Statistics Dictionary
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Pearson ProductMoment Correlation
Correlation coefficients measure the strength of
association between two variables. The most common correlation
coefficient, called the
Pearson productmoment correlation coefficient,
measures the strength of the
linear association between variables.
The sign and the
absolute value
of a Pearson correlation coefficient
describe the direction and the magnitude of the relationship
between two variables.

The value of a correlation coefficient ranges between 1 and
1.

The greater the absolute value of a correlation coefficient,
the stronger the linear relationship.

The strongest linear relationship is indicated by a correlation
coefficient of 1 or 1.

The weakest linear relationship is indicated by a correlation
coefficient equal to 0.

A positive correlation means that if one variable gets bigger,
the other variable tends to get bigger.

A negative correlation means that if one variable gets bigger,
the other variable tends to get smaller.
Keep in mind that the Pearson correlation coefficient only measures
linear relationships. Therefore, a correlation of 0 does not
mean zero relationship between two variables; rather, it means
zero linear relationship. (It is possible for two
variables to have zero linear relationship and a strong
curvilinear relationship at the same time.)
A formula for computing a Pearson correlation coefficient is given below.
Correlation coefficient.
The correlation r between two variables is:
r = Σ (xy) / sqrt [ ( Σ x^{2} ) * ( Σ y^{2} ) ]
where Σ is the summation symbol,
x = x
_{i} 
x,
x
_{i} is the x value for observation i,
x is the mean x value,
y = y
_{i} 
y,
y
_{i} is the y value for observation i,
and
y is the mean y value.
Fortunately, you will rarely have to compute a correlation
coefficient by hand. Many software packages (e.g., Excel) and most
graphing calculators
have a correlation function that will do the job for you.