Data and statements: what must be true

What the item type asks

Bocconi describes its Data/Statements items as giving some data and some statements, and asking you to decide whether each statement is true or false. In this course's checks the same skill appears in a single-answer form: "Which statement is necessarily true?" or "Which statement must be false?", with four options. Either way, the job is the same: judge each statement only against the data given.

Three verdicts, not two

Given a set of data, a statement can be:

Verdict	Meaning	How you show it
Necessarily true	It holds in every situation consistent with the data	A short proof from the data
Necessarily false	It fails in every situation consistent with the data	A contradiction with the data
Not determined	It holds in some consistent situations and fails in others	Two examples, one each way

The middle ground is where most marks are lost. A statement that could be true is not "true"; a statement that is not necessarily true is not "false". When a question asks which statement must be true, every "not determined" statement is a wrong option, however likely it seems.

The counterexample habit

For each statement, try to build a situation that fits all the data but breaks the statement. If you succeed, the statement is not necessarily true. If you try to break it and the data stop you, find the reason, and you have your proof. The same habit run the other way finds necessarily false statements: try to build a situation where the statement holds; if the data always forbid it, it must be false.

Tool 1: overlapping groups

When the data give the sizes of two groups inside a population of $N$, the overlap is pinned between two limits:

$$\max(0,\ |A| + |B| - N) \le |A \cap B| \le \min(|A|, |B|).$$

The lower limit comes from the fact that the two groups cannot fit into $N$ people without sharing at least the excess. If the data also say how many are in neither group, the overlap is fixed exactly:

$$|A \cap B| = |A| + |B| - (N - \text{neither}).$$

Without the "neither" number, any statement that needs an exact overlap is not determined.

Tool 2: orderings

For rankings, schedules and seating, list the fixed facts first, then enumerate. With four or five items there are rarely more than a handful of arrangements that satisfy the data, and writing them out takes less time than reasoning in your head. A statement true in every arrangement is necessarily true; one true in none is necessarily false; anything else is not determined.

Tool 3: shares and amounts

A share is a percentage of a total, and totals change. "The export share fell" does not mean exports fell; it depends on what happened to the total. Convert every share into an amount on one base (call last year's total 100 or $R$) and compare amounts. This is the same habit that data insight questions reward.

Reading quantifiers exactly

• "Some" means at least one; it does not mean "not all".
• "Most" means more than half.
• "At least 15" allows 15 and any larger number; "exactly 15" allows only 15.
• "No more than" and "at most" are the same limit.

Statements are often wrong only because of their quantifier: the data force "at least 15", the statement says "exactly 15".

A routine (ours)

1. Rewrite the data in compact form on your scratch sheet: a Venn diagram, a line of positions, or a small table of amounts.
2. Work out what the data force (fixed values, limits, arrangements) before reading the statements.
3. For each statement, ask "necessarily?" and look for a counterexample.
4. Use only the data. Real-world likelihood is irrelevant.