the presence of a large and concentrated population of human beings. As for the case of children and test scores, it is certainly plausible to suppose that diet influences school performance, but there may be a common underlying cause in that case as well. For example, it might be that children from better-educated households tend to be more intelligent (if only because intelligent people are more likely to complete a university education), and that better educated households tend to eat better diets (perhaps because they tend to have more money, and hence can afford a better diet). Or it might be that children from poor households tend to eat poorer diets, and that poor households tend to suffer from other sorts of family problems that tend to impair school performance.
Under what circumstances, then, can we legitimately infer the presence of a causal relationship? The answer should be evident from the rough-and-ready definition of causation given above: in order to infer a causal relationship from a correlation between X and Y, we need to know that the correlation holds, or would hold, even when other possible causes of Y are or would be absent. In other words, we need to know that X makes Y more likely regardless of the circumstances in which we find X. We need to rule out other possible causes. So what we would need to know in the case of the children, for example, is whether or not children from well-off households do worse at school if their diets are poor, and likewise for children from well-educated households.
It is important to be aware of these issues, and especially to be able to point it out when a causal relationship is wrongly inferred from a correlation. But it would take us too far into the subject to give a general recipe for validly inferring causal relationships from correlations. Suffice it to say that a causal relationship entails a correlation, but a correlation does not entail a causal relationship.
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