head(murder_cases)

##   case_id offender_race victim_race sentence
## 1       1         white       white     jail
## 2       2         black       black     jail
## 3       3         black       black     jail
## 4       4         black       black     jail
## 5       5         white       black     jail
## 6       6         black       black     jail

str(murder_cases)

## 'data.frame':    4898 obs. of  4 variables:
##  $ case_id      : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ offender_race: Factor w/ 2 levels "black","white": 2 1 1 1 2 1 1 1 1 1 ...
##  $ victim_race  : Factor w/ 2 levels "black","white": 2 1 1 1 1 1 1 1 1 1 ...
##  $ sentence     : Factor w/ 2 levels "death","jail": 2 2 2 2 2 2 2 2 2 2 ...

Here are the same data, summarized in a contingency table:

table(murder_cases$offender_race, murder_cases$sentence)

##        
##         death jail
##   black    28 2498
##   white    49 2323

Let's calculate:

• The joint distribution of the offender's race and the sentence.
• The marginal distribution of the sentence.
• The conditional distribution of the sentence given that the offender was black. (And again, for white offenders)
• Is the sentence independent of the offender's race?

murder_cases_white_victim <- filter(murder_cases, victim_race == "white")
table(murder_cases_white_victim$offender_race,
  murder_cases_white_victim$sentence)

##        
##         death jail
##   black    16  359
##   white    49 2223

murder_cases_black_victim <- filter(murder_cases, victim_race == "black")
table(murder_cases_black_victim$offender_race,
  murder_cases_black_victim$sentence)

##        
##         death jail
##   black    12 2139
##   white     0  100

The original data were published in:

Blume et al., Explaining Death Row's Population and Racial Composition. Journal of Empirical Legal Studies, Vol. 1, Issue 1, p. 165-207, 2004.

The issue of Simpson's paradox in relation to these data was discussed further in:

Norton et al., Simpson's Paradox and How to Avoid It. Significance, p. 40-43, August, 2015.