In applied statistics, particularly in research and data analysis, the concepts of correlation and causation are often mixed up. This tutorial dismantles generalized trends and widespread myths like “correlation equals causation” and “correlation implies causation”, clarifying in an illustrative and example-based fashion these two important statistical concepts.

What is Correlation?

In simple terms, correlation is the strength and direction of a (linear) relationship between two variables X and Y. It is a statistical measure that indicates how the two variables are related to each other, that is, correlated. When there exists a correlation between two variables, changes in an observation under the variable X occur together with changes under the other variable, Y.

Suppose we have collected the maximum daily temperatures throughout the summer (variable X), and the daily ice cream sales throughout the same season (variable Y). Below is a small sample of both data variables, recorded over a week.

As can be observed, as the daily temperature X rises, so do the ice cream sales Y. This means there is a positive correlation or direct relationship between X and Y. Correlation means the two variables vary similarly, but it does not necessarily imply that such changes in one of them cause similar changes the other.

Now suppose we analyze together daily temperatures (X), and daily consumption of hot chocolate (Z):

If we observe a tendency of lower hot chocolate sales on warmer days, then there is a negative correlation and inverse relationship between X and Z. Again, changes co-occur in both variables but this time they occur in opposite directions: when observations in X increase, observations in Z decrease.

What is Causation?

Next, it’s time to define causation. Causation is an indication that an event B is the direct result of the occurrence of another event A. In other words, A is the cause for B to happen.

Imagine you are in a large dark room with several light bulbs and light switches: one for turning each bulb on. Regarding causation between events, pressing the switch (event A) is the direct reason why a light bulb will turn on (event B). This is a clear case of causation: pressing the light turns the light on.

Causation can also be understood in terms of statistical variables. Assuming two variables X and Y, causation implies that one variable X is the cause, and the other Y is the effect, or vice versa. If X represents the number of switches pressed at a given time, and Y is the amount of light (measured in lux) in the room, then there is also causation because pressing more switches on causes a higher light intensity in the room.

Relationship and Differences

In the ice cream example shown earlier, besides the clear correlation, there also exists causation because temperature rises (cause) directly lead to increases in ice cream sales (effect): people tend to consume more ice cream in warmer days, thereby showing a direct causal relationship between the two variables.

However, even though correlation and causation often manifest together, it is important to note that:

1. Correlation does not always imply causation
2. Causation (a causal link between two variables) always implies a correlation between them

Let’s see some examples where correlation between two variables does not imply causation, starting with another summer-themed one. Suppose X is the daily temperature recorded, and Y is the number of visitors to a beach. Even though we observe that as temperature rises the number of visitors to the beach also increases (positive correlation), higher temperature do not directly cause more people to go to the beach: such a decision might be influenced by other factors like day of the week, whether it is holiday or not, whether special events are going on at that beach, and so on. Temperature may partly influence the decision to go to the beach, but it is not the direct cause.

On a last -and more autumn-themed- example, suppose that on days of higher rainfall (X) more people leave home with their umbrella (Y), and at the same time, there is more traffic on the streets (Z). Whilst there is both correlation and causation between X and Y, and there also exists a correlation between X and Z, this does not necessarily mean the rain is the direct cause of the denser traffic. There could be other reasons, like the time of the day. In summary, there is no direct causation between X and Z.

Wrapping Up

Understanding the subtle difference between correlation and causation is crucial for accurate analysis and interpretation of data, helping us avoid misleading conclusions and ensuring that we identify cases when two variables simply related versus when cases when one directly influences the other.