Lesson 35 of 42 · Critical thinking
Drawing valid conclusions from data and statements
Find the one statement the data guarantees: orderings, overlaps, averages and percentages, each tested for must-be-true rather than could-be-true.
About 22 minutes
In this lesson
- Separate statements that must be true from statements that could be true.
- Organise orderings and overlapping groups so that what is fixed and what can vary is visible.
- Combine averages and percentages from groups of different sizes correctly.
The idea
The format
The other critical thinking format Bocconi lists for the standard test gives some data or statements and asks you to choose which statement is true (or which is false). Unlike the three-verdict items, several candidate statements are offered, and your task is to find the one the data guarantees.
Must be true, not could be true
The decisive distinction is between a statement that must be true given the data and one that merely could be true. Several options will usually be consistent with the data; only one will be forced by it. For each option ask: is there any arrangement of the facts, consistent with everything given, in which this statement is false? If there is, it is not the answer to a "must be true" question.
For a "which is false" question, flip the test: the answer is the statement that cannot be true in any arrangement consistent with the data. A statement that merely might be false is not enough.
Orderings
Many items give comparisons: A sold more than B, C sold less than A, and so on. Turn them into a vertical line, highest at the top, and place each item only where the data forces it. Comparisons chain (if A is above B and B is above C, then A is above C), but an item compared with only one other may float. If all you know about an item is that it is below Neri, it could sit anywhere below Neri. Mark its possible positions rather than picking one.
Overlaps
When two groups come from the same total, their overlap has limits even if it is not given. If 130 of 200 firms export and 110 have a website, at least firms do both, because otherwise the two groups would need more than 200 firms between them. The overlap is at most 110, the size of the smaller group. Anything from 40 to 110 is possible, so "at least 40 do both" must be true, while "exactly 40 do both" could be true but need not be.
Averages and totals
An average is a total divided by a count. When data gives averages for groups of different sizes, the overall average is not the simple average of the group averages: go back to totals. Group averages also say nothing about individuals: an average salary of €2,600 in one office and €2,100 in another does not mean that everyone in the first earns more than everyone in the second.
Percentages of different bases
A percentage is always a percentage of something. If 30% of the men and 30% of the women in a firm work part-time, then 30% of everyone does. But if 30% of the north office and 50% of the south office do, the overall rate lies somewhere between 30% and 50%, depending on the sizes of the offices. Without the sizes, a statement fixing the overall rate cannot be guaranteed.
Working method
- Organise the data: a line for orderings, a quick count for overlaps, totals for averages.
- Before reading the options, note what is fixed and what can vary.
- Test each option against the parts that can vary. An option that depends on one of them is not guaranteed.
- Choose the option that holds in every arrangement. If two seem to, re-read both: one usually depends on an assumption you made without noticing.
Why this format rewards care
The wrong options in these items are rarely absurd. Each is usually true in at least one arrangement, often the first arrangement that comes to mind. That is why picking an arrangement and checking options against it fails: you need to check them against all the arrangements the data allows.
Worked examples
Example 1: An ordering with a floating item
Five funds are ranked by last year's return, with no ties. Alfa beat Beta. Gamma beat Alfa. Delta did worse than Beta. Epsilon did worse than Alfa. Which statement must be true: (1) Gamma had the highest return; (2) Delta had the lowest return; (3) Epsilon beat Delta?
Chain the fixed comparisons: Gamma above Alfa, Alfa above Beta, Beta above Delta. So Gamma, Alfa, Beta, Delta from top to bottom.
Epsilon is known only to be below Alfa. It could sit between Alfa and Beta, between Beta and Delta, or below Delta.
Statement 1: everyone else is below Gamma (Alfa directly, Beta and Delta through Alfa, Epsilon through Alfa). It must be true.
Statement 2 fails if Epsilon is below Delta; statement 3 fails in exactly that case too. Both could be true but need not be.
Answer
Only statement 1 must be true. Statements 2 and 3 are true in the arrangement most people draw first, which is why they tempt.
Example 2: How many firms do both?
Of 200 firms surveyed, 130 export and 110 have a website. Which must be true: (1) at least 40 firms both export and have a website; (2) exactly 40 do both; (3) more firms export than have a website?
Minimum overlap: . If fewer than 40 did both, the exporters and the website owners together would exceed 200 firms.
Maximum overlap: 110, if every firm with a website also exports. Any value from 40 to 110 fits the data.
Statement 1 holds for every possible overlap. Statement 2 picks one value from the range. Statement 3 compares 130 with 110 directly: it is given by the data.
Answer
Statements 1 and 3 must be true; statement 2 could be true but need not be. In a single-answer item, "exactly 40" is the classic wrong option: it treats the minimum as the actual value.
Example 3: Average of two groups
Branch 1 has 30 staff with an average monthly salary of €2,000. Branch 2 has 20 staff with an average of €2,500. What is the average salary across both branches?
