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My textbook gives the definition of two independent categorical variables to be:

Two categorical variables are said to be independent if the conditional distributions on one variable are identical for every category of the other variable. Otherwise, the variables have an association, or relationship.

I am trying to grasp the concept of an independent categorical variable with the following example:

Let's suppose that I gather a nationally representative sample of adult Americans, and I ask: "In general, how do you feel about things - would you say you feel optimistic, neutral, or pessimistic most of the time?"

I then compare the responses of men and women on the question.

Now, I have two categorical variables - gender and the response. In order for gender and response to be independent, then would the percent of women who answered "optimistic" have to equal the percent of men who answered "optimistic", the percent of women who answered "neutral" have to equal the percent of men who answered "neutral", and so on?

Any help is greatly appreciated.

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You interpret this correct. Yes, in your example for there to be no gender difference (for gender and survey responding to be independent from one another) the % of different responses should be roughly the same for men and women.

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