# Correlation Analysis

Correlation analysis, expressed by correlation coefficients, measures the degree of linear relationship between two variables.

While in regression the emphasis is on predicting one variable from the other, in correlation the emphasis is on the degree to which a linear model may describe the relationship between two variables.

The correlation coefficient may take on any value between + and - 1. The sign of the correlation coefficient (+, -) defines the direction of the relationship, either positive or negative. A positive correlation coefficient means that as the value of one variable increases, the value of the other variable increases; as one decreases the other decreases. A negative correlation coefficient indicates that as one variable increases, the other decreases, and vice-versa.

The absolute value of the correlation coefficient measures the strength of the relationship. A correlation coefficient of r=0.50 indicates a stronger degree of linear relationship than one of r=0.40. Thus a correlation coefficient of zero (r=0.0) indicates the absence of a linear relationship and correlation coefficients of r=+1.0 and r=-1.0 indicate a perfect linear relationship.

The scatter plots presented below perhaps best illustrate how the correlation coefficient changes as the linear relationship between the two variables is altered. When r=0.0 the points scatter widely about the plot, the majority fall roughly in the shape of a circle. As the linear relationship increases, the circle becomes more and more elliptical in shape until the limiting case is reached (r=1.00 or r=-1.00) and all the points fall on a straight line.

A number of scatter plots and their associated correlation coefficients are presented below:

Correlation analysis is typically used for customer satisfaction and employee satisfaction studies to answer questions such as "which elements contribute most to someone's overall satisfaction or loyalty?" This can lead to a "derived importance versus satisfaction" map. See below.

It is also ideal when sample sizes are too low (e.g. less than 100) to run a regression analysis.

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