Correlation does not imply causation
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"Correlation does not imply causation" is a phrase used in science and statistics to emphasize that correlation between two variables does not automatically imply that one causes the other (though correlation is necessary for linear causation, and can indicate possible causes or areas for further investigation... in other words, correlation can be a hint).[1][2]

The opposite belief, correlation proves causation, is a logical fallacy by which two events that occur together are claimed to have a cause-and-effect relationship. The fallacy is also known as cum hoc ergo propter hoc (Latin for "with this, therefore because of this") and false cause. By contrast, the fallacy post hoc ergo propter hoc requires that one event occur before the other and so may be considered a type of cum hoc fallacy.

In a widely-studied example, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD. But randomized controlled trials showed that HRT caused a small but statistically significant increase in risk of CHD. Re-analysis of the data from the epidemiological studies showed that women undertaking HRT were more likely to be from higher socio-economic groups (ABC1), with better than average diet and exercise regimes. The use of HRT and decreased incidence of coronary heart disease were coincident effects of a common cause (i.e., the benefits associated with a higher socioeconomic status), rather than cause and effect as had been supposed.[3]

In logic, the technical use of the word "implies" means "to be a sufficient circumstance". This is the meaning intended by statisticians when they say causation is not certain. Indeed, p implies q has the technical meaning of logical implication: if p then q symbolized as p → q. That is "if circumstance p is true, then q necessarily follows." In this sense, is always correct to say "Correlation does not imply causation".

However, in casual use, the word "imply" loosely means suggests rather than requires. The idea that correlation and causation are connected is certainly true; where there is causation, there is likely to be correlation. Indeed, correlation is used when inferring causation; the important point is that such inferences are not always correct because there are other possibilities, as explained later in this article.

Edward Tufte, in a criticism of the brevity of Microsoft PowerPoint presentations, deprecates the use of "is" to relate correlation and causation (as in "Correlation is not causation"), citing its inaccuracy as incomplete.[1] While it is not the case that correlation is causation, simply stating their nonequivalence omits information about their relationship. Tufte suggests that the shortest true statement that can be made about causality and correlation is one of the following:[4]

"Empirically observed covariation is a necessary but not sufficient condition for causality."
"Correlation is not causation but it sure is a hint."
[edit]General pattern

The cum hoc ergo propter hoc logical fallacy can be expressed as follows:

A occurs in correlation with B.
Therefore, A causes B.
In this type of logical fallacy, one makes a premature conclusion about causality after observing only a correlation between two or more factors. Generally, if one factor (A) is observed to only be correlated with another factor (B), it is sometimes taken for granted that A is causing B even when no evidence supports it. This is a logical fallacy because there are at least five possibilities:

A may be the cause of B.
B may be the cause of A. some unknown third factor C may actually be the cause of both A and B. there may be a combination of the above three relationships. For example, B may be the cause of A at the same time as A is the cause of B (contradicting that the only relationship between A and B is that A causes B). This describes a self-reinforcing system.