7.1 Statistical Tables: Rapid Extraction, Percentages, and Proportions

7.1 Statistical Tables: Rapid Extraction, Percentages, and Proportions

The Aptitude Test (AT) of the Common Recruitment Examination is a high-speed, 45-minute screening examination comprising 35 multiple-choice questions. Among its five core disciplines, the Interpretation of Tables and Graphs module accounts for 8 questions (approximately 23% of the paper). The CSB's guidance notes allow a permitted calculator (cordless, silent, and without printing, graphic or word-display, or dictionary functions) in the Aptitude Test. Even so, keying in every figure costs time. Strong candidates extract the right cells, estimate to eliminate options, and use the calculator only for the final step.

With an average allocation of only 77.1 seconds per question, success on this module does not depend on laborious, multi-decimal precision. The CSB's own sample uses a labour-force table of unemployment by age group, much like an official statistical publication. The questions test your capacity to isolate the correct data cells rapidly, identify underlying mathematical relationships, apply estimation shortcuts, and reject mathematically flawed distractors.

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The Anatomy of Complex Statistical Tables

Statistical tables in this part resemble real policy briefings. Unlike elementary school math problems with isolated figures, CRE tables feature multi-row, multi-column matrices containing mixed metrics: absolute counts, financial values, index points, and percentage distributions. A typical table exhibits four distinct structural components:

1. The Universe & Title: Defines the geographic scope, target population, and temporal window (e.g., "Quarterly Composite Household Survey: Labor Force and Median Monthly Employment Earnings by Selected Economic Activity, 2021–2025").
2. Column and Row Headers: Categorical taxonomies often organized in hierarchical tiers (e.g., broad economic sector $\rightarrow$ sub-industry $\rightarrow$ gender breakdown).
3. Units and Denominations: Crucial scale markers prominently placed at the top-right corner of the table or within individual column headers (e.g., "In thousands unless otherwise specified", "HK$ Millions", or "Percentage share (%)").
4. Footnotes and Qualifier Symbols: Notations marked by asterisks, daggers, or lowercase letters providing critical exclusion rules, provisional estimates, or base-year shifts.

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The 15-Second Pre-Reading Protocol

Untrained candidates immediately read the question stem, glance frantically at the table, and attempt to search for matching numbers. This undisciplined approach leads to cognitive overload and misreading. High-scoring candidates execute the 15-Second Pre-Reading Protocol before reading Question 1 of any tabular data set:

[Step 1: Scope Scan]     --> Read Title & Time Horizon (e.g., 2021-2025 fiscal years)
[Step 2: Units Check]    --> Verify Measurement Units (HK\$ Millions vs Thousands; % vs Counts)
[Step 3: Axis Inventory] --> Identify Rows (Variables/Sectors) vs Columns (Years/Demographics)
[Step 4: Footnote Alert] --> Read Footnotes for Revisions, Exclusions, or Base Index Years

By spending 15 seconds establishing the mental geography of the exhibit, you avoid most misread row and column errors and know where to retrieve values when reading the stems.

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Core Mental Arithmetic & Shortcut Formulas

Whether or not you use a calculator, you need the core formulas of statistical analysis and the shortcuts that make them fast.

1. Percentage Change

The standard formula for percentage change measures the relative shift from a baseline period to a terminal period:

$\text{Percentage Change} = \left( \frac{\text{Value}_{\text{New}} - \text{Value}_{\text{Old}}}{\text{Value}_{\text{Old}}} \right) \times 100\% = \left( \frac{\Delta V}{V_{\text{Old}}} \right) \times 100\%$

2. Percentage Points vs. Percentage Change

A distinction that table and graph questions often test is the difference between an absolute change in percentage rates and a relative percentage change:

• Percentage Points (p.p.): The simple arithmetic difference between two percentage values: $\Delta_{\text{p.p.}} = \text{Rate}_{\text{New}} - \text{Rate}_{\text{Old}}$
• Relative Percentage Change (%): The proportional growth of the rate relative to its initial value: $\text{Relative Change} = \left( \frac{\text{Rate}_{\text{New}} - \text{Rate}_{\text{Old}}}{\text{Rate}_{\text{Old}}} \right) \times 100\%$

Illustration: If the unemployment rate in a district rises from $4.0\%$ in 2022 to $5.0\%$ in 2023:

• The rate rose by $1.0$ percentage point ($5.0\% - 4.0\% = 1.0\text{ p.p.}$).
• The rate rose by $25.0\%$ in relative terms ($\frac{5.0 - 4.0}{4.0} \times 100\% = \frac{1.0}{4.0} \times 100\% = 25.0\%$).
• Confusing these two descriptions is among the most common distractor traps in civil service aptitude tests.

