In most statistics courses, students learn about **linear relationships** between variables.

These are relationships where an increase in one variable is associated with a predictable increase in another variable.

One example of this might be minutes played in a basketball game vs. total points scored:

Players who play more minutes tend to score more points.

However, there can also exist **nonlinear relationships** between variables and these appear all the time in the real world.

This tutorial provides five examples of nonlinear relationships between variables in the real world.

**Example 1: Quadratic Relationships**

One of the most common nonlinear relationships in the real world is a **quadratic relationship** between variables.

When plotted on a scatterplot, this relationship typically exhibits a “U” shape.

One example might be total working hours per week vs. overall happiness:

As working hours increase from zero, overall happiness tends to increase, but beyond a certain threshold more working hours actually leads to decreased happiness.

This upside down “U” shape is the signature shape of a quadratic relationship between two variables.

**Example 2: Cubic Relationships**

Another common nonlinear relationship in the real world is a **cubic relationship** between variables.

When plotted on a scatterplot, this relationship typically has two distinct curves.

This type of relationship exists often between variables in the field of thermodynamics:

Notice that there are two distinct curves on the plot and the relationship between variable X and variable Y is clearly not linear.

**Example 3: Exponential Relationships**

Another common nonlinear relationship in the real world is an **exponential relationship** between variables.

When plotted on a scatterplot, this relationship exhibits a single curve that becomes more pronounced as the variable on the x-axis increases.

One well-known example of an exponential relationship is the lifespan of bamboo plants and their yearly growth:

During the first few years of growth, a bamboo plant grows very slowly but once it reaches a certain age it explodes in height and grows at a rapid pace.

**Example 4: Logarithmic Relationships**

Another common nonlinear relationship in the real world is a **logarithmic relationship** between variables.

When plotted on a scatterplot, this relationship exhibits a single curve that becomes less pronounced as the variable on the x-axis increases.

One example of a logarithmic relationship is between the efficiency of smart-home technologies and time:

When a new smart-home technology (like a self-operating vacuum or self-operating AC unit) is installed in a home, it learns rapidly how to become more efficient, but then once it reaches a certain point it hits a maximum threshold in efficiency.

**Example 5: Cosine Relationships**

Another common nonlinear relationship in the real world is a **cosine relationship** between variables.

When plotted on a scatterplot, this relationship exhibits a “wave” shape.

One example of a cosine relationship is between the frequency of sound waves and time:

Notice how the relationship exhibits a “wave” shape, which is highly nonlinear.

**Additional Resources**

The following tutorials explain how to perform different types of nonlinear regression in Excel:

How to Perform Quadratic Regression in Excel

How to Perform Cubic Regression in Excel

How to Perform Exponential Regression in Excel

How to Perform Logarithmic Regression in Excel