Intro to Linear Regression

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- [Thomas Stratmann] Hi!

In the upcoming series of videos

we're going to give you
a shiny new tool

to put into your
Understanding Data toolbox:

linear regression.

Say you've got this theory.

You've witnessed
how good-looking people

seem to get special perks.

You're wondering -- "Where else
might we see this phenomenon?"

What about full professors?

Is it possible
good-looking professors

might get special perks too?

Is it possible
students treat them better

by showering them
with better student evaluations?

If so, is the effect of looks

on evaluation score
big or [inaudible]?

And say there is a new professor
starting at the university.

What can we predict
about his evaluation

simply by his looks?

Given that these evaluations
can determine pay raises,

if this theory were true
we might see professors resort

to some surprising tactics
to boost their scores.

Suppose you wanted to find out

if evaluations really improve
with better looks.

How would you go about
testing this hypothesis?

You could collect data.

First you would have students rate
on a scale from 1 to 10

how good looking a professor was,

which gives you
an average beauty score.

Then you could retrieve
the teacher's teaching evaluations

from 25 students.

Let's look at these two variables
at the same time

by using a scatterplot.

We'll put beauty
on the horizontal axis,

and teacher evaluations
on the vertical axis.

For example, this dot
represents Professor Peate,

who received a beauty score of 3

and an evaluation of 8.425.

This one way out here
is Professor Helmchen.

- [Professor Helmchen]
Ridiculously good-looking!

- [Thomas] Who got
a very high beauty score,

but not such a good evaluation.

Can you see a trend?

As we move from left to right
on the horizontal axis,

from the ugly to the gorgeous,

we see a trend upwards
in evaluation scores.

By the way, the data
we're exploring in this series

is not made up --
it comes from a real study

done at the University of Texas.

If you're wondering, "pulchritude"
is just the fancy academic way

of saying beauty.

With scatterplots
it can sometimes be hard

to make out the exact relationship
between two variables --

especially when the variables
bounce around quite a bit

as we go from left to right.

One way to cut through
this bounciness

is to draw a straight line
through the data cloud

in such a way that this line
summarizes the data

as closely as possible.

The technical term for this
is "linear regression."

Later on we'll talk about
how this line is created,

but for now we can assume
that the line fits the data

as closely as possible.

So, what can this line tell us?

First, we immediately see

if the line is sloping
upward or downward.

In our data set we see
the [fitted] line slopes upward.

It thus confirms what
we have conjectured earlier

by just looking at the scatterplot.

The upward slope means
that there is a positive association

between looks
and evaluation scores.

In other words, on average,

better-looking professors
are getting better evaluations.

For other data sets we might see
a stronger positive association.

Or, you might see
a negative association.

Or perhaps no association at all.

And our lines
don't have to be straight.

They can curve to fit the data
when necessary.

This line also gives us
a way to predict outcomes.

We can simply take a beauty score
and read off the line

what the predicted
evaluation score would be.

So, back to our new professor.

- [Professor Lloyd] Look familiar?

- [Thomas] We can precisely
predict his evaluation score.

"But wait! Wait!" you might say.

"Can we trust this prediction?"

How well does
this one beauty variable

really predict evaluations?

Linear regression gives us
some useful measures

to answer those questions

which we'll cover
in a future video.

We also have to be aware
of other pitfalls

before we draw
any definite conclusions.

You could imagine a scenario

where what is driving
the association

we see is really a third variable
that we have left out.

For example,
the difficulty of the course

might be behind
the positive association

between beauty ratings
and evaluation scores.

Easy intro. courses
get good evaluations.

Harder, more advanced courses
get bad evaluations.

And younger professors might
get assigned to intro. courses.

Then, if students judge
younger professors more attractive,

you will find
a positive association

between beauty ratings
and evaluation scores.

But it's really
the difficulty of the course.

The variable that we've left out,
not beauty,

that is driving evaluation scores.

In that case, all the primping
would be for naught --

a case of mistaken correlation
for causation,

something we'll talk about further
in a later video.

And what if there were
other important variables

that affect both beauty ratings
and evaluation scores?

You might want to add
considerations like skill,

race, sex, and whether English
is the teacher's native language

to isolate more cleanly the effect
of beauty on evaluations.

When we get
into multiple regression

we will be able to measure
the impact of beauty

on teacher evaluations

while accounting
for other variables

that might confound
this association.

Next up, we'll get our hands dirty
by playing with this data

to gain a better understanding
of what this line can tell us.

- [Narrator] Congratulations!

You're one step closer
to being a data ninja!

However, to master this you'll need
to strengthen your skills

with some practice questions.

Ready for your next mission?
Click "Next Video."

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