In short
An index number is a statistical device that measures the relative change in a variable, or a group of variables, between a current period and a chosen base period, expressed as a percentage of the base. It converts many unlike items — wheat in quintals, cloth in metres, electricity in kWh — into one comparable number, which is why it is the standard tool for measuring inflation, industrial growth, real wages and cost of living. In Indian practice, index numbers drive the CPI (the RBI's inflation-targeting anchor), the WPI, the IIP and the dearness allowance of government employees. The topic carries guaranteed marks in BA Economics quantitative methods papers and appears in UGC-NET, Indian Economic Service, RBI Grade B, bank PO and analyst placement tests, usually as a Laspeyres/Paasche/Fisher computation or a deflating problem.
Explained
What an index number is and how it is classified
- An index number is a weighted average of relatives: it expresses the current value as a ratio to the base value × 100. Base period index = 100 by construction.
- By variable measured: price index (CPI, WPI), quantity/volume index (IIP, agricultural production index), value index (value of exports), and special-purpose indices (stock price index like Nifty 50, human development-type composite indices).
- By scope: wholesale/producer level versus retail/consumer level; general index versus group index (food, fuel, manufactured products).
- By number of items: simple (unweighted — all items given equal importance) versus weighted (each item given a weight reflecting its economic importance).
- By base: fixed base (all periods compared with one base year) versus chain base (each period compared with the preceding one).
- Index numbers are *specialised averages*: they measure net relative change in a group of magnitudes that are not directly addable, and they are expressed in no unit — they are pure numbers.
Steps in constructing an index number
- Define the purpose — a cost-of-living index for industrial workers needs a different basket from a wholesale price index. Purpose decides every later choice.
- Select the base period — it must be normal (no war, famine, hyperinflation), not too distant, and may be a single year or an average of years. Base revision is periodic.
- Select items (the basket) — representative, comparable over time, reasonably stable in specification, and limited in number; selection is usually by sampling from a consumer expenditure survey.
- Collect price data — from the same markets, same grade, same unit, at the same time of month; decide whether to use retail or wholesale prices.
- Choose weights — quantity weights or value (expenditure) weights, taken from the base year, the current year or a typical year.
- Choose the average — arithmetic mean is common and simple; geometric mean is theoretically superior for ratios (it is reversible and less affected by extreme relatives) but harder to compute.
- Choose the formula — simple aggregative, simple average of relatives, Laspeyres, Paasche, Fisher, Marshall-Edgeworth, Dorbish-Bowley, Walsh, Kelly.
Unweighted and weighted formulae
- Simple aggregative index: total of current prices ÷ total of base prices × 100. Defect: it is dominated by high-priced items and depends on the units in which each item is quoted.
- Simple average of price relatives: compute p₁/p₀ × 100 for each item, then average (AM or GM). Unit-free, but treats a matchbox as important as rice.
- Laspeyres (base-year weights, q₀): the most used in practice because base-year quantities need to be collected only once. It has an upward bias — consumers substitute away from items whose prices rose most, so the old basket overstates the cost increase.
- Paasche (current-year weights, q₁): requires fresh quantity data every period, which is costly. It has a downward bias — the current basket already reflects substitution away from dearer goods.
- Fisher's Ideal index: geometric mean of Laspeyres and Paasche. Called "ideal" because it uses both base- and current-year weights, uses the GM, and satisfies both the time reversal and factor reversal tests.
- Marshall-Edgeworth: weights are (q₀ + q₁); satisfies time reversal, nearly satisfies factor reversal.
- Dorbish-Bowley: arithmetic mean of Laspeyres and Paasche; fails both time reversal and factor reversal (the AM of the two indices and the AM of their reciprocals multiply to (L+P)²/4LP, which exceeds 1 unless L = P).
- Walsh index: weights are the geometric mean √(q₀q₁); satisfies time reversal.
- Kelly's fixed-weight aggregative: weights q are quantities of some typical period, held fixed across many periods — the practical form used by official agencies between base revisions.
Tests of adequacy of an index number formula
- Unit test: the formula must be independent of the units in which prices or quantities are quoted. All formulae except the simple aggregative satisfy it.
- Time reversal test: P₀₁ × P₁₀ = 1 (omitting 100). The index must work backwards consistently. Satisfied by Fisher, Marshall-Edgeworth, Walsh, simple GM of relatives, simple aggregative and Kelly; failed by Laspeyres, Paasche and Dorbish-Bowley.
- Factor reversal test: P₀₁ × Q₀₁ = V₀₁, i.e. the price index multiplied by the quantity index must equal the true value index. Only Fisher's ideal index satisfies it (among the common formulae).
