Stat! Quest! Stat! Quest! Stat! Quest! StatQuest!
Stat! Quest! Stat! Quest! Stat! Quest! StatQuest!

StatQuest is brought to you by
StatQuest is brought to you by

the friendly people in the genetics
the friendly people in the genetics

department at the University of North
department at the University of North

Carolina at Chapel Hill. Hello and
Carolina at Chapel Hill. Hello and

welcome to StatQuest in this video
welcome to StatQuest in this video

we're going to talk about R-squared.
we're going to talk about R-squared.

R-squared is a metric of correlation that
R-squared is a metric of correlation that

is easy to compute and intuitive to
is easy to compute and intuitive to

interpret. Most of us are already
interpret. Most of us are already

familiar with correlation and the
familiar with correlation and the

standard metric of it plain old 'r'.
standard metric of it plain old 'r'.

Correlation values that are close to 1
Correlation values that are close to 1

or negative 1 are good and tell you that
or negative 1 are good and tell you that

two quantitative variables for example
two quantitative variables for example

weight and size are strongly related.
weight and size are strongly related.

Correlation values close to zero are
Correlation values close to zero are

lame. Some of you may be asking why
lame. Some of you may be asking why

should we care about R-squared; we
should we care about R-squared; we

already have regular 'r'. Some of you
already have regular 'r'. Some of you

might just be asking what is R-squared.
might just be asking what is R-squared.

R-squared is very similar to its hipper
R-squared is very similar to its hipper

cousin 'r', but interpretation is easier.
cousin 'r', but interpretation is easier.

For example it's not obvious that when 'r'
For example it's not obvious that when 'r'

equals 0.7 that's twice as good a
equals 0.7 that's twice as good a

correlation as when 'r' equals 0.5.
correlation as when 'r' equals 0.5.

However R-squared equals 0.7 is what it
However R-squared equals 0.7 is what it

looks like
looks like

it's 1.4 times as good as R-squared
it's 1.4 times as good as R-squared

equals 0.5. The other thing that I like
equals 0.5. The other thing that I like

about R-squared is that it's easy and
about R-squared is that it's easy and

intuitive to calculate. Let's start with
intuitive to calculate. Let's start with

an example.
an example.

Here were plotting Mouse weight on the
Here were plotting Mouse weight on the

y-axis with high weights towards the top
y-axis with high weights towards the top

and low weights towards the bottom and
and low weights towards the bottom and

mouse identification numbers on the
mouse identification numbers on the

x-axis with ID numbers 1 through 7.
x-axis with ID numbers 1 through 7.

We can calculate the mean or average of
We can calculate the mean or average of

the mouse weights and plot it as a line
the mouse weights and plot it as a line

that spans the graph. We can calculate
that spans the graph. We can calculate

the variation of the data around this
the variation of the data around this

mean as the sum of the squared
mean as the sum of the squared

differences of the weight for each mouse
differences of the weight for each mouse

'i', where 'i' as an individual mouse
'i', where 'i' as an individual mouse

represented by a red dot, and the mean.
represented by a red dot, and the mean.

The difference between each data point
The difference between each data point

is squared so that the points below the
is squared so that the points below the

mean don't cancel out the points above
mean don't cancel out the points above

the mean. Now, what if instead of ordering
the mean. Now, what if instead of ordering

our mice by their identification number
our mice by their identification number

we ordered them by their size? Instead of
we ordered them by their size? Instead of

using identification number on the
using identification number on the

x-axis we have mouse size, with the
x-axis we have mouse size, with the

smallest size on the left side and the
smallest size on the left side and the

largest size on the right side. All we
largest size on the right side. All we

have done is reorder the data on the
have done is reorder the data on the

x-axis, the mean and variation are the
x-axis, the mean and variation are the

exact same as before. Here we show the
exact same as before. Here we show the

mean again as a black bar that spans the
mean again as a black bar that spans the

graph in the exact same location as it
graph in the exact same location as it

was before.
was before.

Also the distances between the dots and
Also the distances between the dots and

the line have not changed, just the order
the line have not changed, just the order

of the dots. Here's a question for you:
of the dots. Here's a question for you:

Given that we know an individual Mouse's
Given that we know an individual Mouse's

size, is the mean, or average weight, the
size, is the mean, or average weight, the

best way to predict that individual
best way to predict that individual

Mouse's weight? Well the answer is 'no'. We
Mouse's weight? Well the answer is 'no'. We

can do way better. All we have to do is
can do way better. All we have to do is

fit a line to the data. Now we can
fit a line to the data. Now we can

predict weight with our line. You tell me
predict weight with our line. You tell me

you have a large Mouse, I can look at my
you have a large Mouse, I can look at my

line and make a good guess about the
line and make a good guess about the

weight. Here's another question: Does the
weight. Here's another question: Does the

blue line that we just drew fit the data
blue line that we just drew fit the data

better than the mean? If so how much
better than the mean? If so how much

better? By eye, it looks like the blue
better? By eye, it looks like the blue

line fits the data better than the mean.
line fits the data better than the mean.

