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234 Part IV: Building Strong Connections with Chi-Square Tests

another variable. However, to cover all the bases and make sure you
can answer this very popular question, here’s the official definition of
independence, straight from the statistician’s mouth: Two categories are
independent if their joint probability equals the product of their marginal
probabilities. The only caveat here is that neither of the categories can be
completely empty.

For example, if being female is independent of being a Democrat, then
P(F + D) = P(F) * P(D), where D = Democrat and F = Female. So, to show that
two categories are independent, find the joint probability and compare it
to the product of the two marginal probabilities. If you get the same answer
both times, the categories are independent. If not, then the categories are
dependent.
You may be wondering: Don’t all probabilities work this way, where the joint
probability equals the product of the marginals? No, they don’t. For example,
if you draw a card from a standard 52-card deck, you get a red card with prob-
ability . You draw a heart with probability . The chance of drawing both
a heart and a red card with one draw is still (because all hearts are red).
However, the product of the individual probabilities for red and heart
comes out to which is not equal to . This tells you that the
categories “red” and “heart” aren’t independent (that is, they’re dependent).

Now the joint probability of a red two is , or . This equals the probability
of a red card, , times the probability of a two (because ). This
tells you that the categories “red” and “two” are independent.

Another way to check for independence is to compare the conditional prob-
ability to the marginal probability. Specifically, if you want to check whether
being female is independent of being Democrat, check either of the following
two situations (they’ll both work if the variables are independent):

✓ Is P(F|D) = P(F)? That is, if you know someone is a Democrat, does that
affect the chance that they’ll also be female? If yes, F and D are indepen-
dent. If not, F and D are dependent.
✓ Is P(D|F) = P(D)? This question is asking whether being female changes
your chances of being a Democrat. If yes, D and F are independent. If
not, D and F are dependent.

Is knowing that you’re in one category going to change the probability of being
in another category? If so, the two categories aren’t independent. If knowing
doesn’t affect the probability, then the two categories are independent.

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