Now that we've included these in our model, we see the original correlation between shoe size (x) and math test scores (y) essentially dissolves to r(xy)=.02. Where did it go? It was mathematically absorbed by the real associations between grade level and math performance [r(wy)=.64] and between study time and math performance [r(zy)=.31]. Although that original association between shoe size and math performance was there in a statistical sense, it was not really there in a practical sense because it was actually just a proxy for the other variables that really mattered in this association. 
Given all this, we can think of correlation as a necessary (in most cases) but insufficient condition for causation. Associations like the shoe-math example presented here are known as spurious correlations - real in a statistical sense but meaningless in a practical sense. Absurd examples like this are easy to spot because the statistical relationship between the variables makes no logical sense. Subtler examples can be much harder to spot, and are frequently accepted as fact. There are other ways of assessing causality beyond those presented here, but always beware when a causal claim is presented from merely correlational evidence.
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