18.12 Correlation

Describing correlation

To help us identify any correlation between the two variables, there are three things we focus on when analysing a scatter diagram:

• Direction: positive or negative
• Form: linear or non-linear
• Strength: strong, moderate, weak

Direction

The direction of the scatter diagram refers to the pattern shown by the data points. We can describe the direction of the pattern as having positive correlation, negative correlation or no correlation:

• Positive correlation
  • From a graphical perspective, this occurs when the $y$y-coordinate increases as the $x$x-coordinate increases, which is similar to a line with a positive gradient.
• Negative correlation
  • From a graphical perspective, this occurs when the $y$y-coordinate decreases as the $x$x-coordinate increases, which is similar to a line with a negative gradient.
• No correlation
  • No correlation describes a data set that has no relationship between the variables.

Form

When we are looking at the form of a scatter diagram we are looking to see if the data points show a pattern that has a linear form. If the data points lie on or close to a straight line, we can say the scatterplot has a linear form. 

Forms other than a line may be apparent in a scatter diagram. If the data points lie on or close to a curve, it may be appropriate to infer a non-linear form between the variables. 

Strength

The strength of a linear correlation relates to how closely the points reassemble a straight line. 

• If the points lie exactly on a straight line then we can say that there is a perfect correlation. 
• If the points are scattered randomly then we can say there is no correlation.

Most scatter diagrams will fall somewhere in between these two extremes and will display a weak, moderate or strong correlation. 

Worked example

Example 1

Identify the type of correlation in the following scatter diagram.

Do: The line fits quite closely to all of the points, so it is a strong correlation. In summary, we would say that this scatter diagram indicates a strong, positive correlation.

Practice questions

Question 1

Question 2