Index Numbers

2.1 Core concepts

• An index number is a statistical device for measuring changes in the magnitude of a group of related variables; it represents the general trend of diverging ratios from which it is calculated and is a measure of the average change in a group of related variables over two different situations (NCERT §2, p. 92).
• Index numbers are conventionally expressed in percentage; the value in the base period is set at 100 and any other period's index is in proportion to it — an index of 250 means the value is two-and-a-half times the base (NCERT §2, p. 92).
• Price index numbers measure and permit comparison of prices of certain goods, while quantity index numbers measure changes in the physical volume of production, construction or employment; price indices are more widely used but a production index is also an important indicator of output (NCERT §2, p. 92).
• An index can be computed by the aggregative method or the method of averaging relatives (NCERT §3, p. 92).
• Simple aggregative price index: P₀₁ = (ΣP₁ ÷ ΣP₀) × 100. It is unweighted and of limited use because units of measurement of different commodities are not the same and the relative importance of items is not reflected (NCERT §3, p. 92).
• Weighted aggregative price index: P₀₁ = (ΣP₁q ÷ ΣP₀q) × 100. A well-specified basket is taken and its worth in each year computed, so the change is purely due to price change (NCERT §3, p. 93).
• Laspeyre's price index uses base-period quantities (q₀) as weights; it answers — if ₹100 was spent on the base-period basket, how much is needed in the current period for the same basket (NCERT §3, p. 93).
• Paasche's price index uses current-period quantities (q₁) as weights; it answers — if the current-period basket had been consumed in the base period at ₹100, how much is needed in the current period for the same basket (NCERT §3, pp. 93–94).
• Method of averaging relatives (simple): P₀₁ = (1/n) × Σ(P₁/P₀) × 100; the ratio (P₁/P₀) × 100 is the price relative (NCERT §3, p. 94).
• Weighted price relatives index: P₀₁ = Σ Wᵢ × (P₁ᵢ/P₀ᵢ × 100) ÷ ΣWᵢ; weights are usually the proportion of expenditure on each item in total expenditure in the base period (or current period); base-period weights are generally preferred for convenience and comparability (NCERT §3, p. 94).
• Consumer Price Index (CPI), also called the cost-of-living index, measures the average change in retail prices of a fixed basket. CPI for industrial workers (2001 = 100) of 277 in Dec 2014 means ₹100 of 2001 spending would need ₹277 to buy the same basket — what matters is the capability to buy it (NCERT §4, p. 95).
• The government compiles several CPIs — CPI-IW (base 2001 = 100), CPI for Agricultural Labourers (1986-87 = 100), CPI for Rural Labourers (1986-87 = 100), All-India Rural, Urban and Combined CPIs (2012 = 100); the RBI uses the All-India Combined CPI as the main measure of consumer-price change (NCERT §4, p. 96).
• CPI (2012 = 100) major-group weights (from CES 2011-12, NSS 68th Round): Food and beverages 45.86, Pan/tobacco/intoxicants 2.38, Clothing & footwear 6.53, Housing 10.07, Fuel & light 6.84, Miscellaneous 28.32 (NCERT §4, p. 96).
• Consumer Food Price Index (CFPI) equals the CPI 'Food and Beverages' group excluding alcoholic beverages and prepared meals/snacks/sweets (NCERT §4, p. 96).
• Wholesale Price Index (WPI) indicates the change in the general price level; unlike CPI it has no reference consumer category and excludes services (barber charges, repairing etc.) — only the prices of goods at the wholesale level are included (NCERT §4, pp. 96–97).
• WPI is now compiled with base 2011-12 = 100; major-group weights — Primary Articles 22.62, Fuel & Power 13.15, Manufactured Products 64.23; "All Commodities" is the Headline Inflation rate; "WPI Food Index" 24.23 (NCERT §4, p. 97).
• Index of Industrial Production (IIP), base 2011-12 = 100, is unlike CPI/WPI in that it measures quantities; it is a weighted arithmetic mean of quantity relatives using Laspeyre's formula, with weights proportional to value added by manufacture in the base year — IIP₀₁ = (Σq₁ᵢWᵢ ÷ ΣWᵢ) × 100 (NCERT §4, pp. 97–98).
• IIP sectoral weights: Mining 14.4, Manufacturing 77.6, Electricity 8.0; the Eight Core Industries (coal, crude oil, natural gas, refinery products, fertilisers, steel, cement, electricity) have a combined weight of 40.27% in IIP (NCERT §4, p. 98).
• SENSEX = Bombay Stock Exchange Sensitive Index, base 1978-79, comprising 30 stocks across 13 sectors of leading companies; a rising SENSEX indicates investor optimism about the economy (NCERT §4, pp. 98–99).
• Issues in construction of an index number: clarity of purpose; representative item selection; choice of a normal (non-extreme), not-too-distant base year that is routinely updated; choice of formula (Laspeyre vs Paasche differ only in weights used); reliability of data (NCERT §5, p. 99).
• Economic uses: CPI guides wage negotiation, income/price/rent/taxation policy; WPI eliminates the price effect on national income/capital formation and measures the rate of inflation; IIP gauges the industrial sector quantitatively; the Agricultural Production Index is a ready reckoner of agriculture; SENSEX guides stock-market investors (NCERT §6, pp. 99–100).
• Real values: Purchasing power of money = 1 ÷ cost-of-living index; Real wage = (Money wage ÷ Cost-of-living index) × 100. With CPI (1982 = 100) = 526 in Jan 2005, one rupee of 2005 is worth 100/526 ≈ ₹0.19 of 1982; ₹10,000 money wage equals ₹1,901 of 1982 purchasing power (NCERT §6, p. 100).
• Why a single number for many prices: an economy has thousands of commodities, each with its own price path; an index number compresses this vector of changes into one comparable scalar — enabling sentences like "prices rose 8% last year" that a non-statistician can understand (NCERT §1, p. 91).
• Why express as percentage of 100: setting the base to 100 makes any later value immediately readable — 110 = 10% rise, 90 = 10% fall, 250 = 2.5× the base. The "% of 100" convention removes the need to remember absolute base values (NCERT §2, p. 92).
• Aggregative vs averaging-of-relatives — conceptual contrast: the aggregative method sums prices (or quantities) directly across items and is sensitive to units of measurement (kg vs tonne, litre vs barrel); the relatives method first converts each item's price into a unit-free ratio P₁/P₀ and then averages — making it invariant to units, a major advantage (NCERT §3, pp. 92–94).
• Laspeyre upward-bias intuition: Laspeyre keeps the base-period basket fixed; consumers, in reality, substitute away from goods whose prices have risen most, so the fixed basket overstates the cost-of-living increase. Paasche has the opposite (downward) bias, using the current basket that already reflects substitution. The true index lies between them, motivating Fisher's "ideal" index (√(L × P)) — not in NCERT but a natural extension (NCERT §3, pp. 93–94).
• CPI-IW worked illustration (Table 7.4-style): five items — Food, Fuel, Cloth, Rent, Misc — with weights W and price relatives R. ΣWR / ΣW = 9786.85 / 100 = 97.86 → CPI of 97.86 (just below the base 100) suggests prices fell marginally; if instead ΣWR = 12500, CPI = 125, signalling a 25% rise (NCERT §3, p. 95).
• CPI-IW is used to determine Dearness Allowance (DA) for Central Government employees — a real-world payoff that makes the index politically salient. The CPI-IW base shift from 2001 to 2016 (recent) changed many DA calculations (NCERT §6, p. 99, contextual).
• WPI inflation = Headline Inflation in India: WPI is the most-quoted inflation measure in older RBI documents (the headline rate); but since 2014 the RBI's Monetary Policy Committee uses CPI-Combined as the inflation-targeting anchor (4% ± 2%) — a contemporary CUET fact often appearing in current-affairs hybrids (NCERT §4, pp. 96–97).
• IIP's role in business cycle dating: a sustained fall in IIP for two or more consecutive quarters is a warning of industrial recession; conversely, a rising IIP signals expansion. CUET sometimes pairs an IIP table with a "which sector contributed most" question (NCERT §4, pp. 97–98).
• Use-based IIP classification weights (Table 7.6, p. 98): Primary 34.1, Capital Goods 8.2, Intermediate 17.2, Infrastructure/Construction 12.3, Consumer Durables 12.8, Consumer Non-durables 15.3. The weights sum to 100 (within rounding) — a quick CUET check.
• SENSEX historical perspective: launched in 1986 with the 1978-79 financial-year base = 100, SENSEX is widely treated as a barometer of investor sentiment about Indian listed companies. A SENSEX of, say, 60000 means the basket is worth 600× its 1978-79 value — a graphic illustration of compounded capital-market returns (NCERT §4, pp. 98–99).
• Index-number formulas — quick reference: simple aggregative ignores quantity; Laspeyre fixes q₀; Paasche fixes q₁; simple average of relatives uses (1/n) Σ(P₁/P₀)×100; weighted relatives use expenditure shares as weights — five formulas, each with a one-line difference (NCERT §3, pp. 92–94).
• Issues in construction (NCERT §5): a good index must (i) have a clear purpose, (ii) include items representative of that purpose, (iii) use a normal, non-extreme base year that is updated periodically as consumer patterns shift, (iv) employ a formula suited to the data, and (v) rest on reliable underlying data (NCERT §5, p. 99). Each of these can become an MCQ on "which of the following is NOT an issue in index construction".
• Real wage worked example: a worker earns ₹20000/month in 2024 when CPI = 280 (base 2012). Real wage in 2012 prices = (20000/280) × 100 = ₹7142.86 — i.e., the 2024 nominal salary commands the same basket as ₹7143 would have done in 2012, exposing how much inflation has eroded purchasing power.
• Quantity vs price index — applications: agriculture and industry are measured by quantity indices (output, value-added) because the policy interest is in physical production; consumer welfare and wage bargaining are measured by price indices because what matters there is purchasing power; both belong to the same statistical family (NCERT §2, p. 92).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Table 7.1 (p. 92): four-commodity price data demonstrating the simple aggregative method.
• Table 7.2 (p. 93): base- and current-period prices and quantities of four commodities used to compute Laspeyre's (135.3) and Paasche's (132.1) indices.
• Table 7.3 (p. 94): weighted price relatives example — weighted index 156 vs unweighted 149, illustrating the impact of weighting.
• Table 7.4 (p. 95): CPI construction through a five-item basket (Food, Fuel, Cloth, Rent, Misc.) showing ΣWR = 9786.85 and CPI = 97.86.
• CPI (2012=100) weight chart (p. 96): Food & beverages 45.86, Misc. 28.32, Housing 10.07, Fuel & light 6.84, Clothing & footwear 6.53, Pan/tobacco/intoxicants 2.38.
• WPI (2011-12=100) major-group weights (p. 97): Primary Articles 22.62, Fuel & Power 13.15, Manufactured Products 64.23, WPI Food Index 24.23.
• Tables 7.5 & 7.6 (p. 98): IIP weight pattern — Mining 14.4, Manufacturing 77.6, Electricity 8.0; use-based — Primary 34.1, Consumer Non-durables 15.3, Consumer Durables 12.8, Intermediate 17.2, Infrastructure/Construction 12.3, Capital 8.2.
• Index construction flow: select purpose → choose items → choose base period → collect prices/quantities → select formula (aggregative vs relatives; weighted vs unweighted) → compute → interpret as percentage of 100.
• Worked Laspeyre vs Paasche numerical (Table 7.2 logic): four commodities with base-period prices p₀ = 4, 6, 5, 2 and quantities q₀ = 10, 5, 8, 15; current-period prices p₁ = 6, 8, 5, 3 and quantities q₁ = 5, 10, 8, 12. Σp₀q₀ = 40 + 30 + 40 + 30 = 140. Σp₁q₀ = 60 + 40 + 40 + 45 = 185. Laspeyre = (185/140) × 100 = 132.14. Σp₀q₁ = 20 + 60 + 40 + 24 = 144. Σp₁q₁ = 30 + 80 + 40 + 36 = 186. Paasche = (186/144) × 100 = 129.17. Both > 100 ⇒ prices have risen on average; Laspeyre slightly higher than Paasche, consistent with Laspeyre's known upward bias.
• Worked weighted-relatives example: three commodities with weights W = 50, 30, 20 (Σ = 100) and price relatives R = P₁/P₀ × 100 = 120, 140, 160. ΣWR = 50×120 + 30×140 + 20×160 = 6000 + 4200 + 3200 = 13400. Weighted index = 13400/100 = 134 — i.e., a 34% rise in the weighted basket cost.
• Worked CPI-based real-wage problem: money wage = ₹15000, CPI (base year) = 250. Real wage = (15000/250) × 100 = ₹6000 in base-year prices — students should be able to do this calculation in seconds for a CUET MCQ.
• Worked purchasing-power example: if CPI rises from 100 to 200 over a decade, the purchasing power of ₹1 halves — 100/200 = 0.5 — meaning the rupee buys only half what it did a decade ago, a vivid one-line summary of long-run inflation.
• Cross-check identity: when weights are expenditures (P₀q₀), the weighted-relatives method exactly equals Laspeyre's aggregative formula — proving that aggregative and relatives families coincide when consistently weighted. A subtle but high-yield NCERT identity (NCERT §3, p. 94, implicit).

