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# Correlation (Mathematics)

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Correlation refers to the degree of correspondence or relationship between two variables. Correlated variables tend to change together. If one variable gets larger, the other one tends (though not necessarily all the time) to become either larger or smaller. For example, we would expect to find such a relationship between scores on an arithmetic test taken three months apart. We could expect high scores on the first test to go with (predict) high scores on the second test, and low scores on the first test to predict low scores on the second test. Yet for some students, this relationship might not hold. Correlation is therefore a statistical property.

In the above example the scores on the first test are known as the independent or predictor variable (designated X) while the scores on the second test are known as the dependent or response variable (designated Y). The relationship between the two variables X and Y is a positive relationship or positive correlation when high measures of X correspond with high measures of Y and low measures of X with low measures of Y. It is also possible for the relationship between variables X and Y to be an inverse relationship or negative correlation. This occurs when high measures of variable X are associated with low measures of variable Y and low measures on variable X are associated with high measures of variable Y. For example, if variable X is school attendance and variable Y is the score on an achievement test we could expect a negative correlation between X and Y. High measures of X (absence) would be associated with low measures of Y (achievement) and low measures of X with high measures of Y.

The correlation coefficient tells us that a relationship exists. The + or - sign indicates the direction of the relationship while the number indicates the

### KEY TERMS

Correlation coefficient The numerical index of a relationship between two variables.

Negative correlation The changes in one variable are reflected by inverse changes in the second variable.

Positive correlation The changes in one variable are reflected by similar changes in the second variable.

magnitude of the relationship. This relationship should not be interpreted as a causal relationship. Variable X is related to variable Y, and may indeed be a good predictor of variable Y, but variable X does not cause variable Y although this is sometimes assumed. For example, there may be a positive correlation between head size and IQ or shoe size and IQ. Yet no one would say that the size of ones head or shoe size causes variations in intelligence. However, when two more likely variables show a positive or negative correlation, many interpret the change in the second variable to have been caused by the first.

## Resources

### BOOKS

Barbaoianu, Catalin. Understanding and Calculating the Odds: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Life. Upper Saddle River, NJ: Infarom, 2006.

Kumar, K. Vijaya, et al. Correlation Pattern Recognition. Cambridge, UK: Cambridge University Press, 2006.

Ross, Sheldon. A First Course in Probability. 7th ed. Upper Saddle River, NJ: Prentice Hall, 2005.

Ross, Sheldon. Introduction to Probability and Statistics for Engineers and Scientists. 3rd ed. San Diego, CA: Academic Press, 2004.

Selma Hughes