If y is usually less than x, the correlation coefficient between x and y will be negative. True or false and explain

Question

(A)If y is usually less than x the correlation coefficient between x and y will be negative. True or false. And explain (b). If the correlation coefficient is -0.80, below average values of the dependent variable Are associated with below average value of the independent variable. True or false and explain.

Answer 1

Peace,

 

Look at how you define the correlation coefficient in relationship to the covariance of X & Y, and look at how it is calculated.  For a sample and using lower case x and y for the individual values of x and y for each data point in the sample, capital X as the mean of X and capital Y as the mean of Y and s as the sample standard deviations, r can be calculated as

 

r = (1/(n-1)) Σ (x_i - X)(y_i - Y)]/s_xs_y

 

So for each set of data points x,y a positive amount is added to r if both x and y are either both greater or both smaller than their respective means.  If both are larger x-X is positive and y-Y is positive and thus their product must be positive.  Similarly, if both are smaller x-X is negative and y-Y is negative and thus their product is also positive.  Conversely, if one is positive and one is negative, i.e. one is larger than its mean and the other is smaller, then their product must be negative.

 

With this fact in mind, lets look at your questions:

 

A)If y is usually less than x the correlation coefficient between x and y will be negative. True or false. And explain.

 

 

Looking at the expressions, the size of x relative to y does not matter, it is their respective distances from their respective means that result in a positive and negative r.  Imagine if this was not the case, suppose I am measuring the correlation between x, lets say weight (in kilograms) and y, lets say height (in meters).  In kilograms, most of our sample is between 50 and 150 kg (~100-300 lbs), but their heights range from 1.5 and 2.0 meters (4.5 to 7 feet).  Measured in kg and m every y is smaller than every x, but if I converted y to centimeters, most of the y's would be larger (150-200 cm).  Rather regardless of units, I would still get the same r because each respective X and Y would still have the same relative relationship to their respective means.

 

(b). If the correlation coefficient is -0.80, below average values of the dependent variable are associated with below average values of the independent variable.

 

This asks the same as (a) but only in reverse.  What has to happen for any individual value of the summation to calculate r to be negative?  The x and y values have to be on opposite sides of their respective means, or put another way, if x is larger than its mean, then y has to be smaller or vice versa.  So if a below average value of the dependent variable (i.e. y minus its mean is negative) is associated with a below average value of the independent variable (i.e. x minus its mean is negative), we would have a negative number times a negative number or a positive contribution to r, not a negative contribution!

 

I hope this helps you answer your questions.