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Correlation coefficients measure the strength of
association between two variables. The most common correlation
coefficient, called the
Pearson product-moment correlation coefficient,
measures the strength of the
linear association between variables.
The sign and the
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
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.
The correlation r between two variables is:
r = Σ (xy) / sqrt [ ( Σ x2 ) * ( Σ y2 ) ]
where Σ is the summation symbol,
x = xi
is the x value for observation i,
is the mean x value,
y = yi
is the y value for observation i,
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
have a correlation function that will do the job for you.