[T11] Correlation


Two things are correlated if the presence of one thing makes the other thing more likely, or less likely. For example, in humans being female is correlated with having a long life (say, over 80 years). A woman is more likely to live over 80 years than a man. This is a positive correlation, since the first property (being female) makes the second property (living over 80 years) more likely. There is a negative correlation between smoking and long life; if you smoke, you are less likely to live over 80 years than if you don't smoke.

If two things are uncorrelated, the presence or absence of the first thing has no effect of the probability of the second thing. For example, if I say that the day of the week is uncorrelated with the weather, I am saying that the probability of rain is unaffected by what day it is; the probability of rain is the same on a Sunday as on a Wednesday.

To say that two things are uncorrelated is the same as saying that they are independent. Recall from before that if A and B are independent, the conditional probabilities P(A|B) and P(A|not B) are the same--they are both equal to P(A). That is, the probability of A given the presence of B is just the same as the probability of A given the absence of B. On the other hand, if A is positively (or negatively) correlated with B, the probability of A given the presence of B will be greater than (or less than) the probability of A given the absence of B. We can use these facts to construct a precise definition of correlation. A and B are uncorrelated if P(A|B)=P(A|not B), or equivalently, if P(A|B) = P(A). They are positively correlated if P(A|B) > P(A|not B), or equivalently, if P(A|B)>P(A). They are negatively correlated if P(A|B) < P(A|not B), or equivalently, if P(A|B) < P(A).

  1. You throw two dice, one after the other, and you want the sum to be 7. Is this outcome correlated with whether the first die shows a 3?
  2. What if you want the sum to be 4?
  3. Suppose we collect data for 10 weeks, and we find that it rains on 5 out of 10 Sundays and 3 out of 10 Wednesdays. Does it follow that the day of the week is correlated with the weather?
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