2.4 Pie Charts & Proportional Representation

2.4 Pie Charts & Proportional Representation

A pie chart is a circular statistical graphic divided into slices (sectors) to illustrate numerical proportions. The arc length, central angle, and area of each sector are directly proportional to the quantity it represents.

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Geometric Principles of Circular Charts

A complete circle contains exactly $360^{\circ}$, corresponding to $100\%$ of the represented total. All proportional calculations derive from this fundamental geometric relationship.

Core Conversion Formulas:

1. Angle from Percentage: $$\text{Central Angle (Degrees)} = \frac{\text{Percentage}}{100} \times 360^{\circ} = \text{Percentage} \times 3.6^{\circ}$$
2. Percentage from Angle: $$\text{Percentage (\%)} = \frac{\text{Central Angle}}{360^{\circ}} \times 100 = \frac{\text{Central Angle}}{3.6}$$
3. Absolute Value from Angle and Total Budget: $$\text{Absolute Value} = \frac{\text{Central Angle}}{360^{\circ}} \times \text{Total Baseline Value}$$
4. Absolute Value from Percentage and Total Budget: $$\text{Absolute Value} = \frac{\text{Percentage}}{100} \times \text{Total Baseline Value}$$

Mental Estimation: Sector Benchmark Cheat Sheet

Memorizing these angular-to-percentage benchmarks allows you to verify options on the CSNT within seconds:

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The Multi-Pie Chart Baseline Trap

The most pervasive psychometric trap involving pie charts occurs when two separate pie charts are presented side-by-side representing different time periods or different departments.

[!CRITICAL] The Percentage Share Fallacy: A sector's visual size or percentage share does not indicate its absolute monetary value unless the baseline totals of the two charts are identical. If the total baseline expands, a smaller percentage slice can represent a larger absolute sum of money!

Consider the Cabinet Office Central Digital & Data Office (CDDO) budget across two consecutive financial years:

Table: Multi-Year Budget Transformation (2024/25 vs. 2025/26)

Critical Insights from the Comparison:

1. Cloud Infrastructure Paradox: In the 2024/25 pie chart, Cloud Infrastructure represented $35\%$ of the pie. In 2025/26, its share shrank to $25\%$ (a drop of $10$ percentage points, with its central angle contracting from $126^{\circ}$ to $90^{\circ}$). Visually, the slice appears substantially smaller. However, because the total budget expanded from £60m to £96m, actual cash expenditure on Cloud Infrastructure rose from £21.0m to £24.0m (a $+£3.0\text{m}$ increase, or $+14.29\%$)!
2. Constant Percentage vs. Growing Expenditure: Digital Skills maintained an identical $15.0\%$ share in both pie charts ($54^{\circ}$ sector). However, its funding expanded from £9.0m to £14.4m, an absolute gain of $+£5.4\text{m}$ ($+60\%$ growth), perfectly matching the expansion of the overall budget.

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Data Table Equivalence & CSNT Accessibility

The Civil Service states plainly that the Numerical Test "presents a range of tables and graphs to show numerical information" and that "there are alternative table versions available by default", because "questions do not rely on users viewing visual content as this content will always be provided in a table format". Consequently, on the CSNT, pie charts are paired with the underlying figures in tabular form. You will often need to take a percentage or angle from the pie chart to complete an empty cell in the accompanying table, and then use that completed table to answer a subsequent financial question.

Worked Step-by-Step Examples

Worked Example 1: Angle-to-Value Conversion

Scenario: On a pie chart representing the 2024/25 central digital budget of £60.0m, the sector for Legacy IT Remediation has a measured central angle of exactly $72^{\circ}$. How much funding was allocated to Legacy IT Remediation?

Step 1: Convert central angle to proportion of the circle

$$\text{Fraction of Circle} = \frac{72^{\circ}}{360^{\circ}} = \frac{1}{5} = 0.20\text{ (or 20\%)}$$

Step 2: Multiply proportion by total budget

$$\text{Funding} = 0.20 \times £60,000,000 = £12,000,000\text{ (or £12.0m)}$$

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Worked Example 2: Successive Multi-Chart Budget Shifts

Scenario: Using the multi-year budget table above, by what percentage did total expenditure on Cyber Security & Resilience increase between 2024/25 and 2025/26?

Step 1: Extract actual expenditure for both years

• 2024/25 Spend ($25\%$ of £60m) = $£15.0\text{m}$
• 2025/26 Spend ($30\%$ of £96m) = $£28.8\text{m}$

Step 2: Calculate absolute monetary increase

$$\Delta \text{Spend} = £28.8\text{m} - £15.0\text{m} = £13.8\text{m}$$

Step 3: Calculate percentage increase relative to the 2024/25 baseline

$$\text{Percentage Increase} = \frac{£13.8\text{m}}{£15.0\text{m}} \times 100 = 92.0\%$$

(Note: Although Cyber Security's budget share increased by only $5$ percentage points—from $25\%$ to $30\%$—its actual expenditure nearly doubled, surging by $92.0\%$ due to the compounding effect of the overall departmental budget increase).