How do we quantify that difference?
How do we quantify that difference?

R-squared. In the bottom of the graph
R-squared. In the bottom of the graph

I've drawn the equation for R-squared.
I've drawn the equation for R-squared.

We're going to walk through it one step
We're going to walk through it one step

at a time.
at a time.

The first part of the equation is just
The first part of the equation is just

the variation around the mean. We already
the variation around the mean. We already

calculated that. It's just the sum of the
calculated that. It's just the sum of the

squared differences of the actual data
squared differences of the actual data

values from the mean. The second part of
values from the mean. The second part of

the equation is the variation around our
the equation is the variation around our

new blue line. This is calculated in a
new blue line. This is calculated in a

very similar way. Here, we just want the
very similar way. Here, we just want the

sum of the squared differences between
sum of the squared differences between

the actual data points and our new blue
the actual data points and our new blue

line. The numerator, which is the
line. The numerator, which is the

difference between the variation around
difference between the variation around

the mean and the variation around the
the mean and the variation around the

blue line, is then divided by the
blue line, is then divided by the

variation around the mean. This makes
variation around the mean. This makes

R-squared
R-squared

range from zero to one, because the
range from zero to one, because the

variation around the line will never be
variation around the line will never be

greater than the variation around the
greater than the variation around the

mean and it will never be less than zero.
mean and it will never be less than zero.

This division also makes our squared a
This division also makes our squared a

percentage and we'll talk more about
percentage and we'll talk more about

that in just a second. Now, we'll walk
that in just a second. Now, we'll walk

through an example where we calculate
through an example where we calculate

things one step at a time.
things one step at a time.

First, we'll start with the variation
First, we'll start with the variation

around the mean. In this case that equals
around the mean. In this case that equals

32. The variation around the blue line is
32. The variation around the blue line is

only six, which is what we suspected
only six, which is what we suspected

since it appears to fit the data much
since it appears to fit the data much

better. Once we've calculated the
better. Once we've calculated the

variation around the mean and the
variation around the mean and the

variation around our blue line we can
variation around our blue line we can

plug these values in to our formula for
plug these values in to our formula for

R-squared. After plugging in our values,
R-squared. After plugging in our values,

we get R-squared equals 32 minus 6 over
we get R-squared equals 32 minus 6 over

32. After subtracting 6 from 32, we get 26.
32. After subtracting 6 from 32, we get 26.

Doing the division, 26 divided by 32,
Doing the division, 26 divided by 32,

gives us 0.81, or 81%.
gives us 0.81, or 81%.

This means that there is 81% less
This means that there is 81% less

variation around the line than the mean.
variation around the line than the mean.

In other words, the size-weight
In other words, the size-weight

relationship accounts for 81% of
relationship accounts for 81% of

the total variation. This means that most
the total variation. This means that most

of the variation in the data is
of the variation in the data is

explained by the size-weight
explained by the size-weight

relationship.
relationship.

Here's another example. In this example
Here's another example. In this example

we're comparing two possibly
we're comparing two possibly

uncorrelated variables. On the y-axis we
uncorrelated variables. On the y-axis we

have mouse weight again, but on the
have mouse weight again, but on the

x-axis we now have "time spent sniffing a rock".
x-axis we now have "time spent sniffing a rock".

Like before, we calculate the
Like before, we calculate the

variation around the mean and just like
variation around the mean and just like

before we got 32. However, this time when
before we got 32. However, this time when

we calculated the variation around the
we calculated the variation around the

blue line, we got a much larger value, 30.
blue line, we got a much larger value, 30.

Now we just plug those values into our
Now we just plug those values into our

formula for R-squared. By doing the math
formula for R-squared. By doing the math

we see that R-squared equals 0.06 or 6%.
we see that R-squared equals 0.06 or 6%.

