what I can get from that
is then if I tell you,

you have symmetric--
your histogram is symmetric,

and your mean is, uh,

48 points, then you know,
your median and your mean,

or, sorry, your median and your mode
are around 48 points.

They might not be exactly 48,

but they're going to be
pretty darn close.

Whereas...

this one...

for your skewed
to the left,

this tail down here, this guy,

he's messing up my average.

He's pulling my average down, actually.

So that's why-- totally eyeballing,

--this is where your mean would be,

because this tail is bringing it down.

And so that's why I have right here
that your mean...

is less than your median.

Where skewed to the right,
your tail is over here, right?

So it's bringing your averages up.

So your total approximation,
mean is about right there.

So that means, huh,
your mean [chuckles]

is greater than your median.

So let's put this information
into play; let's see.

So I'm going to do our last example:

"Suppose you know that a sample
of 6 children's heights

has a mean of 55 inches
and a median of 45 inches."

Well, I know 55 inches is bigger

than 45 inches, right?

And my 55

is the median.

Okay, this isn't the way I wrote it,

but it's saying here--
hopefully, you're okay with this,

--this is the same thing
as saying... this.

I just wrote it,
how it was given to me

in the order, but, hopefully,
you're okay with that.

This is saying the same thing as this.

So I'm saying my mean
is less than my median.

So remember that means my average
is being--

something is pulling down my average.

So I like thinking about
like that, so I don't have to

completely memorize things.

But if I go up here, where is your mean
less than your median?

Yeah, it would be skewed left.
And again, because...

the tail is pulling it down.

Again, if you're visual...

it would look like that, right,
because this stuff

would make your mean be over here

versus your median is up here.

So because of this,
we know the distribution

would have to be skewed left.

All right.
See you next time.