Lurking variables and relationships between categorical variables

When the direction of the relationship reverses, the effect is called Simpson's paradox. As with other 'paradoxes', there is no real contradiction; it just takes a bit more thought to understand why your initial intuition is wrong.

College admission and gender

The table below describes admission rates of male and female students into the Business and Law Schools of a university.

  Admission    
Gender   Admit     Deny     Total     P(Admit)  
  Male 490 210 700 0.700
  Female   280 220 500 0.560

A naive examination of the data suggests that females have a lower probability of being admitted, but the opposite is true if the applicants are split into the two different degrees.

Business
    Admission    
  Gender   Admit     Deny     Total     P(Admit)  
    Male 480 120 600 0.800
    Female   180 020 200 0.900
Law
    Admission    
  Gender   Admit     Deny     Total     P(Admit)  
    Male 010 090 100 0.100
    Female   100 200 300 0.333

Proportional Venn diagram

Simpson's paradox is explained in the proportional Venn diagram below — in it, each rectangle is proportional to the number of women with these values for the variables.

Most of the males applied to the Business school (where admittance rates were high) and this increased their overall admittance rate compared to the females.