# Statistics Dictionary

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### Correlation

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 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 x2 ) * ( Σ y2 ) ]

where Σ is the summation symbol, x = xi - x, xi is the x value for observation i, x is the mean x value, y = yi - y, yi 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.