Describe correlation and association

Detailed Explanation: Describe correlation and association

Association describes how two variables tend to vary together. For two quantitative variables, use a scatterplot and, when appropriate, the correlation coefficient $r$.

• $r$ is between $-1$ and $1$.
• $r>0$: positive association; larger values of one variable tend to go with larger values of the other.
• $r<0$: negative association; larger values of one tend to go with smaller values of the other.
• $\vert r \vert$ close to $1$: strong linear association.
• $\vert r \vert$ close to $0$: weak or no linear association.

Worked example

A teacher records how many hours six students studied and their test scores.

Step 1: Identify the variables

• Explanatory variable: hours studied
• Response variable: test score

Both variables are quantitative, so a scatterplot and correlation are appropriate.

Step 2: Examine the scatterplot

Plot hours studied on the horizontal axis and test score on the vertical axis. The points generally rise from left to right and lie close to a straight line.

This suggests:

• Direction: positive
• Form: approximately linear
• Strength: strong
• Outliers: no obvious unusual points

Step 3: Find the correlation

Using a calculator, the correlation is approximately

Because $r$ is positive and very close to $1$, there is a strong positive linear association between hours studied and test score.

Step 4: State the conclusion carefully

Students who studied more hours tended to have higher test scores in this group. However, this association does not prove that studying caused the higher scores. Other factors, such as prior knowledge or access to help, could also affect test performance.

When describing any association, check the scatterplot first. Look for its direction, strength, shape, and possible outliers. Also remember that correlation measures only a linear relationship: a curved relationship may have a correlation near $0$ even when the variables are clearly related.