# Does Causation Imply Correlation? (3 Examples)

It’s well-known that correlation does not imply causation.

As a simple example, if we collect data for the total number of high school graduates and total pizza consumption in the U.S. each year, we would find that the two variables are highly correlated: This doesn’t mean that an increased number of high school graduates is causing more pizza consumption.

The more likely explanation is that U.S. population has been increasing over time, which means that the number of people receiving a high school degree and the total pizza being consumed are both increasing as population increases.

But what about the reverse statement: Does causation imply correlation?

If one variable causes another variable, does it necessarily mean that the two variables will be correlated?

The following examples show why.

Suppose some variable, X, causes variable Y to take on a value equal to X2.

For example:

• If X = -10 then Y = -102 = 100
• If X = 0 then Y = 02 = 0
• If X = 10 then Y = 102 = 100

And so on.

If we plotted the relationship between X and Y, it would look like this: If we calculated the Pearson correlation coefficient between the two variables, we would find that the correlation is zero.

Although X causes Y, the linear correlation between the two variables is zero.

### Example 2: Quartic Relationship

Suppose some variable, X, causes variable Y to take on a value equal to X4.

For example:

• If X = -10 then Y = -104 = 10,000
• If X = 0 then Y = 04 = 0
• If X = 10 then Y = 104 = 10,000

And so on.

If we plotted the relationship between X and Y, it would look like this: If we calculated the Pearson correlation coefficient between the two variables, we would find that the correlation is zero.

We know that X causes Y, but the linear correlation between the two variables is zero.

### Example 3: Cosine Relationship

Suppose some variable, X, causes variable Y to take on a value equal to cos(X).

For example:

• If X = -10 then Y = cos(-10) = -0.83907
• If X = 0 then Y = cos(0) = 1
• If X = 10 then Y = cos(10) = -0.83907

And so on.

If we plotted the relationship between X and Y, it would look like this: If we calculated the Pearson correlation coefficient between the two variables, we would find that the correlation is zero.

We know that X causes Y, but the linear correlation between the two variables is zero.