Psych 100A Week 8 Discussion Notes
Dawn Chen
November 20, 2009
Correlation
When it is impractical or unethical to manipulate one variable (e.g., gender, level of
smoking) to see if it causes a change in another variable, we cannot determine if there is a
causal
relationship between the two variables.
However, we can still measure the two variables and
determine whether they are simply
related
by calculating the correlation coefficient between
them.
The sample correlation coefficient
, denoted by
r
, tells us the direction and strength of the
linear
relationship between two variables
X
and
Y
in a sample of (
x
,
y
) values
1
.
r
can take values
in the range [–1, 1].
Below are some possible (though sometimes unlikely) scenarios for the
sample data and their corresponding
r
values.
Note that
r
can be 0 even if there is a relationship
between
X
and
Y
, if that relationship does not have a linear component (as in scenario (d)).
Hypothesis testing also applies to
r
.
In this case, we usually want to know whether
r
is
significantly different from 0, or in other words, whether there exists a linear relationship
between
X
and
Y
.
This hypothesis is expressed in terms of
,
ρ
the population correlation
coefficient.
0
:
0 and
:
0.
A
H
H
=
≠
The equation for
r
given in lecture was:
( 29
( 29
1
1
1
n
i
i
i
x
y
x
x
y
y
r
n
s s
=


=

∑
Doing some algebraic manipulations, we have:
( 29
( 29
( 29
( 29
1
1
1
1
1
1
n
i
i
n
i
i
i
i
x
y
x
y
x
x
y
y
x
x
y
y
r
n
s s
n
s s
=
=




=
=


∑
∑
( 29
( 29
( 29
( 29
( 29
( 29
( 29
( 29
1
1
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
n
n
i
i
i
i
i
i
n
n
n
n
i
i
i
i
i
i
i
i
x
x
y
y
x
x
y
y
n
n
x
x
y
y
x
x
y
y
n
n
n
=
=
=
=
=
=




=
=









∑
∑
∑
∑
∑
∑
1
In these notes, an uppercase letter (such as
X
) denotes a variable and the corresponding lowercase letter (such as
x
)
denotes a particular value of the variable.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(a)
r
= 1
(b)
r
= –1
(c)
r
= 0
(d)
r
= 0
(e)
r
= 0.8
.
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( 29
( 29
( 29
( 29( 29
1
2
2
1
1
n
i
i
XY
i
n
n
X
Y
i
i
i
i
x
x
y
y
CP
SS
SS
x
x
y
y
=
=
=


=
=


∑
∑
∑
In the last formula,
( 29
( 29
1
n
XY
i
i
i
CP
x
x
y
y
=
=


∑
is called the cross products of
X
and
Y
,
( 29
2
1
n
X
i
i
SS
x
x
=
=

∑
is the sum of squares for
X
, and similarly,
( 29
2
1
n
Y
i
i
SS
y
y
=
=

∑
is the sum of
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 Spring '10
 Chen
 Statistics, Correlation, Regression Analysis, Quadratic equation, daytime energy levels

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