# Simpson's paradox in Freedman, Pisani and Purves book

In this book, there is an example of sex bias in graduate admissions.

Major Men Women
A 825 62 108 82
B 560 63 25 68
C 325 37 593 34
D 417 33 375 35
E 191 28 393 24
F 373 6 341 7
Total 2691 45 1835 30

The total # applicants is just the sum of the entries above. The Total % admitted, 45 for men, is

$$\frac{62\cdot825 }{2691} + \cdots + \frac{6\cdot373}{2691}$$

Likewise, for women.

The authors then make the argument that a statistician would do the following:

Major Total number of applicants
A 933 = 825 + 108
B 585
C 918
D 792
E 584
F 714 = 373 + 341
Total 4526

They then state that the weighted average admission rate for men is:

$$\frac{62\cdot933}{4526} + \cdots + \frac{6\cdot714}{4526} \approx 39$$

and likewise for women, it is:

$$\frac{82\cdot933}{4526} + \cdots + \frac{7\cdot714}{4526} \approx 43$$

I am having a tough time interpreting the last pair of calculations "weighted average admission rate for men/women".

They state:

The weighted averages control for the confounding factor -- choice of major. These averages suggest that if anything, the admissions process is biased against the men.

How exactly do the weighted averages of 39% and 43% mean in this context? What is the right way to interpret these numbers? The original percentages of 45% and 30% seem much easier for me to understand and interpret.

• This is one version of a dataset often used to illustrate this paradox, frequently referred as the Berkeley admissions data. The original paper by Bickel, Hammel and O'Connell from 1975 is often quoted (and misquoted). refsmmat.com/posts/2016-05-08-simpsons-paradox-berkeley.html is one of many commentaries. Commented Aug 2 at 9:39

This is explained on same page (p. 19):

Technical note. Table 2 is hard to read because it compares 12 admission rates. A statistician might summarize table 2 by computing one overall admissions rate for men and another for women, but adjusting for the sex differences in application rates. An ordinary average ignores the differences in size among the departments. Instead a weighted average of the admission rates could be used, the weights being the total number of applicants (male and female) to each department; see table 3.

In other words, the weighting is done by the number of applicants per department: $$\frac{0.62 \times933 + 0.63\times585 + 0.37 \times 918 + 0.33 \times 792 + 0.28 \times 584 + 0.06 \times 714}{4,\!526} \approx 39\%.$$

You can interpret the equation from the book as follows: $$\textrm{weighted average rate} = \sum_{i=1}^k \frac{\textrm{male admissions in }k}{\textrm{male applicants in }k} \times \frac{\textrm{total applicants in }k}{\textrm{total applicants}},$$ where $$k \in \{ \textrm{A}, \textrm{B}, \dots, \textrm{F} \}$$ is a given major.

Note that the authors don't say "a statistician would," but that "a statistician might." Personally, I disagree that 12 numbers is too much to display. This obsession with reducing results to a single metric is what led to the paradox in the first place. A simple visualization can summarize all data:

Left: We see the overall admissions and rejections when only considering sex.
Right: We see many more male applicants in the blue groups, which have a low rejection rate.

### Reference

• David Freedman, Robert Pisani, and Roger Purves (2007), Statistics (4th edition), W. W. Norton. ISBN 0-393-92972-8.

### Code to produce the figure

DF <- data.frame(
major      = factor(rep(LETTERS[1:6], 2)),
sex        = factor(rep(c("men", "women"), each = 6)),
applicants = c(825, 560, 325, 417, 191, 373,
108,  25, 593, 375, 393, 341),
admitted   = c(512, 353, 120, 138,  54,  22,
89,  17, 202, 131,  94,  24)
)
DF$$rejected <- DF$$applicants - DF$$admitted DFlong <- rbind(DF[, 1:2], DF[, 1:2]) DFlong$$status <- factor(rep(c("admitted", "rejected"), each = nrow(DF)))
DFlong$$count <- c(DF$$admitted, DF$$rejected) DFlong$$total  <- c(DF$$applicants, DF$$applicants)
Men           <- DFlong[DFlong$$sex == "men", ] Women <- DFlong[DFlong$$sex == "women", ]

cols <- c("#a6cee3", "#1f78b4", "#b2df8a", "#33a02c", "#fb9a99", "#e31a1c")
layout(matrix(c(1, 1, 4, 5, 5,
2, 3, 4, 6, 7), nrow = 2, byrow = TRUE),
heights = c(1, 4),
widths = c(1, 1, 0.5, 1, 1))
par(mar = c(0, 0, 0, 0))
plot(NA, xlim = c(0, 1), ylim = c(0, 1), type = "n", axes = FALSE, ann = FALSE)
text(0.5, 0.5, "Admissions by sex", font = 2, cex = 1.5)
par(mar = c(5, 4, 4, 0) + 0.1)
barplot(count ~ major + status, Men, ylim = c(0, 1500),
border = NA, col = rep(1, 6), las = 1, main = "Men")
barplot(count ~ major + status, Women, ylim = c(0, 1500),
border = NA, col = rep(1, 6), las = 1, main = "Women")
par(mar = c(0, 0, 0, 0))
plot(NA, xlim = c(0, 1), ylim = c(0, 1), type = "n", axes = FALSE, ann = FALSE)
plot(NA, xlim = c(0, 1), ylim = c(0, 1), type = "n", axes = FALSE, ann = FALSE)
text(0.5, 0.5, "Admissions by sex and major", font = 2, cex = 1.5)
par(mar = c(5, 4, 4, 0) + 0.1)
barplot(count ~ major + status, Men, ylim = c(0, 1500),
border = NA, col = cols, las = 1, main = "Men")
barplot(count ~ major + status, Women, ylim = c(0, 1500),
border = NA, col = cols, las = 1, main = "Women")

• Is it not true that P(admission | male) = unconditional probability of admission of a male = 0.45? So, the 39% is evaluating something else. It is, in words, supposing all applicants were male, the unconditional probability of admission of a male is 39%. Likewise, supposing all applicants were female, the unconditional probability of admission of a female is 43% . Is this the right interpretation apart from the formula used to calculate these numbers? Commented Aug 2 at 9:35
• @One_Cable5781 Yes indeed, they are both valid values, but one is telling something different than the other Commented Aug 2 at 19:27
• You could produce a figure just as nice with a lot less code with ggplot.
– qwr
Commented Aug 3 at 5:19
• @qwr If you're interested why some people prefer base graphics, there's a nice write-up here. Commented Aug 3 at 5:40