** Module: Basic statistics**

- T00. Introduction
- T01. Basic concepts
- T02. The rules of probability
- T03. The game show puzzle
- T04. Expected values
- T05. Probability and utility
- T06. Cooperation
- T07. Summarizing data
- T08. Samples and biases
- T09. Sampling error
- T10. Hypothesis testing
- T11. Correlation
- T12. Simpson's paradox
- T13. The post hoc fallacy
- T14. Controlled trials
- T15. Bayesian confirmation

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- William James

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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).

- 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?
- What if you want the sum to be 4?
- 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?