- Circular test: P₀₁ × P₁₂ × P₂₀ = 1 — an extension of time reversal that allows base shifting without recomputation. Satisfied only by the simple aggregative, the simple GM of price relatives and Kelly's fixed-weight index; Fisher fails it.
- Because Fisher satisfies both time and factor reversal, it is theoretically best; because it needs current quantities every period, official agencies still use Laspeyres or Kelly.
Consumer price index and the cost-of-living index
- A cost-of-living index (CLI) measures the change in money a specified class of consumers must spend to maintain a given standard of living. The CPI is its operational form.
- Two methods: (a) aggregative expenditure method — identical to Laspeyres with base-year quantities; (b) family budget / weighted average of relatives method — weights are base-year expenditures (W = p₀q₀) applied to price relatives. The two give the same answer.
- India's CPI series (NSO, Ministry of Statistics and Programme Implementation): CPI (Rural), CPI (Urban) and CPI (Combined), base 2024 = 100 (weights from the Household Consumption Expenditure Survey 2023-24), published monthly; the new series runs from January 2026 data (first released 12 February 2026), and the old 2012 = 100 series ends at December 2025, with MoSPI linking factors to join the two. This is the inflation measure the RBI targets under the flexible inflation-targeting framework — 4% CPI inflation with a ±2% tolerance band, mandated under the amended RBI Act.
- CPI-IW (Industrial Workers) is compiled by the Labour Bureau, base 2016 = 100, and is the index used to fix dearness allowance for central government employees and PSU workers, and for minimum-wage revision under the Code on Wages (the four Labour Codes came into force on 21 November 2025, replacing the Minimum Wages Act, 1948 among others). Separate series exist for agricultural and rural labourers (CPI-AL/RL, base 1986-87 = 100).
- WPI (Office of the Economic Adviser, DPIIT), base 2022-23 = 100 since 15 June 2026 (replacing the 2011-12 series; first print for May 2026), covers primary articles, fuel & power and manufactured products at the wholesale stage; it excludes services and does not include GST/indirect taxes. India's first official Producer Price Indices were launched alongside it on the same date.
- IIP (NSO, base 2022-23 = 100 since 1 June 2026, first used for April 2026; earlier 2011-12) is a quantity index of industrial production with mining, manufacturing and electricity; its use-based classification splits output into primary, capital, intermediate, infrastructure, consumer durable and consumer non-durable goods.
Base shifting, splicing, deflating and uses
- Base shifting: new index = old index of the year ÷ old index of the new base year × 100. Used when the base becomes too old or when comparing two series.
- Splicing: joining an old discontinued series to a new one by dividing the old series by (old index value in the overlap/link year ÷ 100) to put it on the new base, or scaling the new series up by (old index of the link year ÷ 100) to put it on the old base.
- Deflating: real (constant-price) value = money value ÷ price index × 100. Real wage, real income and real GDP are all deflated nominal figures; the GDP deflator = (nominal GDP ÷ real GDP) × 100.
- Purchasing power of the rupee = 100 ÷ price index, i.e. the reciprocal of the index.
- Inflation rate from an index = (Iₜ − Iₜ₋₁)/Iₜ₋₁ × 100. Year-on-year compares the same month a year earlier; base effect means a low index in the earlier period mechanically raises the current inflation print.
- Limitations: no index is unique or perfectly accurate; arbitrary choice of base, basket and formula; quality change and new goods are not captured (substitution bias and quality bias); the basket becomes unrepresentative between revisions; a general index can mislead for a particular household; errors in sampling and price collection; it measures net relative change only, not absolute levels.
Key points
- Index number - a pure number measuring relative change in one or more variables against a base period taken as 100.
- Base period - the reference period; must be normal, recent and stable; base index is always 100.
- Price relative - p₁/p₀ × 100 for a single item; the building block of unweighted indices.
- Simple aggregative - ratio of price totals; the only common formula that fails the unit test.
- Laspeyres - base-year quantity weights (q₀); cheap to compute, has upward bias.
- Paasche - current-year quantity weights (q₁); costly data, has downward bias.
- Fisher's ideal - GM of Laspeyres and Paasche; satisfies time reversal and factor reversal.
- Marshall-Edgeworth - weights (q₀+q₁); satisfies time reversal, close on factor reversal.
- Kelly's index - fixed weights of a typical period; satisfies the circular test, used officially.
- Time reversal test - P₀₁ × P₁₀ = 1; checks consistency when base and current period swap.
- Factor reversal test - P₀₁ × Q₀₁ = V₀₁; satisfied only by Fisher.
