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What does it mean for two variables or a set of points on a graph to be positively correlated? What does positive correlation NOT mean?

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If two variables are positively correlated, then in general, as one increases, the other increase. A graph would have a general positive-slope trend:

Notice that, although the general trend is upward to the right, we could easily pick higher point to the left and one lower point to the right. Correlation is true at the population scale, but not at the scale of one individual to another.

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