The correlation,
r
is a pure, unitless number.
17 / 29
Today’s Goals
Summary Statistics
Association
Correlation
Properties
Interchange Variables
r
= the average of the products of the data in standard units
=
1
n
n
X
i
=1
x
i

¯
x
sd
x
×
y
i

¯
y
sd
y
=
1
n
n
X
i
=1
y
i

¯
y
sd
y
×
x
i

¯
x
sd
x
Another way: the correlation is all about the linear clustering of
points.
The correlation between
x
and
y
is the same as the correlation
between
y
and
x
18 / 29
Today’s Goals
Summary Statistics
Association
Correlation
Properties
Change of Location
Recall that if each entry in a list gets increased by a number,
b
,
the mean gets increased by
b
, and the sd does not change
new
x
i
= old
x
i
+ b
new ¯
x
= old ¯
x
+ b
new
sd
x
= old
sd
x
Applies exactly the
same to
y
new
x
i
in s.u. =
new
x
i

new ¯
x
new
sd
x
=
(old
x
i
+
b
)

( old ¯
x
+
b
)
old
sd
x
=
old
x
i

old ¯
x
old
sd
x
= old
x
i
in s.u.
Values in s.u. don’t change, correlation doesn’t change
19 / 29
Today’s Goals
Summary Statistics
Association
Correlation
Properties
Change of Scale
Recall that if each entry in a list gets multiplied by a number,
d
,
the mean gets multiplied by
d
, and the sd gets multiplied by

d

new
x
i
= old
x
i
×
d
new ¯
x
= old ¯
x
×
d
new
sd
x
=
old
sd
x
× 
d

Applies exactly the
same to
y
new
x
i
in s.u. =
new
x
i

new ¯
x
new
sd
x
=
old
x
i
×
d

old ¯
x
×
d
old
sd
x
× 
d

= sign of
d
×
old
x
i

old ¯
x
old
sd
x
= sign of
d
×
old
x
i
in s.u.
Values in s.u. don’t change, correlation value doesn’t change, but
sign of correlation
might
20 / 29
Today’s Goals
Summary Statistics
Association
Correlation
Properties
Affect of Outliers
Outliers ruin a linear pattern, making correlation drop
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
2
1
0
1
2
0
5
10
Correlation = 0.95
x
y
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
2
1
0
1
2
5
0
5
10
Correlation = 0.69
x
y
21 / 29
Today’s Goals
Summary Statistics
Association
Correlation
Properties
Ecological Correlation
Calculating a correlation between group averages (instead of raw
data) is called an ecological correlation
400
500
600
700
800
900
200
400
600
800
Correlation = 0.631
Verbal Score
Math Score
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●