10.1 Multi-Source Analysis, Set Data & Arrangement

Combine sources before calculating

Multi-source data analysis means answering a question using information distributed across a passage, table or graphic. The difficult step is often deciding which figures describe the same population. Adding unrelated totals quickly produces a precise but meaningless answer. Before any arithmetic, identify the entity, unit, time period, category definition and relationship between the sources.

For the written CBT, practise translating each source into a small common working table. Read chart labels as carefully as numerical values. “Trips,” “passengers,” “tonnes” and “tonne-kilometers” measure different quantities. Similarly, a quarterly total cannot be compared directly with one month's total without an explicit conversion or additional information. A statement that one quantity rose does not establish why it rose.

Worked example: passage plus table

The following dataset is fictional. A depot handles two classes of shipments. A passage states that each Class A shipment carries 12 tonnes and each Class B shipment carries 8 tonnes. All recorded shipments are completed, and no shipment appears in both classes.

January's shipment count is $40+30=70$, while its tonnage is $40(12)+30(8)=720$. February has 75 shipments and $50(12)+25(8)=800$ tonnes. March has 80 shipments and $45(12)+35(8)=820$ tonnes. Keep both count and mass in separate columns.

Across the quarter there are $70+75+80=225$ shipments carrying $720+800+820=2340$ tonnes. The average load per shipment is $2340/225=10.4$ tonnes. It is not the unweighted mean of 12 and 8, because the numbers of shipments differ. Nor is it the mean of the three monthly averages unless monthly shipment counts happen to be equal.

Tonnage rises from 720 to 800 between January and February. The increase is 80 tonnes, so the percentage increase is $80/720\times100\approx11.11\%$. Shipment count rises by only $5/70\times100\approx7.14\%$. These answers differ because the composition changed. The table supports the numerical comparison; it does not establish a real-world cause such as better staff productivity.

Use Venn diagrams for overlapping data

A Venn diagram records membership, not physical location or a required hierarchy. For two sets A and B:

$$|A\cup B|=|A|+|B|-|A\cap B|$$

The overlap is subtracted once because adding the two totals counted it twice. If a survey of 120 people finds 70 using A, 55 using B and 30 using both, A-only is 40, B-only is 25, and at least one is 95. The remaining 25 use neither. Check: $40+30+25+25=120$.

“At least one” includes the overlap. “Exactly one” excludes it and equals $70+55-2(30)=65$. “Both” means intersection, not union. A question about the share of A-users who also use B uses denominator 70, giving $30/70$; the share of all surveyed people using both uses denominator 120, giving $30/120$.

For three sets, add individual totals, subtract pairwise intersections and add the triple intersection back. Check whether a phrase such as “A and B” includes people in all three sets or means “A and B only.” If 10 belong to A and B inclusive and 4 belong to all three, the A-and-B-only region is 6. Work from the innermost region outward rather than subtracting overlaps repeatedly without labeling them.

Arrange data using constraints

Data arrangement converts scattered conditions into an organized table. Suppose four fictional consignments P, Q, R and S use four distinct slots numbered 1–4. P is before Q, R is immediately after P, S is not last, and Q is in slot 4.

Fix Q in slot 4. The adjacent P–R block could occupy slots 1–2 or 2–3. If P–R occupies 1–2, S takes slot 3. If P–R occupies 2–3, S takes slot 1. Both arrangements satisfy every clue. Therefore, P's exact slot is not determined, while Q's slot is determined. Recording one valid table is not proof of a unique solution.

If an extra clue says S is immediately before Q, then S must occupy slot 3 and P–R must occupy 1–2. The unique order is P, R, S, Q. Notice how the added clue resolves the uncertainty. Do not import unstated rules about shipment priority or railway practice to fill a blank.

A reliable source-merging routine

1. List the exact quantity requested, including its unit and denominator.
2. Extract only the relevant facts from each source.
3. Establish whether categories overlap, are disjoint or are exhaustive.
4. Convert compatible units and periods before combining numbers.
5. Calculate and test totals against the original sources.
6. State whether the requested answer is unique or whether information is missing.

A final inference should be no stronger than the evidence. “February carried more tonnage” follows from the fictional table. “February was more profitable” does not follow without revenue and cost data. Treating absent data as zero, confusing percentages with counts, or selecting one arbitrary arrangement are reasoning errors even when the arithmetic is otherwise correct.