Go back to totals: Branch 1 pays euros; Branch 2 pays euros.
Overall: euros.
Check: the answer lies between 2,000 and 2,500 and nearer to 2,000, because Branch 1 is larger.
Answer
€2,200. The simple average of the two averages, €2,250, is the tempting wrong answer.
Common traps
Could be true taken for must be true
An option fits the arrangement you drew, so you choose it. But the data may allow other arrangements. Before choosing, try to build one in which the option fails.
The minimum overlap taken as the overlap
From 130 and 110 out of 200 you work out 40 and pick "exactly 40". The data fixes only a range. "At least" statements are guaranteed; "exactly" statements usually are not.
Averaging averages
Two group averages are averaged as if the groups were the same size. Multiply each average by its group size, add, and divide by the total count.
From averages to individuals
A higher average in one group is read as "everyone in that group earns more". Averages describe groups, not their members.
Check yourself
Answer each question before you open its answer.
Question 1: Ranking five sales agents
Critical thinking
Five sales agents, Fabbri, Greco, Leone, Moro and Neri, each closed a different number of contracts in March. Greco closed more than Fabbri but fewer than Neri. Moro closed more than Neri. Leone closed fewer than Neri.
Which of the following statements must be true?
- A.Leone closed fewer contracts than Greco.
- B.Moro closed the most contracts.
- C.Fabbri closed the fewest contracts.
- D.Greco ranked third.
Show the answer
Answer: B. Moro closed the most contracts.
Working
Fixed chain: Moro above Neri, Neri above Greco, Greco above Fabbri. Leone is below Neri but otherwise free: above Greco, between Greco and Fabbri, or below Fabbri. Everyone is below Neri except Moro, and Moro is above Neri, so Moro is at the top in every arrangement. The statements about Leone, Fabbri and Greco each fail in at least one arrangement.
Why the other options are wrong
- A. Leone is known only to be below Neri; Leone could sit between Neri and Greco, above Greco.
- C. Leone could be below Fabbri, in which case Leone closed the fewest.
- D. Greco is third only if Leone is below Greco. If Leone is between Neri and Greco, Greco is fourth.
What carries over
Place each item only where the data forces it, and test every option against all the positions a floating item can take.
Question 2: Students who passed both
Critical thinking
All 120 students in a course sat two exams. 85 passed the economics exam and 70 passed the statistics exam.
Which of the following statements must be true?
- A.Exactly 35 students passed both exams.
- B.At least 15 students failed both exams.
- C.At least 35 students passed both exams.
- D.At least half the students passed both exams.
Show the answer
Answer: C. At least 35 students passed both exams.
Working
The two pass groups add up to places among 120 students, so at least students are in both. The overlap can be as large as 70 (every statistics pass also passed economics). So "at least 35" holds in every case; "exactly 35" and "at least 60" do not. The number failing both is minus those who passed at least one, which can be anything from 0 to 35.
Why the other options are wrong
- A. 35 is the smallest possible number who passed both. The number could be anything from 35 to 70.
- B. 15 is the difference between the two pass counts, not a count of failures. If every student passed at least one exam, none failed both.
- D. Half the course is 60. The overlap could be as low as 35, so this need not hold.
What carries over
Two groups from the same total overlap by at least their sum minus the total; the exact overlap is usually not fixed.
Question 3: Salaries at two branches
Critical thinking
| Branch | Employees | Average monthly salary (€) |
|---|---|---|
| Riva | 40 | 2,100 |
| Sole | 10 | 2,600 |
Which of the following statements must be true?
- A.Every employee at Sole earns more than every employee at Riva.
- B.Sole's total monthly salary bill is larger than Riva's.
- C.The average monthly salary across both branches is €2,350.
- D.The average monthly salary across both branches is €2,200.
Show the answer
Answer: D. The average monthly salary across both branches is €2,200.
Working
Totals: Riva euros, Sole euros. Overall average: euros. The result sits nearer Riva's average because Riva is the larger branch.
Why the other options are wrong
- A. Averages describe groups, not individuals; a Riva employee could earn more than a Sole employee.
- B. Sole's bill is euros against Riva's euros.
- C. This is the simple average of the two averages, which ignores that Riva has four times as many employees.
What carries over
Combine group averages through totals, weighting each group by its size.
Key points
Choose the statement that holds in every arrangement the data allows, not the one that fits your first drawing.
Two groups from one total overlap by at least their sum minus the total; "exactly" is rarely guaranteed.
Combine averages through totals, and never read a group average as a fact about each member.
All questions here are original and written for UniExamer; none comes from the test maker.
Written by the UniExamer team with AI assistance. Every worked example and check question was re-solved by an independent checker before publication. No subject expert has reviewed this lesson yet. If you find a mistake, write to support@uniexamer.com.