Wolfram Alpha:

Pearson Correlation
-------------------

Linear regression is similar to correlation in that the purpose
is to measure to what extent there is a linear relationship
between two variables. The major difference between the two
is that correlation makes no distinction between independent
and dependent variables while linear regression does.

ρ = population correlation coefficient.

ρ = (1/N)Σ[(xi - μx)/σx][(yi - μy)/σy]

Pearson r = sample correlation coefficient.
_            _
r = (1/(n - 1))Σ[(xi - x)/sx][(yi - y)/sy]

SSxy
= -------------------------------------
(n - 1)√[(SSxx/(n - 1)]√[SSyy/(n - 1)]

SSxy
= ----------
√(SSxxSSyy)

Alternatively,

r = √(SSM/Total SS)

Where:
_       _
SSxy = Σ(xi - x)(yi - y)
_
SSxx = Σ(xi - x)2
_
SSyy = Σ(yi - y)2

Total SS = Model SS + Error SS

= SSM + SSE
_            _
Σ(yi - y)2 = Σ(yp - y)2 + Σ(yi - yp)2
_
= Σ(yp - y)2 + Σ(yi - (βxi + α))2

Coefficient of Determination:

r2  = Proportion of the variance of the dependent variable (y)
that is explained by the independent variable (x).

= (SSyy - SSE)/SSyy

Hypotheses:

If there is a significant correlation between the variables,
r will not equal zero.

H0: ρ = 0
H1: ρ ≠ 0

Test statistic:  t test = r√[(n - 2)/(1 - r2)]

= β/sβ

=  β/(s/√SSxx)

The test has n - 2 df.

General rule of thumb:

ρ:  0.10 - 0.29 small correlation
0.30 - 0.49 medium correlation
0.50 - 1.0 strong correlation

A simple example:
_         _         _       _
xi  yi  (xi - x)2 (yi - y)2 (xi - x)(yi - y)
-  ---  -------  -------   --------------
1  1.9   2.25     0.8556       1.3875
2  2.3   0.25     0.2756       0.2625
3  2.8   0.25     0.0006      -0.0125
4  4.3   2.25     2.1756       2.2125
-  ---   ----     ------       ------
Σ          5.00     3.3075       3.8500
_
x = 2.5000
_
y = 2.8250
SSxx = 5.0000
SSxy = 3.8500
SSyy = 3.3075
β = SSxy/SSxy = 3.8500/5.0000 = 0.7700
SSE = SSyy - βSSxy = 3.3075 - 0.7700*3.8500 = 0.3430
= Σ(yi - yp)2 = 0.3430
_
yi    yp  (yi - yp)2  (yp - y)2
---  ----  ------     ------
1.9  1.67  0.0529     1.3340
2.3  2.44  0.0196     0.1482
2.8  3.21  0.1681     0.1482
4.3  3.98  0.1024     1.3340
------     ------
Σ            0.3430     2.9645

Total SS = SSM + SSE
_            _
Σ(yi - y)2 = Σ(yp - y)2 + Σ(yi - yp)2

3.3075 = 2.9645 + 0.3430

Coefficient of Correlation:

r = √(SSM/Total SS)

= √(2.9645/3.3075)

= 0.9466

= SSxy/√(SSxxSSyy)

= 3.8500/√(5.0000*3.3075)

= 0.9466

Coefficient of Determination:

r2 = (SSyy - SSE)/SSyy

= (3.3075 - 0.3430)/3.3075

= 0.8963

s2 = SSE/(n - 2) = 0.3430/2 = 0.1715 ∴ s = 0.4141

t = β/(s/√SSxx) = 0.77/(0.4141/√5.0000) = 4.158

t test = r√[(n - 2)/(1 - r2)]

= 0.9466√[2/(1 - 0.8962)]

= 4.155

t0.05/2 for 4 - 2 = 2 df is 4.303 (2-tailed). Therefore,
reject H0.

SPSS:  Analyze>Correlate>Bivariate

Multiple Correlation
--------------------

Because we now have multiple variables, we have more than one
correlation value to consider => partial correlation coefficient.
PCC is the measure of the correlation between the dependent
variable and one independent variable holding constant the
influence of all other independent variables.

Correlation matrix:

x1  x2  x3  x4
x2  1   ?   ?
x3  ?   1   ?
x4  ?   ?   1

Test statistic:  One-sample t test = r√[(n - 2)/(1 - r2)]

SPSS:  Analyze>Correlate>Bivariate