Maths IA · Statistics and probability · topic 37 of 97
Simpson's paradox: when the overall trend lies · IB Maths IA idea: guide, sample graph and step-by-step plan
overall rate = weighted average of the subgroupsIn 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.
- ✓ you like results that look impossible until you do the arithmetic
- ✓ you want to question a headline number rather than repeat it
- ✓ you prefer an IA built on a clear, checkable table of real figures
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.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Take a published two-way table, compute the overall rates and the rates within each subgroup, and show the reversal.
→ Conditional probabilities for each cell. Check every percentage against the counts.Write each overall rate as a sum of subgroup rates times subgroup shares and show which shares cause the reversal.
→ A formula that explains the paradox, not just an example of it. Show the share of each group in each subgroup.Give both groups the same subgroup weights (for instance the total applicants) and recompute the overall rates.
→ A standardised rate. Compare it with the raw overall rate and decide which one answers the question.Derive what must be true of the rates and weights for the reversal, with two groups and two subgroups.
→ Inequalities in four rates and two shares. Test them on tables you invent and on real ones.Simulate many random two-by-two-by-two tables and count how often a reversal occurs.
→ A probability for a surprising event, and a clear assumption about how the tables are generated.Find a second real table, such as medical treatments, and test whether the lesson is the same.
→ A comparison: in one case the subgroup rates are the right guide, in another the overall rate may be. Say why.Your Maths IA research question: run the check
which table? which question?Write your own question first. Then make sure it ticks every box:
- 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.
The model
Two groups, two subgroups, and the number who enter each. The overall rate depends on those numbers. Draw your own table this way.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
The question behind the data, why a single overall rate might mislead, and what you expect before calculating.
Where the table comes from, what each number counts, and a check that the subgroups add up to the totals.
Conditional probabilities for the whole and for each subgroup, in a table, with the reversal clear.
The overall rate as a weighted average, and the weights that cause the reversal.
A standardised rate or another method that removes the effect of the different weights, and its result.
When the reversal can occur, tested on invented tables, and a second real example.
Back to your aim: which comparison is fair for this question, with numbers.
What the table cannot show: causes, hidden variables, the choice of subgroups. Say which limitation matters most.
Where every table and result came from.
What your Maths IA graph could look like
mark both groups on their linesMade-up numbers to show the form of a strong graph, not the answer: admission rates for two groups in two departments, with the overall rates drawn as lines. example only
- Two weighted-average lines on the same axes: the group with the higher line can still end up lower overall.
- Axis labels say what is plotted: a share on x, a rate on y.
- Mark where each real group sits on its line, and say what the gap between them means.
- Calculate each overall rate as a weighted average first, then plot it.
- Another graph to consider: subgroup rates as dots sized by the number of applicants.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind 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 tablePick a published two-way table with counts, not just percentages. Say what the groups, subgroups and outcome are.
- Check the arithmeticCheck 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 weightsWrite each overall rate as a weighted average of subgroup rates and show which weights cause the difference.
- StandardiseRecompute 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 conditionsDerive when a reversal can happen for two groups and two subgroups, and test the conditions on invented tables.
- Go further, with a reasonAdd a second table, or simulate how often reversals occur. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter 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 sendCheck every table is labelled, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Conditional probability – Find and from counts, and compare each pair.
- Law of total probability – Write for each group and read the weights 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
Syllabus links
- 4.5Basic probability and expected frequency
- 4.6Venn and tree diagrams, combined events
- 4.11Conditional probability and independence
- 4.13Bayes' theoremHL
- 4.1Populations, samples, bias and sampling
- 4.5Basic probability and expected frequency
- 4.6Venn and tree diagrams, conditional probability
- 4.11Hypothesis tests: chi-squared and t-test
Read up first
Data and sources to start from
Use the published Berkeley admission counts by department and sex, and a second table such as the kidney stone treatment data.
References to cite
- Bickel, P. J., Hammel, E. A., & O'Connell, J. W. (1975). Sex bias in graduate admissions: Data from Berkeley. Science, 187(4175), 398–404.
- Charig, C. R., Webb, D. R., Payne, S. R., & Wickham, J. E. A. (1986). Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy. British Medical Journal, 292(6524), 879–882.
- Pearl, J. (2014). Understanding Simpson's paradox. The American Statistician, 68(1), 8–13.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
weights, not plain averages- 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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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.