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10.2: The Linear Correlation Coefficient

  • Page ID
    543
  • skills to develop

    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 Strengths

    Definition: 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 \(\PageIndex{2}\)

    1. The value of \(r\) lies between \(−1\) and \(1\), inclusive.
    2. The sign of \(r\) indicates the direction of the linear relationship between \(x\) and \(y\):
    3. The size of \(|r|\) indicates the strength of the linear relationship between \(x\) and \(y\):
      1. If \(|r|\) is near \(1\) (that is, if \(r\) is near either \(1\) or \(−1\)), then the linear relationship between \(x\) and \(y\) is strong.
      2. If \(|r|\) is near \(0\) (that is, if \(r\) is near \(0\) and of either sign). then the linear relationship between \(x\) and \(y\) is weak.

    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

      \(x\) \(y\) \(x^2\) \(y^2\)
    68 151 4624 10268 22801
    69 146 4761 10074 21316
    70 157 4900 10990 24649
    70 164 4900 11480 26896
    71 171 5041 12141 29241
    72 160 5184 11520 25600
    72 163 5184 11736 26569
    72 180 5184 12960 32400
    73 170 5329 12410 28900
    73 175 5329 12775 30625
    74 178 5476 13172 31684
    75 188 5625 14100 35344
    859 2003 61537 143626 336025

    Key Takeaway

    • The linear correlation coefficient measures the strength and direction of the linear relationship between two variables \(x\) and \(y\).
    • The sign of the linear correlation coefficient indicates the direction of the linear relationship between \(x\) and \(y\).
    • When \(r\) is near \(1\) or \(−1\) the linear relationship is strong; when it is near \(0\) the linear relationship is weak.

    Contributor

    • Anonymous