Statistics for Environmental Engineers

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30.3 Metal Inhibition. Solve Exercise 27.5 using regression.

31

Correlation


KEY WORDS BOD, COD, correlation, correlation coefficient, covariance, nonparametric correlation, Pearson product-moment correlation coefficient, R2, regression, serial correlation, Spearman rank correlation coefficient, taste, chlorine.


Two variables have been measured and a plot of the data suggests that there is a linear relationship between them. A statistic that quantifies the strength of the linear relationship between the two variables is the correlation coefficient.


Care must be taken lest correlation is confused with causation. Correlation may, but does not necessarily, indicate causation. Observing that у increases when x increases does not mean that a change in x causes the increase in у. Both x and у may change as a result of change in a third variable, z.

Covariance and Correlation


A measure of the linear dependence between two variables x and у is the covariance between x and у. The sample covariance of x and у is:


N-


where px and ру are the population means of the variables x and у, and N is the size of the population. If x and у are independent, Cov(x, у) would be zero. Note that the converse is not true. Finding Cov(x, у) = 0 does not mean they are independent. (They might be related by a quadratic or exponential function.)


The covariance is dependent on the scales chosen. Suppose that x and у are distances measured in inches. If x is converted from inches to feet, the covariance would be divided by 12. If both x and у are converted to feet, the covariance would be divided by 122 = 144. This makes it impossible in practice to know whether a value of covariance is large, which would indicate a strong linear relation between two variables, or small, which would indicate a weak association.

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