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

The correlation coefficient is calculated as

r = n Σ ( x y ) ( Σ x ) ( Σ y ) [ n Σ x 2 ( Σ x ) 2 ] [ n Σ y 2 ( Σ 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 negativecorrelation. 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."
Three scatter plots with lines of best fit. The first scatterplot shows points ascending from the lower left to the upper right. The line of best fit has positive slope. The second scatter plot shows points descending from the upper left to the lower right. The line of best fit has negative slope. The third scatter plot of points form a horizontal pattern. The line of best fit is a horizontal line.
(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 (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).

The coefficient of determination

The variable r 2 is called the coefficient of determination and is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. It has an interpretation in the context of the data:

  • r 2 , when expressed as a percent, represents the percent of variation in the dependent (predicted) variable y that can be explained by variation in the independent (explanatory) variable x using the regression (best-fit) line.
  • 1 – r 2 , when expressed as a percentage, represents the percent of variation in y that is NOT explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.

Consider the third exam/final exam example introduced in the previous section

  • The line of best fit is: ŷ = –173.51 + 4.83x
  • The correlation coefficient is r = 0.6631
  • The coefficient of determination is r 2 = 0.6631 2 = 0.4397
  • Interpretation of r 2 in the context of this example:
  • Approximately 44% of the variation (0.4397 is approximately 0.44) in the final-exam grades can be explained by the variation in the grades on the third exam, using the best-fit regression line.
  • Therefore, approximately 56% of the variation (1 – 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best-fit regression line. (This is seen as the scattering of the points about the line.)

Chapter review

A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the x and y variables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also called “errors,” measure the distance from the actual value of y and the estimated value of y . The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. Regression lines can be used to predict values within the given set of data, but should not be used to make predictions for values outside the set of data.

The correlation coefficient r measures the strength of the linear association between x and y . The variable r has to be between –1 and +1. When r is positive, the x and y will tend to increase and decrease together. When r is negative, x will increase and y will decrease, or the opposite, x will decrease and y will increase. The coefficient of determination r 2 , is equal to the square of the correlation coefficient. When expressed as a percent, r 2 represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression line.

Use the following information to answer the next five exercises . A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).

x y x y
0 2 5 12
3 8 4 9
2 7 3 9
1 3 0 3
5 13 4 10

Draw a scatter plot of the data.

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Use regression to find the equation for the line of best fit.

ŷ = 2.23 + 1.99 x

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Draw the line of best fit on the scatter plot.

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What is the slope of the line of best fit? What does it represent?

The slope is 1.99 ( b = 1.99). It means that for every endorsement deal a professional player gets, he gets an average of another $1.99 million in pay each year.

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What is the y -intercept of the line of best fit? What does it represent?

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What does an r value of zero mean?

It means that there is no correlation between the data sets.

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When n = 2 and r = 1, are the data significant? Explain.

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When n = 100 and r = -0.89, is there a significant correlation? Explain.

Yes, there are enough data points and the value of r is strong enough to show that there is a strong negative correlation between the data sets.

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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