- Circular test - chaining across periods returns to 1; Fisher fails it, Kelly and simple aggregative pass.
- Cost-of-living index - money needed to maintain a fixed standard of living; computed by aggregative expenditure or family budget method.
- CPI (Combined), base 2024=100 - NSO's retail inflation index and the RBI's 4% ±2% target variable; new base from January 2026 data (old 2012 series to December 2025).
- CPI-IW, base 2016=100 - Labour Bureau index used for dearness allowance and minimum-wage revision.
- WPI, base 2022-23=100 (from June 2026) - wholesale-stage index excluding services and indirect taxes.
- Deflating - real value = money value ÷ price index × 100; purchasing power of the rupee = 100 ÷ index.
- Chain base - link relatives of successive years multiplied cumulatively to give a chain index.
Formulas and examples
Price relative = (p₁ / p₀) × 100 — p₀, p₁ = price of the item in base and current period (₹ per unit)
Simple aggregative index P₀₁ = (Σp₁ / Σp₀) × 100
Simple average of price relatives (AM) = (1/N) Σ (p₁/p₀) × 100; (GM) = antilog[ Σ log(p₁/p₀ × 100) / N ]
Laspeyres PL = (Σ p₁q₀ / Σ p₀q₀) × 100 — q₀ = base-year quantity
Paasche PP = (Σ p₁q₁ / Σ p₀q₁) × 100 — q₁ = current-year quantity
Fisher's ideal PF = √( PL × PP ) = √[ (Σp₁q₀ · Σp₁q₁) / (Σp₀q₀ · Σp₀q₁) ] × 100
Marshall-Edgeworth PME = [ Σ p₁(q₀+q₁) / Σ p₀(q₀+q₁) ] × 100
Dorbish-Bowley PDB = (PL + PP)/2; Walsh PW = [ Σ p₁√(q₀q₁) / Σ p₀√(q₀q₁) ] × 100
Kelly's fixed weights PK = (Σ p₁q / Σ p₀q) × 100, q = quantity of a typical period
Cost-of-living (family budget) CLI = Σ(P·W) / ΣW, where P = price relative, W = p₀q₀ (₹)
Value index V₀₁ = (Σ p₁q₁ / Σ p₀q₀) × 100; Quantity index (Laspeyres form) Q₀₁ = (Σ q₁p₀ / Σ q₀p₀) × 100
Time reversal: P₀₁ × P₁₀ = 1; Factor reversal: P₀₁ × Q₀₁ = V₀₁ (indices as ratios, not ×100)
Real value = (Money value / Price index) × 100; Purchasing power of ₹1 = 100 / Price index
Inflation rate (%) = [(Iₜ − Iₜ₋₁) / Iₜ₋₁] × 100; Base shifting: new index = (old index / old index of new base year) × 100
Example 1 — Laspeyres, Paasche, Fisher, Marshall-Edgeworth from one table. Four commodities, 2020 (base) and 2026 (current):
| Item | p₀ (₹) | q₀ | p₁ (₹) | q₁ |
|---|---|---|---|---|
| Rice (kg) | 30 | 10 | 45 | 8 |
| Wheat (kg) | 20 | 5 | 30 | 6 |
| Oil (litre) | 100 | 2 | 150 | 2 |
| Sugar (kg) | 40 | 4 | 50 | 5 |
Step 1 — the four sums:
- Σp₀q₀ = 300 + 100 + 200 + 160 = 760
- Σp₁q₀ = 450 + 150 + 300 + 200 = 1,100
- Σp₀q₁ = 240 + 120 + 200 + 200 = 760
- Σp₁q₁ = 360 + 180 + 300 + 250 = 1,090
Step 2 — indices:
- Laspeyres = 1,100/760 × 100 = 144.74
- Paasche = 1,090/760 × 100 = 143.42
- Fisher = √(144.74 × 143.42) = √20,758.3 = 144.08
- Marshall-Edgeworth = (1,100+1,090)/(760+760) × 100 = 2,190/1,520 × 100 = 144.08
- Dorbish-Bowley = (144.74 + 143.42)/2 = 144.08
Step 3 — interpretation: prices rose about 44% over 2020-26. Laspeyres > Paasche, as theory predicts.
Example 2 — Fisher satisfies both reversal tests (same data).