3. Base-Period Indexing

Statistical tables frequently express economic time series as an index, where a designated baseline period is assigned a value of $100.0$:

$\text{Index}_{t} = \left( \frac{\text{Value}_{t}}{\text{Value}_{\text{Base}}} \right) \times 100$

To find the percentage change between any two non-base periods $t_1$ and $t_2$, you cannot simply subtract the index points unless $t_1$ is the base year! For any two arbitrary years:

$\text{Percentage Change}_{t_1 \rightarrow t_2} = \left( \frac{\text{Index}_{t_2} - \text{Index}_{t_1}}{\text{Index}_{t_1}} \right) \times 100\%$

4. Estimation Shortcuts

Estimation is often faster than keying numbers into a calculator, and it is the quickest way to eliminate options:

The Official Sample

The CSB's sample table shows unemployment by age group from 1992 to 2000. Bold figures are the numbers of unemployed persons in thousands, and the figures in brackets are unemployment rates. A note defines the unemployment rate as the number unemployed in an age group divided by the labour force of the same age group, times 100%.

• Question 1: During the period from 1992 to 2000, which age group consistently had the highest unemployment rate? Compare the bracketed rates within each year: the 15-19 group is highest in every year, so the answer is 15-19. The absolute numbers are a distractor, because the 20-29 group has the most unemployed people.
• Question 2: Estimate the total number of persons in the labour force in 1999. The overall row shows 57.3 thousand unemployed at a 1.9% rate. Labour force = 57,300 ÷ 1.9% ≈ 3,015,789, so the nearest option is 3,020,000.

The sample shows the two habits this part rewards: read the note (units and definitions), and reverse a formula when the question asks for the base rather than the rate.

Realistic Examination Scenario & Worked Examples

To practise statistical interpretation problems, examine the following synthetic data set modeled on employment and earnings surveys conducted across major industries in Hong Kong.

Exhibit 1: Employment and Earnings in Selected Economic Sectors (2022 vs. 2025)

(Note: Data modeled for civil service examination preparation purposes)

Footnote: Persons engaged counts are rounded to the nearest hundred persons. Median monthly earnings include basic salary, cost-of-living allowances, and year-end bonuses, rounded to the nearest hundred HK dollars.

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Worked Problem 1: Percentage Change in Sectoral Employment

Question: Which economic sector experienced the highest percentage growth in persons engaged between 2022 and 2025?

Step-by-Step Mathematical Derivation:

1. Isolate Target Data Cells: We examine the "Persons Engaged" columns for 2022 and 2025 across all four sectors.
2. Compute Absolute Changes ($\Delta V = V_{2025} - V_{2022}$):
   • Financing and Insurance: $297.0 - 270.0 = +27.0$ thousand
   • Information and Communications: $138.0 - 120.0 = +18.0$ thousand
   • Retail, Accommodation & Food Services: $475.0 - 500.0 = -25.0$ thousand (Contracted, eliminate immediately)
   • Public Administration & Social Services: $451.0 - 410.0 = +41.0$ thousand
3. Compute Percentage Change ($\frac{\Delta V}{V_{2022}} \times 100\%$):
   • Financing and Insurance: $\frac{27.0}{270.0} \times 100\% = \frac{1}{10} \times 100\% = 10.0\%$
   • Information and Communications: $\frac{18.0}{120.0} \times 100\% = \frac{18}{120} = \frac{3}{20} = 15.0\%$
   • Public Administration: $\frac{41.0}{410.0} \times 100\% = \frac{1}{10} \times 100\% = 10.0\%$
4. Distractor Analysis: Many candidates mistakenly choose Public Administration & Social Services because its absolute gain ($41{,}000$ persons) was the largest in the table. However, because its baseline was also large ($410{,}000$), its percentage growth was only $10.0\%$. Information and Communications achieved a $15.0\%$ growth rate, making it the highest.

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Worked Problem 2: Sector Share and Percentage Point Shift

Question: By how many percentage points did the employment share of Financing and Insurance within the total selected sectors change between 2022 and 2025?

Step-by-Step Mathematical Derivation:

1. Determine 2022 Employment Share: $\text{Share}_{2022} = \frac{270.0}{1{,}300.0} \times 100\% = \frac{27}{130} \times 100\%$ Mental division: $130 \times 0.20 = 26.0$. Remaining $1.0 / 130 \approx 0.0077$. Thus, $\text{Share}_{2022} \approx 20.77\%$.
2. Determine 2025 Employment Share: $\text{Share}_{2025} = \frac{297.0}{1{,}361.0} \times 100\%$ Mental division: $1{,}361 \times 0.20 = 272.2$. Difference: $297.0 - 272.2 = 24.8$. $24.8 / 1{,}361 \approx 0.0182$. Thus, $\text{Share}_{2025} \approx 21.82\%$.
3. Calculate Percentage Point Difference: $\Delta_{\text{p.p.}} = 21.82\% - 20.77\% = +1.05\text{ percentage points}$
4. Distractor Analysis: A distractor will often offer $\frac{21.82 - 20.77}{20.77} \times 100\% \approx +5.06\%$. That represents the relative percentage change in share, not the change in percentage points requested by the stem.