Thus there is only 6% less variation
Thus there is only 6% less variation

around the line than the mean. In other
around the line than the mean. In other

words, the sniff-weight relationship
words, the sniff-weight relationship

accounts for only 6% of the total
accounts for only 6% of the total

variation. This means that hardly any of
variation. This means that hardly any of

the variation in the data is explained
the variation in the data is explained

by the sniff-weight relationship.
by the sniff-weight relationship.

Now, when someone says "The statistically
Now, when someone says "The statistically

significant R-squared was 0.9", you can
significant R-squared was 0.9", you can

think to yourself "Very good! The
think to yourself "Very good! The

relationship between the two variables
relationship between the two variables

explains 90% of the variation in the
explains 90% of the variation in the

data." And when someone else says, "The
data." And when someone else says, "The

statistically significant R-squared was
statistically significant R-squared was

0.01." You can think to yourself, "Dag! Who
0.01." You can think to yourself, "Dag! Who

cares if that relationship is
cares if that relationship is

significant? It only accounts for 1% of
significant? It only accounts for 1% of

the variation in the data. Something else
the variation in the data. Something else

must explain the remaining 99%."
must explain the remaining 99%."

What about plain old 'r'? How is it related to R-squared?
What about plain old 'r'? How is it related to R-squared?

R-squared is just the square of 'r'
R-squared is just the square of 'r'

Now, when someone says
Now, when someone says

"The statistically significant 'r' was 0.9", and
"The statistically significant 'r' was 0.9", and

we're talking about just plain old 'r',
we're talking about just plain old 'r',

you can think to yourself "0.9 times
you can think to yourself "0.9 times

0.9 equals 0.81.
0.9 equals 0.81.

Very good! The relationship between the
Very good! The relationship between the

two variables explains 81% of the
two variables explains 81% of the

variation in the data." And when someone
variation in the data." And when someone

else says "The statistically significant
else says "The statistically significant

'r'", that's plain old 'r', "was 0.5." You can
'r'", that's plain old 'r', "was 0.5." You can

think to yourself "0.5 times 0.5 equals
think to yourself "0.5 times 0.5 equals

0.25. The relationship accounts for 25%
0.25. The relationship accounts for 25%

of the variation in the data. That's good
of the variation in the data. That's good

if there are a million other things
if there are a million other things

accounting for the remaining 75% and bad
accounting for the remaining 75% and bad

if there's only one thing." I like
if there's only one thing." I like

R-squared more than just plain old 'r'
R-squared more than just plain old 'r'

because it's easier to interpret. Here's
because it's easier to interpret. Here's

an example: How much better is 'r' equals
an example: How much better is 'r' equals

0.7 then 'r' equals 0.5? Well if we convert
0.7 then 'r' equals 0.5? Well if we convert

those numbers to R-squared, we see that
those numbers to R-squared, we see that

when R-squared equals 0.7
when R-squared equals 0.7

squared it actually equals 0.5,
squared it actually equals 0.5,

which means 50% of the original
which means 50% of the original

variation is explained by the
variation is explained by the

relationship. When R-squared equals 0.5
relationship. When R-squared equals 0.5

squared which equals 0.25, we see that
squared which equals 0.25, we see that

only 25% of the original variation is
only 25% of the original variation is

explained by the relationship. With
explained by the relationship. With

R-squared it's easy to see that the first
R-squared it's easy to see that the first

correlation is twice as good as the
correlation is twice as good as the

second. Explaining 50% of the original
second. Explaining 50% of the original

variation is twice as good as only
variation is twice as good as only

explaining 25% of the original variation.
explaining 25% of the original variation.

That said our squared does not indicate
That said our squared does not indicate

the direction of the correlation because
the direction of the correlation because

squared numbers are never negative. If
squared numbers are never negative. If

the direction of the correlation isn't
the direction of the correlation isn't

obvious, you can say "the two variables
obvious, you can say "the two variables

were positively or negatively correlated
were positively or negatively correlated

with R squared equals dot dot dot",
with R squared equals dot dot dot",

whatever that value may be. These are the
whatever that value may be. These are the

two main ideas for R-squared. R-squared
two main ideas for R-squared. R-squared

is the percentage of variation explained
is the percentage of variation explained

by the relationship between two
by the relationship between two

variables, and also, if someone gives you
variables, and also, if someone gives you

a value for plain old are just square it
a value for plain old are just square it

in your head, you'll understand what's
in your head, you'll understand what's

going on a whole lot better. We've
going on a whole lot better. We've

reached the end of our StatQuest. Tune
reached the end of our StatQuest. Tune

in next time for an exciting adventure
in next time for an exciting adventure

into the land of statistics.
into the land of statistics.