Berkeley Admissions by Sex (1973)
Reviewed by the crosstabs.com methods team · Last updated
In this table, sex is significantly associated with admission — a small association: χ²(1, N = 4526) = 92.21, p < .001, V = .14.
The data
| Sex \ Admission | Admitted | Rejected | Total |
|---|---|---|---|
| Men | 1,198 | 1,493 | 2,691 |
| Women | 557 | 1,278 | 1,835 |
| Total | 1,755 | 2,771 | 4,526 |
Background
In autumn 1973, the University of California, Berkeley admitted about 44% of its male graduate applicants but only about 35% of its female applicants. The aggregate figures looked like prima facie evidence of sex discrimination, and the university asked statisticians to investigate.
This table is the aggregate crosstab: 4,526 applicants to the six largest departments, by sex and admission decision. A chi-square test on it is decisively significant — sex and admission outcome are clearly associated in the pooled data.
The twist is what Bickel, Hammel, and O'Connell found when they disaggregated by department: within most individual departments, women were admitted at equal or higher rates than men. Women had simply applied in larger numbers to the most competitive departments. The pooled table answers a real question, but not the causal one people care about.
Results
Chi-square test
χ² = 92.21
df = 1, p < .001
Effect size
Cramér's V = 0.143
a small association
Fisher's exact test
p < .001
two-sided, exact for this 2×2 table
Odds ratio
OR = 1.84
95% CI [1.62, 2.09]
APA-style report: χ²(1, N = 4526) = 92.21, p < .001, V = .14. N = 4,526.
Interpretation
The chi-square test rejects independence at the conventional 0.05 level (p < .001): a pattern this strong is unlikely if sex and admissionwere unrelated. Cramér's V of 0.143 puts this in the small range — the association is real but modest — knowing one variable tells you only a little about the other.
Because this is a 2×2 table, Fisher's exact test (p < .001) provides an exact significance check, and the odds ratio of 1.84 (95% CI [1.62, 2.09]) summarizes the strength of the relationship in odds terms.
Caveats
- This aggregate table is the classic example of Simpson's paradox. When the same data are broken out by department, the apparent direction of the association reverses: most departments admitted women at rates equal to or higher than men. The pooled association is largely explained by women applying to more competitive departments, so do not read this table as evidence of admissions bias.
- A significant chi-square on aggregated data tells you the variables are associated in the pooled table — it cannot tell you why, and it can be misleading when a lurking variable (here, department) drives the pattern.
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