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Learning Objectives To learn what the linear correlation coefficient is, how to compute it, and what it tells us about the relationship between two variables \(x\) and \(y\) Figure \(\PageIndex{1}\) illustrates linear relationships between two variables \(x\) and \(y\) of varying strengths. It is visually apparent that in the situation in panel (a), \(x\) could serve as a useful predictor of \(y\), it would be less useful in the situation illustrated in panel (b), and in the situation of panel (c) the linear relationship is so weak as to be practically nonexistent. The linear correlation coefficient is a number computed directly from the data that measures the strength of the linear relationship between the two variables \(x\) and \(y\). Figure \(\PageIndex{1}\): Linear Relationships of Varying StrengthsDefinition: linear correlation coefficient The linear correlation coefficient for a collection of \(n\) pairs \(x\) of numbers in a sample is the number \(r\) given by the formula The linear correlation coefficient has the following properties, illustrated in Figure \(\PageIndex{2}\)
so that \[ r= \dfrac{SS_{xy}}{\sqrt{SS_{xx}SS_{yy}}}=\dfrac{2.44.583}{\sqrt{(46.916)(1690.916)}}=0.868\] The number quantifies what is visually apparent from Figure \(\PageIndex{2}\) weights tends to increase linearly with height (\(r\) is positive) and although the relationship is not perfect, it is reasonably strong (\(r\) is near \(1\)). Figure \(\PageIndex{2}\): Linear Correlation Coefficient \(r\). Pay particular attention to panel (f), which shows a perfectly deterministic relationship between \(x\) and \(y\), but \(f=0\) because the relationship is not linear. (In this particular case the points lie on the top half of a circle.)
Example \(\PageIndex{1}\) Compute the linear correlation coefficient for the height and weight pairs plotted in Figure \(\PageIndex{2}\). Solution: Even for small data sets like this one computations are too long to do completely by hand. In actual practice the data are entered into a calculator or computer and a statistics program is used. In order to clarify the meaning of the formulas we will display the data and related quantities in tabular form. For each
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