Two-way Tables

There's a relationship if

conditional distributions are different or conditional distribution is different from marginal distribution

There's no relationship if

conditional distributions are the same or conditional distribution is the same as marginal distribution

Marginal distributions

distribution of each variable separately, totals add up to all in the table, i.e. marginal distributions of gender splits up gender into males and females

Conditional distributions

distribution of other variable when looking at one variable, totals add up to a marginal distribution, i.e. conditional distributions of majors for gender

Simpson's paradox

original two-way table shows one relationship but if we add a third variable, the relationship reverses

Two-way tables

show relationships between two categorical variables