Which of the following statements accurately describes a correlation coefficient?

Question

(3) No matter how close to zero the coefficient may be. SOMETIMES there is no significance to the correlation between the variables.

OR

(4) The more neg. the coefficient become, the linear relationship between the two variables ALWAYS becomes stronger.

Answer 1

I'd like to see the other answers.  

 

Okay, I can't reject #3 because it's true as stated, but there's really no relationship between the clauses of the sentence.  A coefficient close to zero means that the MAGNITUDE of the relationship is low.  That could be ALWAYS present (highly significant), or NEVER present (not significant).  

 

Take, the total weight of a battleship compared to the number of personnel on board.  It doesn't go up MUCH, but it goes up pretty consistently every time another person gets on or off.  SO, small coefficient, but high significance.

 

Likewise, there could be a situation with a small coefficient but no real relationship at all - the weight of a battleship and the number of ships in port with it.  So SOMETIMES is correct.

 

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The next one is closer to being relevant, since high coefficients (positive or negative) indicate a strong numerical relationship.  However, once again, the two readings could have a high correlation, but not be SIGNIFICANT considering the data available.  For example, the weight of the battleship in Puerto Rico, and the weight of the same battleship six months later in Guam.  If there are only two data points, you will undoubtedly have a 100% correlation coefficient, and in one direction or the other it will be negative, but it has no real significance.  

 

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Between the two, number 3 is a reasonable statement.  It's only the idea of getting closer to zero having anything to do with the significance that bothers me.  Closer to zero is only important when you are testing probability, not the correlation coefficient.