2.5 Key formulas

2.4 Common confusions / NTA trap points

• Laspeyre vs Paasche weights: Laspeyre uses base-year (q₀) quantities; Paasche uses current-year (q₁) quantities. NTA often swaps the two.
• CPI vs WPI scope: CPI tracks retail prices for a specific consumer category; WPI tracks wholesale prices, has no consumer reference, and excludes services. Inflation is generally measured by WPI.
• Base years differ across indices — CPI-IW: 2001=100; CPI-AL & CPI-RL: 1986-87=100; Rural/Urban/Combined CPI: 2012=100; WPI: 2011-12=100; IIP: 2011-12=100; SENSEX: 1978-79.
• Largest weight in CPI (2012=100) is Food & beverages (45.86%) — followed by Miscellaneous (28.32%).
• IIP measures quantities, not prices, and uses Laspeyre's formula on quantity relatives with value-added weights.
• Eight Core Industries weight in IIP = 40.27%, not equal to Manufacturing's 77.6%.
• An index of 150 means a 50% rise, not a 150% rise.
• CFPI excludes alcohol and prepared meals/snacks from CPI Food & Beverages.
• RBI uses the All-India Combined CPI as its main consumer-price measure.
• SENSEX comprises 30 stocks across 13 sectors, not 50 or 100.
• Real wage formula has CPI in the denominator, not the numerator.
• Headline Inflation refers to the WPI All-Commodities rate.