Index numbers: meaning, uses, types and steps in construction

1. Meaning and definitions

An index number is a statistical device that measures the relative change in a variable or group of related variables over time, place or other characteristic. It is expressed as a percentage of a base, the base being 100.

• Croxton and Cowden: “Index numbers are devices for measuring differences in the magnitude of a group of related variables.”
• Spiegel: “An index number is a statistical measure designed to show changes in a variable or a group of related variables with respect to time, geographic location or other characteristics.”

Features: expressed in percentages (base = 100, but the % sign is dropped); a measure of relative change, so items in different units (rice per kg, cloth per metre) can be combined; a special kind of average; measures changes that cannot be observed directly (the “general price level” or “cost of living”).

2. Uses (importance)

1. Economic barometers: they measure the pulse of the economy — price levels, industrial production, share prices.
2. Measure the purchasing power of money and convert nominal values into real values (deflating): real wage = money wage ÷ CPI × 100.
3. Wage and dearness-allowance fixation: consumer price indices are used to adjust wages, pensions and allowances.
4. Policy-making: government and the central bank watch price indices to decide fiscal and monetary measures; the RBI’s inflation target under the RBI Act, 1934 is set in terms of consumer price inflation.
5. Business decisions: forecasting demand, fixing prices, studying trends.
6. Comparison of places and periods (cost of living in two cities).
7. Indexation in law and contracts: escalation clauses in long contracts, and the cost inflation index for capital gains under income-tax law.

3. Kinds

• Price index: changes in prices (wholesale price index, consumer price index).
• Quantity index: changes in physical volume (index of industrial production, agricultural production).
• Value index: changes in total value, Σp₁q₁ / Σp₀q₀ × 100 (sales, exports).
• Special-purpose indices: cost of living, share price (Sensex, Nifty).

By method: unweighted (simple) — simple aggregative and simple average of price relatives; weighted — weighted aggregative (Laspeyres, Paasche, Fisher and others) and weighted average of relatives (family budget method). By base: fixed base and chain base (each year on the previous year).

4. Steps in construction (January 2026 essay)

1. Define the purpose. A cost-of-living index for factory workers needs different items, prices and weights from a wholesale index.
2. Select the base period. It should be a normal year, free from war, famine, boom or depression, and not too distant. The chain base is an alternative.
3. Select the commodities. Representative items, of standard quality, in reasonable number, consumed by the group concerned.
4. Collect prices. From representative markets and sources, retail or wholesale as the purpose requires, at regular intervals.
5. Choose the average. Arithmetic mean is common; the geometric mean is theoretically better (it satisfies the time reversal test for relatives).
6. Assign weights. Items differ in importance (rice matters more than salt); weights may be quantities or expenditure shares.
7. Select the formula. Laspeyres, Paasche or Fisher according to data and purpose; Fisher is the “ideal”.

5. Simple (unweighted) methods: worked

Using the January 2026 prices (base and current) for A, B, C, D: p₀ = 12, 4, 9, 12; p₁ = 20, 4, 12, 24.

Simple aggregative: P₀₁ = Σp₁ / Σp₀ × 100 = 60/37 × 100 = 162.16.

Simple average of price relatives: P₀₁ = Σ(p₁/p₀ × 100) / n.

P₀₁ = 600 / 4 = 150.00.

The two results differ because the simple aggregative is dominated by high-priced items and by the units in which prices are quoted. Neither allows for the importance of the items; that is why weighted indices (next note) are preferred.

6. Problems and limitations

• They are only approximate indicators of change; errors creep in at every step.
• Choice of items, base year, weights and formula is subjective, and different choices give different answers.
• Changes in quality, tastes and new products are hard to reflect; a fixed basket becomes dated.
• An index made for one purpose cannot be used for another (a wholesale index does not measure the cost of living of students).
• International and inter-regional comparisons are unreliable because consumption patterns differ.
• They can be manipulated by a convenient base year.

Tests of adequacy (Irving Fisher): unit test (the index must not depend on the units of price); time reversal test; factor reversal test; circular test. The TRT and FRT are worked in the next note.

How this is examined

“Define index number; explain the different steps involved in the construction of index number” (January 2026); “What is index number? Explain the uses” (June 2025); “Explain the importance and limitations” (January 2025); “What is index number? Explain the advantages” (August 2024, six marks); “Explain the types of index number” (January 2025, six marks).