IB Maths IA idea · Probability and statistics

Simpson's paradox: when the overall trend lies

In a famous set of university admission figures, men were admitted at a clearly higher rate than women overall, yet in most departments women did as well or better. Both statements are true.

Probability and statistics
SL ideal
HL ideal
Maths AA Maths AI Published data
Syllabus: AA 4.5, AA 4.6, AA 4.11, AA 4.13

By Photon Academy · Updated 13 November 2026

These ideas are starting points, not an IA to copy. Never copy an idea blindly: make the question, method and analysis your own, and check your plan with your teacher.

The big idea

A group's overall rate is a weighted average of its rates in each subgroup, and the weights are the numbers of people in each. Two groups can have different weights. If one group applies mostly to subgroups with low rates, its overall rate drops even when it does better inside every subgroup. That is Simpson's paradox: a trend that reverses when groups are combined. You write conditional probabilities for each subgroup and for the whole, show exactly how the weights cause the reversal, and find what is needed for it to happen. The real question is which number answers your question. Decide which comparison is fair, and justify it; that judgement, not the arithmetic, is where the high marks are.

The maths you will use

Conditional probability. Find $P(\text{admitted} \mid \text{man}, \text{dept } d)$ and $P(\text{admitted} \mid \text{man})$ from counts, and compare each pair.
Law of total probability. Write $P(A) = \sum_d P(A \mid d)P(d)$ for each group and read the weights $P(d)$ off the table.
Weighted averages. Show how a group's overall rate changes with the share of applicants to the hard department, as a straight line in that share.
Inequalities. Derive the conditions on four rates and two shares that give a reversal, and test them with numbers.
Standardisation. Apply the same weights to both groups and compare the standardised rates; optionally test independence with a chi-squared test (AI).

Key equations

P(A) = \sum_{d} P(A \mid d)\,P(d)

\frac{a_1}{n_1} > \frac{b_1}{m_1},\ \frac{a_2}{n_2} > \frac{b_2}{m_2} \quad\text{but}\quad \frac{a_1+a_2}{n_1+n_2} < \frac{b_1+b_2}{m_1+m_2}

The maths in action

Left: dots for men and women in each of six departments, sized by number of applicants, with dashed lines for the overall rates; women's dots are at or above men's in four departments. Right: department admission rate against the share of women applicants, falling. — (a) Berkeley 1973, six largest departments: women's admission rate is higher in 4 of 6 departments, yet the overall rate is 44.5% for men and 30.4% for women (dashed). (b) Women applied more to the departments with low admission rates.

Example research question

To adapt, never to copy

In published admission figures for six university departments, how can men's overall admission rate be higher than women's while women's rate is higher in most departments, and does giving both groups the same department weights remove the gap?

Change the context, the data or the method so the question is yours. Our Maths IA guide shows how to check it.

Check your research question

Names the table or tables, the groups being compared and the outcome.
Says which rates you will compute: overall, by subgroup, and a standardised rate.
Says what question the data are meant to answer, because that decides which rate is the fair one.
Goes beyond showing one reversal: there is a formula, a condition or a fair-comparison method.
Is careful about causes: the data show associations, and you say what you cannot conclude.

Which course it fits

AA SL ideal
AA HL ideal
AI SL ideal
AI HL ideal

Syllabus links in Maths AA:

AA 4.5 Basic probability and expected frequency
AA 4.6 Venn and tree diagrams, combined events
AA 4.11 Conditional probability and independence
AA 4.13 Bayes' theorem (HL only)

Steps

Read and predict. Find out what Simpson's paradox says. Before any calculation, guess whether the overall trend in your chosen table will match the subgroup trends.
Choose the table. Pick a published two-way table with counts, not just percentages. Say what the groups, subgroups and outcome are.
Check the arithmetic. Check that the subgroup counts add up to the totals, and recompute two of the quoted percentages by hand.
Trial run (1 lesson). Compute overall and subgroup rates in a spreadsheet and see whether the trend reverses. If it does not, choose a different table now.
Write the weights. Write each overall rate as a weighted average of subgroup rates and show which weights cause the difference.
Standardise. Recompute both groups' overall rates with the same weights for both. Compare with the raw rates and decide which is fairer for your question.
Find the conditions. Derive when a reversal can happen for two groups and two subgroups, and test the conditions on invented tables.
Go further, with a reason. Add a second table, or simulate how often reversals occur. HL: this is where the rigour for criterion E comes from.
Reflect as you go. After every result, write one line on what it means for the original question and what it assumes. These lines become your evaluation.
Cut, check, then send. Check every table is labelled, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.

Make it yours

Rebuild the paradox from a table. Take a published two-way table, compute the overall rates and the rates within each subgroup, and show the reversal.
Weighted averages. Write each overall rate as a sum of subgroup rates times subgroup shares and show which shares cause the reversal.
Fair-comparison rate. Give both groups the same subgroup weights (for instance the total applicants) and recompute the overall rates.
Find the conditions. Derive what must be true of the rates and weights for the reversal, with two groups and two subgroups.
How common is it?. Simulate many random two-by-two-by-two tables and count how often a reversal occurs.
A second example. Find a second real table, such as medical treatments, and test whether the lesson is the same.

HL extension

Prove the conditions for a reversal in general, derive the standardised rate, and discuss when the subgroup comparison or the overall comparison is the causally appropriate one.

The data you need

Use the published Berkeley admission counts by department and sex, and a second table such as the kidney stone treatment data.

Bickel, Hammel and O'Connell (1975), Science: Sex bias in graduate admissions, data from Berkeley

Traps

Giving the paradox as a trick without showing the weights that cause it.
Saying the overall rate is simply wrong. It is correct; it answers a different question.
Concluding bias or discrimination from rates alone. Rates show associations, not reasons.
Using percentages from a source without the counts, so the weights cannot be checked.
Averaging the subgroup percentages instead of weighting them by the numbers in each subgroup.
Choosing subgroups after seeing the data so the reversal appears.

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