- Time reversal: P₀₁ = √[(1,100 × 1,090)/(760 × 760)] = √(1,199,000/577,600) = √2.0758 = 1.44078. P₁₀ = √[(Σp₀q₁ · Σp₀q₀)/(Σp₁q₁ · Σp₁q₀)] = √[(760 × 760)/(1,090 × 1,100)] = √0.48174 = 0.69407. Product = 1.44078 × 0.69407 = 1.000 ✓
- Factor reversal: Q₀₁ = √[(Σq₁p₀ · Σq₁p₁)/(Σq₀p₀ · Σq₀p₁)] = √[(760 × 1,090)/(760 × 1,100)] = √0.99091 = 0.99545. P₀₁ × Q₀₁ = 1.44078 × 0.99545 = 1.43423. True value index V₀₁ = Σp₁q₁/Σp₀q₀ = 1,090/760 = 1.43421 ✓ (equal to rounding).
- Check Laspeyres fails time reversal: PL₀₁ = 1,100/760 = 1.44737; PL₁₀ = Σp₀q₁/Σp₁q₁ = 760/1,090 = 0.69725; product = 1.00918 ≠ 1.
Example 3 — Cost-of-living index by the family budget method. A worker's base-year monthly expenditure and price rise:
| Group | W = base expenditure (₹) | p₀ | p₁ | P = p₁/p₀ × 100 | P·W |
|---|---|---|---|---|---|
| Food | 6,000 | 100 | 135 | 135 | 810,000 |
| Rent | 3,000 | 100 | 120 | 120 | 360,000 |
| Clothing | 1,000 | 100 | 150 | 150 | 150,000 |
| Fuel & light | 1,500 | 100 | 160 | 160 | 240,000 |
| Miscellaneous | 1,000 | 100 | 125 | 125 | 125,000 |
| Total | 12,500 | 1,685,000 |
- CLI = 1,685,000 / 12,500 = 134.8
- Interpretation: the worker needs 34.8% more money for the same standard of living. To keep real income unchanged, salary must rise from ₹12,500 to 12,500 × 1.348 = ₹16,850 per month.
Example 4 — Deflating a wage series and real income.
| Year | Money wage (₹/month) | CPI (2012 = 100) | Real wage at 2012 prices (₹) |
|---|---|---|---|
| 2021 | 25,000 | 160 | 25,000/160 × 100 = 15,625 |
| 2024 | 32,000 | 185 | 32,000/185 × 100 = 17,297 |
| 2026 | 36,000 | 210 | 36,000/210 × 100 = 17,143 |
- Money wage rose 44% from 2021 to 2026 (25,000 → 36,000), but real wage rose only from ₹15,625 to ₹17,143, i.e. 9.7%.
- Between 2024 and 2026 the money wage rose 12.5% while prices rose 13.5%, so the real wage fell by 0.9%.
- Purchasing power of the rupee in 2026 relative to 2012 = 100/210 = ₹0.476 — the rupee buys less than half of what it did in 2012.
Example 5 — Chain base index and base shifting. Link relatives (each year on the previous year) for 2021-2025: 105, 108, 104, 110, 106.
- Chain index (2020 = 100): 2021 = 105.0; 2022 = 105 × 108/100 = 113.4; 2023 = 113.4 × 104/100 = 117.94; 2024 = 117.94 × 110/100 = 129.73; 2025 = 129.73 × 106/100 = 137.51.
- Shift the base to 2022 (= 100): divide each by 113.4 × 100 → 2020 = 88.18, 2021 = 92.59, 2022 = 100, 2023 = 104.00, 2024 = 114.40, 2025 = 121.26.
- Inflation in 2025 = (137.51 − 129.73)/129.73 × 100 = 6.0% (as the link relative said, 106).
| Formula | Weights | Unit test | Time reversal | Factor reversal | Circular |
|---|---|---|---|---|---|
| Simple aggregative | none | No | Yes | No | Yes |
| Simple GM of relatives | none | Yes | Yes | No | Yes |
| Laspeyres | q₀ | Yes | No | No | No |
| Paasche | q₁ | Yes | No | No | No |
| Fisher ideal | both (GM) | Yes | Yes | Yes | No |
| Marshall-Edgeworth | q₀+q₁ | Yes | Yes | Nearly | No |
| Dorbish-Bowley | both (AM) | Yes | No | No | No |
| Kelly | fixed q | Yes | Yes | No | Yes |
Where it is used
- Monetary policy and macro research: the RBI's Monetary Policy Committee sets the repo rate against CPI (Combined) inflation; economists at RBI, NITI Aayog, finance ministry and rating agencies (CRISIL, ICRA, CARE) build CPI/WPI/IIP nowcasts.
- Official statistics: NSO/MoSPI and the Labour Bureau compile CPI, WPI, IIP and CPI-IW; roles include price-collection supervision, basket revision from the Household Consumption Expenditure Survey and base-year revision.
- Compensation and HR: dearness allowance for central and state government staff and PSUs is linked to CPI-IW; minimum-wage revision under the Code on Wages and long-term wage settlements use the same index.
- Finance and capital markets: Nifty 50 and Sensex are weighted stock price indices (free-float market-capitalisation weighted); index funds and ETFs replicate them; inflation-indexed bonds and CPI-linked swaps need index arithmetic.
- Contracts and construction: price-escalation clauses in EPC and government contracts are indexed to WPI/CPI components of steel, cement and labour; NHAI and railway contracts use published index series.
- Business analytics and consulting: deflating revenue to real terms, constructing company-specific input cost indices, CAGR versus index-based growth in FMCG and retail price tracking.
Interview questions
- Q: What exactly does an index number measure? A: The net relative change in a variable or group of variables between a current and base period, expressed as a percentage of the base; it is a unit-free specialised average, not an absolute level.
- Q: Why does Laspeyres overstate and Paasche understate price change? A: Laspeyres holds the base-year basket fixed and so ignores consumers substituting away from goods whose prices rose; Paasche uses the current basket which already reflects that substitution, so it understates.
- Q: Why is Fisher's index called "ideal"? A: It is the geometric mean of Laspeyres and Paasche, uses both base- and current-year weights, and is the only common formula satisfying both the time reversal and factor reversal tests.
- Q: If Fisher is ideal, why do statistical agencies use Laspeyres? A: Paasche needs current-period quantity data every month, which is expensive and delayed; Laspeyres (or Kelly's fixed-weight form) needs the basket only once per base revision.
- Q: Which formula satisfies the circular test and why does it matter? A: The simple aggregative and Kelly's fixed-weight index, because their weights do not change with the period; it matters when you want to shift the base or chain periods without recomputing from raw data.
- Q: What is the difference between CPI and WPI in India? A: CPI (NSO, base 2024=100 from January 2026) measures retail prices paid by consumers, has a large food and services weight, and is the RBI's target; WPI (base 2022-23=100 from June 2026) measures wholesale transaction prices of goods only, excludes services and indirect taxes.
- Q: Which index fixes dearness allowance? A: CPI for Industrial Workers, compiled by the Labour Bureau with base 2016 = 100 — not the NSO's CPI (Combined).
- Q: CPI rose from 160 to 210 while a salary rose from ₹25,000 to ₹36,000. Is the person better off? A: Yes, marginally — real wage went from ₹15,625 to ₹17,143 at base prices, about 9.7% higher, far less than the 44% nominal rise.
- Q: What is the base effect in an inflation number? A: A low or high index value in the comparison period a year earlier mechanically inflates or depresses the year-on-year inflation rate even when current prices are stable.
- Q: Why is the geometric mean theoretically better than the arithmetic mean for index numbers? A: Index numbers are ratios; the GM gives equal weight to equal proportionate changes, is reversible (base and current periods can be swapped), and is less distorted by extreme price relatives.
- Q: What is splicing and when do you need it? A: Joining a discontinued old series to a new one with a different base, using the overlap year as the link — needed to build a long comparable series across a base revision.
- Q: Name three limitations of any price index. A: It cannot capture quality improvement or new products, the basket becomes unrepresentative as consumption shifts, and the choice of base, items and formula is arbitrary, so no index is unique or exact.
Common mistakes
- Using q₁ with p₀q₀ or mixing weights across formulae — always build the four sums Σp₀q₀, Σp₁q₀, Σp₀q₁, Σp₁q₁ first, then pick the two the formula needs.
- Forgetting to multiply by 100, or multiplying twice, so the answer comes out as 1.44 or 14,474 instead of 144.74.
- Taking the arithmetic mean of Laspeyres and Paasche and calling it Fisher — that is Dorbish-Bowley; Fisher is the geometric mean.
- Applying the reversal tests with indices still multiplied by 100; the tests require the ratio form (product should be 1, not 10,000).
- Confusing CPI (Combined, base 2024 = 100) with CPI-IW (base 2016 = 100) and WPI (base 2022-23 = 100), or quoting a withdrawn base year (CPI 2012, WPI and IIP 2011-12).
- Treating a change in the index value as the inflation rate — a rise from 160 to 170 is 10 points but 6.25%, not 10%.
- Multiplying nominal values by the price index instead of dividing when deflating, which inflates rather than deflates the series.
Related topics
Descriptive statistics · Time series forecasting · Introductory econometrics: regression · Economic modelling with Excel and R · National income accounting and GDP deflator · Inflation and monetary policy.