# Question about correlation in statistics

A Feb 2008, how would the correlation change reported a study on the distance students lived from campus and average GPA. Here is a summary of the result.

Residence / Avg. GPA

Residence hall / 3.33

Walking distance / 3.16

Near campus, long walk or short drive / 3.12

Within the county not near campus / 2.97

Outside the county / 2.94

Q1: Base on these data is the association between the distance a student lives from campus and average GPA positive, negative or near 0?

Q2: Would the association be stronger, weaker, or the same if the data given listed the GPA of individual students (rather than averages) and the distance they lived from campus?

My answer for the Q1 is negative because if the distance a student lived shorter, the GPA gets higher. However, I have no idea for Q2. Anybody can help me? Thanks!

## 1 Answer

Your answer in Q1 is correct based on the data. However, based on this link (or others that you may find about ecological correlation or ecological fallacy)

Ecological correlation is the correlation coefficient calculated for averages of individuals, rather than for individuals. Ecological correlations say little about the (linear) association for individuals; generally, ecological correlations tend to overstate the strength of the association for individuals.

the conclusion of Q1 is not as strong as it seems.

So the answer in Q2 is that the association would be stronger if these scores referred to individual students rather than averages. Still, assuming you had only this data and this referred to individuals, again your conclusion would be not so strong because a sample of $5$ is certainly not enough to draw safe conclusions about a population as big as the students of a university. If you had the individual measurements from which the averages came and these measurements were plenty enough then that would be the best case (concerning the strength of your conclusions).