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The Correlation Coefficient r

26 七月, 2019 - 12:02
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Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Use the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship between x and y.

The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is a numerical measure of the strength of association between the independent variable x and the dependent variable y.
The correlation coefficient is calculated as

r=\frac{n\cdot \sum x\cdot y-(\sum x)\cdot(\sum y)}{\sqrt{[n\cdot \sum x^{2}-(\sum x)^{2}]\cdot[n\cdot\sum y^{2}-(\sum y)^{2}]}}

where n = the number of data points.

If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.

What the VALUE of r tells us:

  • The value of r is always between -1 and +1: −1 ≤ r ≤ 1.
  • The size of the correlation r indicates the strength of the linear relationship between x and y. Values of r close to -1 or to +1 indicate a stronger linear relationship between x and y.
  • If r = 0 there is absolutely no linear relationship between x and y (no linear correlation).
  • If r = 1, there is perfect positive correlation. If r = −1, there is perfect negative correlation. In both these cases, all of the original data points lie on a straight line. Of course, in the real world, this will not generally happen.

What the SIGN of r tells us

  • A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease (positive correlation).
  • A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation).
  • The sign of r is the same as the sign of the slope, b, of the best fit line.
Note: Strong correlation does not suggest that x causes y or y causes x. We say "correlation does not imply causation." For example, every person who learned math in the 17th century is dead. However, learning math does not necessarily cause death!
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Figure 6.11 (a) A scatter plot showing data with a positive correlation. 0 <r< 1 (b) A scatter plot showing data with a negative correlation. −1 <r< 0  
 
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Figure 6.12 (C) A scatter plot showing data with zero correlation. r=0  

The formula for r looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate r. The correlation coefficient r is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).

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