7.4 Distractor Analysis, Unit Verification & Sanity Checking

7.4 Distractor Analysis, Unit Verification & Sanity Checking

A wrong multiple-choice option can correspond to an intermediate total, an inverted ratio, a wrong percentage base or a unit error. The hypothetical examples here show how to recognise those errors. Their positions among the options provide no evidence of correctness or of an exam publisher’s item-writing policy.

Four Calculation Errors to Check

1. Partial Calculation Traps

Candidates complete the first one or two steps of a multi-stage problem, obtain an intermediate figure, spot that number among the answer choices, and immediately select it. Item writers routinely place intermediate values as distractors.

• Administrative Example: A question asks for the "net budget after a 15% efficiency saving on a €600,000 sub-head". The correct net total is €510,000. The distractor €90,000 (the efficiency deduction itself) would answer the saving amount rather than the requested net total.

2. Inverted Operations and Ratio Reversals

Occurs when candidates divide instead of multiply, or invert the numerator and denominator in ratio, exchange rate, or per-capita calculations.

• Administrative Example: In computing caseworker caseload capacity (e.g., 450 cases across 18 officers $\implies 450 / 18 = 25$ cases per officer), the inverted distractor $18 / 450 = 0.04$ or an operational error like $450 \times 18 = 8{,}100$ would have the wrong meaning or unit for the requested result.

3. False Base and Denominator Confusion

A pervasive trap in percentage change and proportional analysis. Candidates divide the change by the new final value ($T_1$) rather than the historical baseline value ($T_0$), or calculate a fraction of the gross fund instead of the "remaining unallocated balance".

• Administrative Example: Capital expenditure rises from €400,000 in 2024 to €500,000 in 2025. The true increase is $\frac{100{,}000}{400{,}000} = +25\%$. The false base distractor uses 2025 as the denominator: $\frac{100{,}000}{500{,}000} = 20\%$.

4. Boundary and Off-by-One Traps

In multiannual projections or indexing questions, candidates miscount the number of growth intervals (e.g., counting both endpoints: from 2020 to 2025 represents 5 annual growth periods, not 6).

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Terminal Digit (Units Digit) and Parity Checks

When an item requires calculating multi-digit products or sums without an onscreen calculator, terminal-digit analysis can eliminate incompatible integer answers. A matching units digit is a necessary check, not usually a complete calculation.

Modulo 10 Terminal Arithmetic

For integer arithmetic, use residues modulo 10. For a non-negative integer this is its ordinary units digit; for subtraction interpret the residue modulo 10 (for example, $3-5\equiv8\pmod{10}$). The rules are:

$\text{Units}(A \times B) = \text{Units}(\text{Units}(A) \times \text{Units}(B))$ $\text{Units}(A \pm B) = \text{Units}(\text{Units}(A) \pm \text{Units}(B))$

Tactical Multiplication Application

Evaluate the total cost of purchasing 367 specialized IT hardware units at €483 each:

1. Units digit of 367 is 7.
2. Units digit of 483 is 3.
3. The units digit of the product must equal $\text{Units}(7 \times 3) = \text{Units}(21) = \mathbf{1}$.
4. If the options are €177,258; €177,261; €177,264; €177,266, only €177,261 ends in 1. It is the only units-digit-compatible choice; direct multiplication confirms €177,261.

Parity Filtering (Even vs. Odd)

Parity rules provide an instantaneous filter for arithmetic expressions:

• $\text{Even} \times \text{Any Integer} = \text{Even}$.
• $\text{Odd} \times \text{Odd} = \text{Odd}$.
• $\text{Even} \pm \text{Even} = \text{Even}$; $\text{Odd} \pm \text{Odd} = \text{Even}$; $\text{Even} \pm \text{Odd} = \text{Odd}$.

An even integer multiplied by an integer rate gives an even integer product in those units. A fractional euro rate does not justify the same conclusion about whole euros: two people at €0.50 gives €1.00. Convert to suitable integer units, such as cents, before using parity.

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Metric and Unit Verification Traps

Civil service tabular exhibits present data across various scales and dimensions. Overlooking unit headings is a major vulnerability under time pressure.

Scale Mismatches (€k vs. €m vs. €bn)

A table column headed "Expenditure (€ 000s)" or "Capital Grants (€ Millions)" requires scaling table entries before combining them with unscaled raw data:

• Misreading 42.5 in a "€ Millions" column as €42,500 instead of €42,500,000 creates a thousand-fold calculation error.
• Always circle or mentally highlight column header units before extracting numbers.

Temporal Conversions

• Annual vs. Monthly: Sum the actual monthly amounts. Multiplication or division by 12 assumes twelve equal periods; an annual total does not prove uniform monthly spending.
• Weekly vs. Annual: Use the number of paid or worked weeks supplied. A 52-week exercise is not proof that every calendar or payroll year contains exactly 52 payment dates.
• Working Days vs. Calendar Days: A stipulated 52-week, five-day schedule gives 260 gross weekdays. Holidays, leave, year boundaries and service schedules can change usable working days. Use the supplied net days rather than treating 260 or 365 as a universal work-year denominator.

Demographic Ratios

Rates expressed "per 1,000 population" or "per 100,000 population" (common in public health and crime statistics exhibits) must never be confused with simple percentages (which represent rates per 100).

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Plausibility & Administrative Sanity Checks

Before confirming an answer, evaluate whether the result makes sense within the real-world context of Irish public administration:

• Salary Checks: Check the scale against the given data. If the exercise stipulates €36,000 annual pay, a derived monthly figure should be €3,000 under twelve equal payments, not €30,000. Do not substitute a remembered grade salary for the stated value.
• Headcount Checks: Can personnel numbers be negative or fractional? If calculating whole-time equivalents (WTE), fractions are normal (e.g., 14.5 WTE); point-in-time counts of physical people are whole numbers, while an average headcount over time can be fractional.
• Directional Sanity: If a departmental efficiency review reduces administrative costs by 8%, the final expenditure must be strictly smaller than the baseline. If your calculated figure is larger, a directional sign error has occurred.

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A Brief Pre-Confirmation Check

When useful within the available time, check the requested metric, units and direction. This is a practice routine, not an official five-second rule:

[Pre-Confirmation Check]
     │
     ├── Check 1: Question Verification
     │   └── Did I answer the exact metric requested?
     │       (Net total vs. saving? % increase vs. % point diff?)
     │
     ├── Check 2: Unit Alignment
     │   └── Do the units match the prompt?
     │       (€ vs. €k vs. €m? Annual vs. monthly? Per 1,000 vs. %?)
     │
     └── Check 3: Magnitude & Direction Sanity
         └── Does the number make administrative sense?
             (Is it in the right ballpark? Did an increase actually go up?)

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Psychometric Distractor Anatomy & Verification Matrix

The following reference table details common distractor profiles and their diagnostic counters: