5.2 Pie Charts & Proportional Sector Expenditure

Proportional Visualisation in Irish Public Finance

A pie chart divides one total into non-overlapping sectors. The hypothetical budget examples below practise converting sector angles and shares into amounts, combining categories and comparing years with different totals. The total and category definitions matter as much as the visible slice sizes.

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The Angle-to-Percentage Conversion Engine

A full turn around a circle’s centre is $360^\circ$, representing the whole. Let $p$ be the numerical percentage (for example, $p=15$ for 15%) and let $\theta$ be the numerical angle measured in degrees. Then:

$\frac{p}{100}=\frac{\theta}{360}$ $\theta=3.6p,\qquad p=\frac{\theta}{3.6}=\frac{5\theta}{18}$

Thus 15% corresponds to 54 degrees. One degree corresponds to exactly $5/18$ of one percentage point, or about 0.2778% of the whole. Use numeric percentage values consistently: 15 multiplied by 3.6 gives 54; multiplying the fraction 0.15 by 3.6 would not give the angle.

Benchmark Angle Conversions

Memorizing key geometric benchmarks allows candidates to evaluate visual options instantly without tedious manual calculation:

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Calculating Sector Monetary Allocations

When a psychometric chart displays sector sizes as central angles or percentages alongside a consolidated financial total, calculating the euro allocation requires a single integrated operation:

$\text{Monetary Allocation} = \text{Total Budget} \times \left(\frac{\theta}{360}\right) = \text{Total Budget} \times \left(\frac{p}{100}\right)$

Step-by-Step Worked Example: Regional Transport Grant

A municipal authority receives a total regional development grant of €720 million. The infrastructure pie chart designates an angle of $54.0^\circ$ for Public Transport Modernisation.

1. Calculate the proportional share: $\frac{54.0^\circ}{360^\circ} = \frac{3}{20} = 0.15 = 15.0\%$
2. Multiply the total budget by the proportional share: $\text{Allocation} = \text{EUR }720{,}000{,}000 \times 0.15 = \text{EUR }108{,}000{,}000$

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The Multi-Year Comparative Fallacy: "Expanding Slice, Shrinking Pie"

Perhaps the most pervasive trap in civil service data interpretation involves comparing two pie charts representing different calendar or fiscal years. Candidates frequently assume that an increase in a sector's percentage share guarantees an increase in its monetary funding.

Mathematical Proof of the Fallacy

Let $S_1$ and $S_2$ represent positive decimal shares (such as 0.20 for 20%) of a sector in Year 1 and Year 2, and let $B_1$ and $B_2$ represent the total budgets: $\text{Euro Spend}_1 = B_1 \times S_1$ $\text{Euro Spend}_2 = B_2 \times S_2$

If the total budget contracts ($B_2 < B_1$), the monetary spend will decrease even when the percentage share expands ($S_2 > S_1$), provided: $\frac{B_2}{B_1} < \frac{S_1}{S_2}$

Civil Service Case Study: Departmental Budget Consolidation

Consider the operational budget of an executive agency across two fiscal years during an expenditure reallocation:

• Year 1 Total Budget: €50.0 million. Information Technology represents a $72.0^\circ$ sector ($20.0\%$). $\text{IT Spend (Year 1)} = \text{EUR }50.0\text{m} \times 0.20 = \text{EUR }10.0\text{m}$
• Year 2 Total Budget: Reduced to €40.0 million. Information Technology's slice expands to $81.0^\circ$ ($22.5\%$). $\text{IT Spend (Year 2)} = \text{EUR }40.0\text{m} \times 0.225 = \text{EUR }9.0\text{m}$

Notice the divergence:

• The central angle expanded by $+9.0^\circ$ ($+2.5$ percentage points).
• The monetary allocation decreased by €1.0 million (a $-10.0\%$ funding contraction).

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Irish Public Finance Example: Gross Voted Expenditure

Consider a hypothetical expenditure table comparing gross voted departmental allocations across two consecutive years:

Analytical Takeaways from the Data Display

1. Social Protection: Its slice shrank from 30.0% ($108.0^\circ$) to 28.0% ($100.8^\circ$). Yet its euro funding increased from €24.00 billion to €24.64 billion (+€0.64 billion) because total public expenditure expanded by 10% (from €80B to €88B).
2. Education: Its percentage share remained strictly constant at 15.0% ($54.0^\circ$). Its funding grew in exact proportion to the national budget (+10.0%, or +€1.20 billion).
3. Combined Portfolios: Combining Social Protection and Health in Year 1 accounted for $108.0^\circ + 90.0^\circ = 198.0^\circ$ ($55.0\%$ of the national budget, or €44.00 billion). In Year 2, their combined angle remained $100.8^\circ + 97.2^\circ = 198.0^\circ$ ($55.0\%$, or €48.40 billion).

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Residual Slices and Data Limitations

A pie chart may combine smaller categories into "Other" or "Miscellaneous". Candidates must respect data sufficiency boundaries:

• Slices marked "Other" cannot be presumed to contain any specific program unless an accompanying explanatory footnote explicitly details its contents.
• A change in an unspecified sub-scheme is generally unknown from the combined slice alone. Check whether additional figures or explicit bounds settle the question; otherwise do not infer how the residual total is split.