LNAT Practice Passage 6: Averages
Read the passage below and answer the three LNAT-style multiple-choice questions. Full answers and explanations follow.
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Averages
A statistical puzzle known as Simpson's paradox occurs when a pattern found inside separate groups reverses after the groups are combined. Suppose two treatments are tested on mild and severe cases. Treatment X succeeds more often than Y among mild patients and also more often among severe patients. It can still have the lower success rate overall if doctors gave X mainly to severe cases and Y mainly to mild ones. Nothing in the arithmetic has contradicted itself. The surprise comes from forgetting that an overall percentage is weighted by the size of the groups from which it is built. The combined result depends both on how well each treatment performs and on how many easy or difficult cases each treatment receives. Edward Simpson described the form of this problem in 1951, although related reversals had been noticed earlier.
The same difficulty can appear when schools are compared. Consider two schools whose pupils begin with different levels of prior attainment.
| Pupil group | Pass results |
|---|---|
| Higher prior attainment | School R: 18/20 (90%); School S: 44/50 (88%) |
| Lower prior attainment | School R: 56/80 (70%); School S: 26/40 (65%) |
| All pupils | School R: 74/100 (74%); School S: 70/90 (78%) |
A league table using only the final row would rank School S above School R. Yet R has the higher pass rate among pupils with higher prior attainment and among those with lower prior attainment. The reversal occurs because the schools contain different proportions of the two groups: more than half of S's pupils are in the higher-attainment group, compared with only one fifth of R's. Since that group has higher pass rates in both schools, S's overall figure is pulled upwards. It would be easy to call one result "the real one", but the figures answer different questions. If we ask what proportion of all pupils passed, S did better. If we ask how pupils with similar starting attainment performed, R did better in each of the two categories shown. Nor does the second comparison automatically prove that R is the better school. Prior attainment may be an important difference to control for, but pupils also differ in subjects taken, absence, family circumstances and many other respects. A school might also influence which courses pupils enter, so even the apparently prior grouping may not be entirely independent of what the school does. Splitting the data into enough groups can make comparisons unstable or leave very few pupils in each. Simpson's paradox therefore warns against trusting an aggregate figure without asking what lies inside it; it does not provide a mechanical instruction to divide every dataset until the preferred answer appears.
School comparisons are vulnerable because a single percentage looks pleasantly decisive. Governments, parents and newspapers can rank institutions in seconds, while explaining why the percentages differ takes longer. A sensible comparison should begin with the question being asked rather than with whichever number produces the neatest order. A parent whose child entered with lower prior attainment might learn more from the 70% and 65% figures than from the two overall percentages. Another parent may still care about the school-wide result.
Questions
Question 1
Which of the following best explains why School S has a higher overall pass rate even though School R has a higher pass rate in both pupil groups?
- ASchool S must be more successful at moving pupils from the lower-attainment group into the higher-attainment group.
- BThe overall percentage is calculated differently from the percentages for the two pupil groups.
- CSchool R's higher-attainment pupils have less influence on its overall result because their individual pass rate is already high in percentage terms.
- DSchool S has a much larger proportion of pupils in the group which has the higher pass rate at both schools.
Question 2
What is the writer's main point about using overall pass rates to compare schools?
- AOverall pass rates should be replaced by comparisons between pupils who began with the same level of attainment, because those subgroup figures show how comparable pupils performed.
- BOverall pass rates answer a legitimate question, but may support a misleading conclusion if they are treated as showing how comparable groups of pupils performed.
- COverall pass rates are useful only when the schools being compared contain equal proportions of different types of pupil.
- DOverall pass rates should not be published because parents and newspapers are likely to interpret them incorrectly.
