In statistics, a monotonic relationship between two variables refers to a scenario where a change in one variable is generally associated with a change in a specific direction in another variable.

There are two types of monotonic relationships:

Positive Monotonic: When the value of one variable increases, the value of the other variable tends to increase as well.

Negative Monotonic: When the value of one variable increases, the value of the other variable tends to decrease.

If two variables don’t generally change in the same direction , then they are said to have a non-monotonic relationship. 

Here’s one example of a non-monotonic relationship between two variables:

And here’s another example of a non-monotonic relationship between two variables:

As the value of x increases, the value of y sometimes increases, but sometimes the value of y decreases.

Strictly Monotonic vs. Non-Strictly Monotonic

Two variables are said to have a strictly monotonic relationship if changes in one variable are always associated with a change in the same direction in another variable.

For example, the following chart illustrates a strictly positive monotonic relationship between two variables:

When the value of x increases, the value of y always increases.

When the value of x increases, the value of y always decreases.

How to Quantify Monotonic Relationships

The most common way to quantify the relationship between two variables is to use the , which measures the linear association between two variables.

This coefficient always takes on a value between -1 and 1 where:

• -1 indicates a perfectly negative linear correlation between two variables
• 0 indicates no linear correlation between two variables
• 1 indicates a perfectly positive linear correlation between two variables

The closer the coefficient is to 1, the stronger the positive relationship between two variables. Conversely, the closer the coefficient is to -1, the stronger the negative relationship between two variables.

However, if the relationship between two variables is monotonic but non-linear (like an exponential relationship) then it’s a good idea to use the , which was designed to handle monotonic relationships well.

No matter which type of correlation coefficient you calculate, it’s always a good idea to create a scatterplot to visualize the relationship between the variables as well.