Interpret correlation coefficients

Detailed Explanation: Interpret correlation coefficients

The correlation coefficient, written as $r$, describes the direction and strength of a linear relationship between two quantitative variables.

• The value of $r$ is always between $(-1)$ and $1$.
• The sign gives the direction:
  • $(r>0)$: the variables tend to increase together.
  • $(r<0)$: as one variable increases, the other tends to decrease.
• The size of $(|r|)$ gives the strength:
  • $(|r|)$ close to $1$: strong linear association.
  • $(|r|)$ close to $0$: weak or no linear association.

Worked example

A study records the number of hours students exercise per week and their resting heart rates. The correlation coefficient is

Interpret this value.

Step 1: Use the sign.

Since $r$ is negative, the variables have a negative linear association. Students who exercise more hours per week tend to have lower resting heart rates.

Step 2: Use the absolute value.

The strength is determined by

Because $(0.82)$ is fairly close to $1$, the association is strong.

Step 3: State the interpretation in context.

There is a strong negative linear association between weekly exercise time and resting heart rate among the students studied. In general, students who exercise more tend to have lower resting heart rates, and the points would be expected to lie fairly close to a downward-sloping line.

Step 4: Check the scatterplot and context.

Before finalizing the interpretation, look for unusual points or outliers. A single outlier could make the correlation appear stronger or weaker than the overall pattern. Also, this correlation does not prove that exercising more causes a lower resting heart rate. Other factors, such as age, fitness level, or health, may also be involved.

Remember that $(r=0)$ means there is no linear association; it does not rule out every possible relationship between the variables.