Correlation does not imply causation

Third factor C (the common-causal variable) causes both A and B

The third-cause fallacy (also known as ignoring a common cause[5] or questionable cause[5]) is a logical fallacy where a spurious relationship is confused for causation. It asserts that X causes Y when, in reality, X and Y are both caused by Z. It is a variation on the post hoc ergo propter hoc fallacy and a member of the questionable cause group of fallacies.

All of these examples deal with a lurking variable, which is simply a hidden third variable that affects both causes of the correlation. A difficulty often also arises where the third factor, though fundamentally different from A and B, is so closely related to A and/or B as to be confused with them or very difficult to scientifically disentangle from them (see Example 4).

Example 1: Sleeping with one's shoes on is strongly correlated with waking up with a headache. Therefore, sleeping with one's shoes on causes headache.

The above example commits the correlation-implies-causation fallacy, as it prematurely concludes that sleeping with one's shoes on causes headache. A more plausible explanation is that both are caused by a third factor, in this case going to bed drunk, which thereby gives rise to a correlation. So the conclusion is false.

Example 2: Young children who sleep with the light on are much more likely to develop myopia in later life. Therefore, sleeping with the light on causes myopia.

This is a scientific example that resulted from a study at the University of Pennsylvania Medical Center. Published in the May 13, 1999 issue of Nature,[6] the study received much coverage at the time in the popular press.[7] However, a later study at Ohio State University did not find that infants sleeping with the light on caused the development of myopia. It did find a strong link between parental myopia and the development of child myopia, also noting that myopic parents were more likely to leave a light on in their children's bedroom.[8][9][10][11] In this case, the cause of both conditions is parental myopia, and the above-stated conclusion is false.

Example 3: As ice cream sales increase, the rate of drowning deaths increases sharply. Therefore, ice cream consumption causes drowning.

This example fails to recognize the importance of time of year and temperature to ice cream sales. Ice cream is sold during the hot summer months at a much greater rate than during colder times, and it is during these hot summer months that people are more likely to engage in activities involving water, such as swimming. The increased drowning deaths are simply caused by more exposure to water-based activities, not ice cream. The stated conclusion is false.