Can correlation ever prove causation?

No, correlation alone can never prove causation. To prove causation, you must conduct a controlled experiment where you actively manipulate one variable while keeping all other variables constant to see if it causes a change in the other variable.

What does a correlation of zero mean?

A correlation of $r = 0$ means there is no linear relationship between the two variables. However, it does not mean the variables are entirely unrelated, as they could still share a strong curved or non-linear relationship.

Why do we even use correlation if it doesn't prove causation?

Correlation is still incredibly useful for making predictions. If we know two variables are correlated, we can guess the value of one based on the other, even if we do not know exactly why they are moving together. It also helps researchers figure out where to focus their controlled experiments.

What Does Correlation Not Tell You?

Correlation tells you if two variables tend to move together, but it definitively does not tell you if one causes the other. When we see a strong correlation, it is incredibly tempting to assume that the first variable is directly influencing the second. However, statistical correlation only measures the mathematical pattern between two sets of data, not the real-world mechanics driving them. Think of a rooster crowing every morning right before the sun comes up. The two events are highly correlated, happening together every single day. But the rooster's crow does not cause the sun to rise. In statistics, we summarize this fundamental rule with a famous phrase: correlation does not imply causation.

What Correlation Actually Measures

Correlation measures the strength and direction of a linear relationship between two quantitative variables. We usually represent this with the correlation coefficient, $r$ , which ranges from $-1$ to $1$ . If $r$ is close to $1$ , as one variable goes up, the other tends to go up. If $r$ is close to $-1$ , as one goes up, the other tends to go down. It is a powerful tool for predicting one variable based on another, but it is strictly a measurement of pattern, not of cause and effect.

The Trap of Confounding Variables

Often, two variables are highly correlated because a third, hidden variable is driving both of them. This invisible third factor is called a confounding variable. A classic example is the strong positive correlation between ice cream sales and shark attacks. Buying ice cream does not make sharks attack you, and surviving a shark attack does not make you crave ice cream. Instead, both are driven by a confounding variable: summer weather. When it is hot, more people buy ice cream and more people swim in the ocean, leading to more shark attacks.

Missing the Big (Non-Linear) Picture

Correlation specifically measures linear relationships (straight lines). It does not tell you if two variables have a strong non-linear relationship. For example, if you plot a person's athletic performance against their stress level, you might get an upside-down U-shape. A little stress helps performance, but too much stress ruins it. If you calculate the correlation coefficient $r$ for this U-shaped curve, it might be exactly $0$ . Correlation fails to tell you that these two things are deeply related just because the relationship isn't a straight line.

Worked through

A researcher collects data from an elementary school and finds a strong positive correlation ( $r = 0.85$ ) between a student's shoe size and their score on a standardized reading test. Based on this correlation, the school proposes a new program to stretch students' feet so they will become better readers. What does the correlation actually tell us here, and why is the school's proposal flawed?

The correlation $r = 0.85$ tells us that as shoe size increases, reading test scores also tend to increase. However, the correlation does not tell us that having larger feet causes better reading ability. The school's proposal assumes causation where none exists. The flawed reasoning ignores a massive confounding variable: the age of the students. Older children (like fifth graders) have larger feet than younger children (like first graders), and they have also had years of additional reading instruction. Age causes both shoe size and reading ability to increase.

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Where this comes from: OpenStax Introductory Statistics, Chapter 12: Linear Regression and Correlation · Khan Academy: Bivariate data, Scatterplots, and Correlation