Question 3
Suppose Firm X promotes 80% of its senior employees and 40% of its junior employees, while Firm Y promotes 75% of its senior employees and 35% of its junior employees. Nevertheless, Firm Y has the higher promotion rate overall. According to the writer's argument, this could occur if:
- AFirm X employs a substantially larger proportion of junior employees than Firm Y, and junior employees are less likely to be promoted at both firms
- BFirm Y employs a substantially larger proportion of junior employees than Firm X
- Cthe percentages for at least one of the four employee groups must have been calculated incorrectly
- DFirm X must be applying a stricter promotion standard than Firm Y to at least one category of employee
Answers & explanations
Question 1: show answer and explanation
Correct answer: D.
School S has a much larger proportion of pupils in the group which has the higher pass rate at both schools.
Higher-attainment pupils have the higher pass rate in both schools. They make up 50 of School S's 90 pupils but only 20 of School R's 100. School S therefore contains many more pupils from the group in which passing is common, which can raise its overall percentage even though R performs slightly better within each group. Trap: The table contains no mathematical inconsistency. The reversal comes from the different composition of the two schools, not from the method used to calculate the percentages.
- A The table contains no information about pupils moving between the two attainment groups. The groups describe their starting attainment.
- B All of the figures are ordinary pass percentages. The apparent contradiction does not arise from using a different method of calculation.
- C The influence of a group on the overall percentage depends principally on how many pupils it contains, not on its pass percentage already being high. Higher-attainment pupils actually make up a much smaller share of School R than of School S.
- D Higher-attainment pupils have the higher pass rate in both schools. They make up 50 of School S's 90 pupils but only 20 of School R's 100. School S therefore contains many more pupils from the group in which passing is common, which can raise its overall percentage even though R performs slightly better within each group.
Question 2: show answer and explanation
Correct answer: B.
Overall pass rates answer a legitimate question, but may support a misleading conclusion if they are treated as showing how comparable groups of pupils performed.
The overall figure correctly answers the question "what proportion of all pupils passed?" On that measure S genuinely does better. The difficulty arises if this result is then treated as evidence that similar pupils fare better at S, because within each of the two attainment groups the opposite pattern appears. Trap: A is attractive because the subgroup figures reveal something the overall figure conceals. But the writer resists replacing one automatic rule with another.
- A The writer does regard prior-attainment comparisons as informative, but explicitly warns against simply declaring the subgroup result to be "the real one". Other differences between pupils may also matter, and dividing the data further is not automatically better.
- B The overall figure correctly answers the question "what proportion of all pupils passed?" On that measure S genuinely does better. The difficulty arises if this result is then treated as evidence that similar pupils fare better at S, because within each of the two attainment groups the opposite pattern appears.
- C Equal group proportions would remove this particular source of reversal, but the writer does not make equality of pupil composition a condition for publishing or using overall results.
- D The writer criticises overly simple interpretations of rankings, not the existence of the figures themselves. The final paragraph even acknowledges that some parents may legitimately care about the school-wide result.
Question 3: show answer and explanation
Correct answer: A.
Firm X employs a substantially larger proportion of junior employees than Firm Y, and junior employees are less likely to be promoted at both firms
Junior employees have the lower promotion rate at both firms. If Firm X contains many more junior employees while Firm Y contains proportionately more senior employees, the different workforce composition can pull X's overall promotion rate below Y's despite X having the higher promotion rate within both categories. This is the same type of reversal illustrated by the school table. Trap: The temptation is to assume that if X performs better in every stated subgroup, its overall figure must also be higher. Simpson's paradox shows why that inference fails when the proportions belonging to the groups differ.
- A Junior employees have the lower promotion rate at both firms. If Firm X contains many more junior employees while Firm Y contains proportionately more senior employees, the different workforce composition can pull X's overall promotion rate below Y's despite X having the higher promotion rate within both categories. This is the same type of reversal illustrated by the school table.
- B If Firm Y contained more of the lower-promotion junior group, that would tend to pull Y's overall rate down rather than explain why it was higher.
- C Simpson's paradox demonstrates precisely why all four subgroup percentages can be correct while the combined figures apparently point in the opposite direction.
- D The information given already says that X promotes a higher percentage within both categories. There is therefore no need to assume that it applies a stricter standard to either group in order to explain the overall reversal.