Linear Regression and Correlation: Facts About the Correlation Coefficient for Linear Regression1.52008/06/23 16:21:17 GMT-52008/07/16 12:18:30.861 GMT-5BarbaraIllowskyillowskybarbara@deanza.eduSusanDeandeansusan@deanza.eduConnexionscnx@cnx.orgelementarystatisticsThis module provides an overview of Facts About the Correlation Coefficient for Linear Regression as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.A positive r means that when x increases, y increases and when x decreases, y decreases (positive correlation).A negative r means that when x increases, y decreases and when x decreases, y increases (negative correlation).An r of zero means there is absolutely no linear relationship between x and y(no correlation).High 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!Positive Correlation
A scatter plot showing data with a positive correlation.
Negative Correlation
A scatter plot showing data with a negative correlation.
Zero Correlation
A scatter plot showing data with zero correlation.
The 95% Critical Values of the Sample Correlation Coefficient Table at the end of
this chapter (before the Summary) may be used to give you a good idea of whether the
computed value of r is significant or not. Compare r to the appropriate critical value in
the table. If r is significant, then you may want to use the line for prediction.Suppose you computed r=0.801 using n=10 data points.
df=n-2=10 -2=8. The critical values associated with df=8 are -0.632 and
+ 0.632. If rnegative critical value or r>positive critical value, then r is
significant. Since r=0.801 and 0.801>0.632, r is significant and the line may be used
for prediction. If you view this example on a number line, it will help you.
r is not significant between -0.632 and +0.632. r=0.801>+0.632. Therefore, r is significant.
Suppose you computed r=-0.624 with 14 data points. df=14-2=12. The critical values are -0.532 and 0.532. Since -0.624-0.532, r is significant and
the line may be used for prediction
r=-0.624-0.532. Therefore, r is significant.
Suppose you computed r=0.776 and n=6. df=6-2=4. The
critical values are -0.811 and 0.811. Since -0.8110.7760.811, r is not significant
and the line should not be used for prediction.
-0.811r=0.7760.811. Therefore, r is not significant.
If r is -1 or r is +1, then all the data points lie exactly on a straight line. If the line is significant, then within the range of the x-values, the line can be used to predict a y value. As an illustration, consider the third exam/final exam example.
The line of best fit is: y^=-173.51+4.83x with
r=0.6631Can the line be used for prediction? Given a third exam score (x value), can we
successfully predict the final exam score (predicted y value). Test r=0.6631
with its appropriate critical value.Using the table with df=11-2=9, the critical values are -0.602 and +0.602. Since
0.6631>0.602, r is significant. Because r is significant and the scatter plot shows a reasonable linear trend, the line can be used to predict final exam scores.
Suppose you computed the following correlation coefficients. Using the
table at the end of the chapter, determine if r is significant and the line of best fit associated
with each r can be used to predict a y value. If it helps, draw a number line.
r=-0.567 and the sample size, n, is 19. The df=n-2=17. The critical value is -0.456. -0.567-0.456 so r is significant.r=0.708 and the sample size, n, is 9. The df=n-2=7. The critical value is 0.666. 0.708>0.666 so r is significant.r=0.134 and the sample size, n, is 14. The df=14-2=12. The critical value is 0.532. 0.134 is between -0.532 and 0.532 so r is not significant.r=0 and the sample size, n, is 5. No matter what the dfs are, r=0 is between the two critical values so r is not significant.