Correlation

In statistics and probability theory, **correlation** refers to how similar two sets of data are. This relationship means that if one set of data changes, the other will change as well (at least more commonly than if it were up to pure chance).

## Overview

Correlation does not always mean that one causes the other. In fact, it is very possible that there is a third factor involved. Correlation can have one of two directions: Positive or negative. If it is positive, then the two sets go up together. If it is negative, then one goes up while the other goes down. Lots of different measurements of correlation are used for different situations. For example, on a scatter graph, people draw a best fit line to show the correlation between two variables.

Correlation can be measured with correlation coefficients. The most common correlation coefficients are Pearson correlation coefficient and Spearman's rank correlation. Correlation is useful in real life. An example of correlation is the height of a parent and their children.

## Correlation

_Strong_ and _weak_ are words used to describe the strength of correlation. If there is strong correlation, then the points are all close together. If there is weak correlation, then the points are all spread apart.
There are ways of making numbers show how strong the correlation is. These measurements are called **correlation coefficients**.

The best known is the Pearson product-moment correlation coefficient, sometimes denoted by  or its Greek equivalent . You plug data into a formula, and it gives you a number between -1 and 1. If the number is 1 or −1, then there is strong correlation. If the answer is 0, then there is no correlation. Another type of correlation coefficient is Spearman's rank correlation coefficient.

## Correlation vs causation

Correlation does not always mean that one thing causes the other (**causation** or **causal** **relationship**), because there might be something else that is at play.

#### Ice cream sales vs shark attacks

On hot days people buy ice cream, and people also go to the beach where some are attacked by sharks. There is a correlation between ice cream sales and shark attacks (they both go up as the temperature goes up in this case). However, ice cream sales going up does not mean ice cream sales cause (causation) more shark attacks or vice versa. However, there is also a correlation between temperature and shark attacks, which are two things that actually do have a causal relationship; higher temperatures cause shark attacks, because they cause more people to go swimming.

This means that while checking for correlation can be (and often is) used to **test** if there **could** be causation (if there is causation, there will probably also be a correlation, at least if you look at enough data), it is not enough to **prove** that there definitely is causation. As in the ice cream example, it is also possible that the correlation is due to an additional factor – in statistics, this is called a confounding variable. Because correlation alone does not prove causation, scientists, economists, and others will test their ideas by creating isolated experiment<nowiki/>al environments where only one factor is changed (if possible).

#### Clinical trials

Clinical trials are another example to illustrate this. Doctors often find two groups of people that are very similar to each other in every possible way (e.g. age, sex, health conditions etc.), but only test the actual drug on one of the two groups.  Afterwards they can check if they can find significant differences between the groups.  **Significance** in statistics means that a difference or relationship is stronger than what might just be caused by pure chance.  If they find such significant differences, they can be relatively sure that this is due to the drug, because they have made sure beforehand that no other factors could cause such a difference between the groups.

## Common misuses

However, politicians, salesmen, news outlets and others often erroneously suggest that a particular correlation implies causation. This may be due to ignorance or dishonesty. Thus, a news report may attract attention by saying that people who consume a particular product more often have a particular health problem, implying a causation that could be actually due to something else.

Another thing to keep in mind is that even if there is causation, just looking at the correlation alone does not tell us which **direction** it is in. For example, if we look at how often people visit hospitals and how often people are sick, we will see a correlation that is, in this case, due to a causation. Yet, it would be wrong to conclude from this that going to the hospital causes people to be sick: The causation here is clearly in the other direction, people go to the hospital because they are sick.  This is why it is always important to check if your interpretation of causality based on a correlation is plausible or not.

## Related pages

- Rank correlation