3.3 Index Numbers & Relative Performance Measures

3.3 Index Numbers & Relative Performance Measures

Statistical reporting in the UK public sector frequently presents expenditure, caseload growth, and cost inflation in the form of index numbers. Published by bodies such as the Office for National Statistics (ONS) and HM Treasury, index numbers standardize complex time-series data so that policymakers can evaluate trends without being distracted by currency scale or population size.

In the Civil Service Numerical Test, questions involving index numbers are designed to test your ability to convert indexed figures into actual cash flows, calculate accurate percentage changes between non-base periods, and deflate nominal budgets into real-terms purchasing power.

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The Concept and Formula of an Index Number

An index number expresses the value of a variable in any given period relative to its value in an agreed base period, which is assigned an arbitrary benchmark value of 100.0.

$\text{Index Number in Period } t = \left( \frac{\text{Value in Period } t}{\text{Value in Base Period } 0} \right) \times 100$

Properties of Index Numbers

1. Base Period Value is Always 100: If 2020 is chosen as the base year, $\text{Index}_{2020} = 100.0$.
2. Direct Percentage Change from Base: If the operational cost index in 2024 stands at $124.6$, costs have increased by exactly $124.6 - 100.0 = 24.6%$ relative to the 2020 base year.
3. Unit Neutrality: Index numbers have no units (£, cases, or tonnes); they are pure ratios multiplied by 100.

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The Non-Base Year Percentage Change Trap

The single most pervasive error candidates make in the CSNT is confusing index points with percentage change when comparing two non-base periods.

⚠️ The Golden Rule of Non-Base Calculations

When comparing Period A and Period B, where neither period is the base period ($100$): $\text{Percentage Change} = \left( \frac{\text{Index}_B - \text{Index}_A}{\text{Index}_A} \right) \times 100\%$ Never simply subtract the two index numbers! Simple subtraction gives the difference in index points, which is not equal to the percentage change.

Mathematical Demonstration of the Trap

Suppose the Civil Service Travel & Subsistence Cost Index (Base Year 2020 = 100.0) displays the following series:

• Year 2022: Index = 110.0

• Year 2024: Index = 126.5

• The Incorrect Calculation (The Distractor): $\Delta = 126.5 - 110.0 = 16.5\% \quad \text{[INCORRECT]}$ $16.5$ is the change in index points, not the percentage change.

• The Correct Calculation: $\text{Percentage Change} = \left( \frac{126.5 - 110.0}{110.0} \right) \times 100\% = \left( \frac{16.5}{110.0} \right) \times 100\% = 15.0\% \quad \text{[CORRECT]}$

Test writers always include $16.5%$ as an attractive distractor choice. A candidate who subtracts index points directly will immediately select this distractor.

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Deflation: Converting Nominal Spend into Real Terms

In government accounting, financial figures are reported either in nominal terms (the actual cash paid in that year) or in real terms (spending adjusted for inflation to reflect constant purchasing power).

When general inflation causes prices to rise, a department whose nominal budget rises by 5% might actually experience a decline in real frontline purchasing power.

The Deflation Formula

To convert nominal cash expenditure into constant base-year prices using a price index (such as the GDP Deflator or Consumer Prices Index):

$\text{Real Expenditure (in Base Period Prices)} = \left( \frac{\text{Nominal Expenditure}}{\text{Price Index}} \right) \times 100$

$\text{Nominal Expenditure} = \frac{\text{Real Expenditure} \times \text{Price Index}}{100}$

Worked Public Sector Scenario: Ministry of Justice Court Administration

To see how the CSNT tests index numbers and deflation, consider this 5-year longitudinal review of court administration expenditure within the Ministry of Justice (MoJ):

Table: MoJ Court Administration Nominal vs. Real Expenditure (Base Year 2020 = 100.0)

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Step-by-Step Analysis of the 2024 Outturn

1. Calculate Real Spend in 2024

In 2024, the department spent £489.6 million in cash, and the Public Sector Price Index reached 127.5: $\text{Real Spend}_{2024} = \left( \frac{\text{Nominal Spend}}{\text{Price Index}} \right) \times 100 = \left( \frac{£489.6\text{m}}{127.5} \right) \times 100 = £384.00\text{ million}$

2. Evaluate the 4-Year Trend (2020 to 2024)

• Headline Nominal Change: $\text{Nominal \% Change} = \left( \frac{£489.6\text{m} - £400.0\text{m}}{£400.0\text{m}} \right) \times 100\% = \frac{+£89.6\text{m}}{£400.0\text{m}} \times 100\% = +22.40\%$ To the public, the court system received a substantial cash budget increase of +22.4% (+£89.6 million).

• True Real-Terms Change (Purchasing Power in Constant 2020 Pounds): $\text{Real \% Change} = \left( \frac{£384.0\text{m} - £400.0\text{m}}{£400.0\text{m}} \right) \times 100\% = \frac{-£16.0\text{m}}{£400.0\text{m}} \times 100\% = -4.00\%$ In real terms, despite the headline £89.6 million cash injection, the court administration's operational purchasing power actually shrank by 4.00% because inflation (+27.5%) outpaced budget growth (+22.4%).

This discrepancy between nominal and real trends is a classic theme in Civil Service numerical reasoning tests.

Composite Index Numbers in Public Procurement

Government commercial teams frequently construct a composite price index to track inflation across a procurement basket containing diverse goods and services. A composite index uses a weighted average of individual component indices:

$\text{Composite Index} = \frac{\sum (\text{Index}_i \times \text{Weight}_i)}{\sum \text{Weight}_i}$

Where weights represent the proportion of expenditure allocated to each category (often summing to 1.0 or 100%).

Worked Example: DWP Regional Office Consumables Basket

Suppose the Department for Work and Pensions tracks an operational consumables basket comprising three expense categories:

To compute the composite procurement index for Year 3: $\text{Composite Index} = (110.0 \times 0.30) + (124.0 \times 0.50) + (105.0 \times 0.20)$ $\text{Composite Index} = 33.0 + 62.0 + 21.0 = 116.0$

Because the baseline index was 100.0, an index of 116.0 represents an overall weighted procurement cost increase of exactly 16.0% across the 3-year period.

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Comparison: Nominal vs. Real Performance Summary