Correlation Coefficients

Correlations have two properties: strength and direction. The strength of a correlation is determined by its numerical (absolute) value. The direction of the correlation is determined by sign of the correlation coefficient ‘r’, whether the correlation is positive or negative.

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Correlation standardizes the measure of interdependence between two variables and, consequently, tells you how closely the two variables move. A correlation coefficient is the covariance divided by the product of each variable’s standard deviation.

The correlation measurement, i.e. correlation coefficient, will always take on a value between 1 and – 1:

• If the correlation coefficient is one, the variables have a perfect positive correlation. This means that if one variable moves a given amount, the second moves proportionally in the same direction. A positive correlation coefficient less than one indicates a less than perfect positive correlation, with the strength of the correlation growing as the number approaches one.
• If correlation coefficient is zero, no relationship exists between the variables. If one variable moves, you can make no predictions about the movement of the other variable; they are uncorrelated.
• If correlation coefficient is –1, the variables are perfectly negatively correlated (or inversely correlated) and move in opposition to each other. If one variable increases, the other variable decreases proportionally.

A negative correlation coefficient greater than –1 indicates a less than perfect negative correlation with the strength of the correlation growing as the number approaches –1.

There are two types of correlation: bivariate and partial. A bivariate correlation is a correlation between two variables whereas a partial correlation looks at the relationship between two variables while ‘controlling’ the effect of one or more additional variables.

Pearson’s product moment correlation coefficient (r): evaluates the linear relationship between two continuous variables. A relationship is linear when a change in one variable is associated with a proportional change in the other variable. Pearson correlation is a parametric statistic and requires interval data for both variables. To test its significance we assume normality of both the variables. For example, you might use a Pearson correlation to evaluate whether increases in temperature at your production facility are associated with decreasing thickness of your chocolate coating.

Spearman’s rank-order correlation coefficient (ρ): Also called Spearman’s rho, the Spearman correlation evaluates the monotonic relationship between two continuous or ordinal variables. In a monotonic relationship, the variables tend to change together, but not necessarily at a constant rate. The Spearman correlation coefficient, a non-parametric statistic, is based on the ranked values (ordinal) for each variable rather than the raw data. Spearman correlation is often used to evaluate relationships involving ordinal variables. For example, you might use a Spearman correlation to evaluate whether the order in which employees complete a test exercise is related to the number of months they have been employed.

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Kendall’s correlation coefficient, tau (τ): non-parametric statistic like Spearman’s rs but probably better for small samples.