What is an index number?
An index number is a statistical tool that measures the relative change in a group of related things (prices, output, wages) over time or between places.
- Base year (0): the year we compare with. Its index is always 100. It should be a normal year: no war, flood or boom.
- Current year (1): the year we are studying.
- P0: price in the base year; P1: price in the current year; q: quantity (used as weight).
An index of 125 means a 25% rise over the base year; 90 means a 10% fall.
Simple aggregative method
P01 = ΣP1 ÷ ΣP0 × 100.
Steps: add the base-year prices, add the current-year prices, divide, multiply by 100.
It is easy, but every item is treated as equally important, and items with high prices (in rupees) dominate the result.
Simple average of price relatives
Price relative of each item = P1 ÷ P0 × 100. Index = Σ(price relatives) ÷ n. It stops high-priced items from dominating, but still gives equal weight to all.
Weighted index numbers
Items differ in importance. A family spends far more on rice than on salt. So we use weights.
- Laspeyres (base-year quantities as weights): P01 = Σp1q0 ÷ Σp0q0 × 100. It tells how much more the same old basket costs now.
- Paasche (current-year quantities as weights): P01 = Σp1q1 ÷ Σp0q1 × 100.
- Weighted average of price relatives: Σ(RW) ÷ ΣW, where R = P1 ÷ P0 × 100 and W = weight.
Important indices in India: CPI, WPI, IIP
Consumer Price Index (CPI)
Measures the change in retail prices of a fixed basket of goods and services that households buy: food, clothing, housing, fuel, education, health. CPI (Rural, Urban and Combined) is published by the NSO with base year 2012; a newer base year is being introduced. Older series such as CPI for Industrial Workers are published by the Labour Bureau. CPI is used to measure inflation and to set dearness allowance.
Wholesale Price Index (WPI)
Measures the change in prices at the wholesale (bulk) level for primary articles, fuel and power, and manufactured products. It does not include services. Published by the Office of the Economic Adviser, base 2011-12.
Index of Industrial Production (IIP)
Measures the change in the volume of production (not prices) in mining, manufacturing and electricity. Published monthly by the NSO, base 2011-12. It shows whether industry is growing.
Other indices
Sensex (BSE, 30 big companies) and Nifty (NSE, 50 companies) show movement of share prices. There are also the index of agricultural production and the service sector index.
Uses of index numbers
- Measure inflation and changes in the cost of living.
- Set dearness allowance and adjust wages and pensions.
- Find real income: real income = money income ÷ CPI × 100.
- Help the government and RBI make policy (interest rates, taxes, subsidies).
- Show trends in industrial output (IIP) and share markets (Sensex).
- Deflate GDP to get real GDP.
Things to take care of
Choose a normal base year, pick items that represent the group, use right weights, and use the right price source (retail or wholesale).
Inflation and index numbers
Inflation is a steady rise in the general level of prices. As prices rise, the same money buys less; its purchasing power falls.
Inflation rate = (Index this year − Index last year) ÷ Index last year × 100.
In India, the CPI is the main measure used by the RBI for inflation. WPI inflation shows price pressure in wholesale markets.
Purchasing power of money = 100 ÷ price index × 100. If CPI is 125, ₹100 buys only ₹80 worth of base-year goods.
Try it: your family's price index
Ask at home the price of 3 things one year ago and today (say, milk, potatoes and cooking gas). Find ΣP0 and ΣP1, then the index. Is your family's index above 100? Then set the same numbers on the sliders in the last 3D step (scale them if needed) and compare.
Board exam pattern
Expect: construct an index by the simple aggregative method or a weighted method (4 marks), difference between CPI and WPI, uses of index numbers, and a short sum on inflation or real income (3 marks).
Key formulas and definitions
- Simple aggregative: P01 = ΣP1 ÷ ΣP0 × 100
- Price relative R = P1 ÷ P0 × 100; simple average = ΣR ÷ n
- Laspeyres: Σp1q0 ÷ Σp0q0 × 100
- Paasche: Σp1q1 ÷ Σp0q1 × 100
- Weighted average of relatives = ΣRW ÷ ΣW
- Inflation rate = (I1 − I0) ÷ I0 × 100
- Real income = money income ÷ CPI × 100
Worked examples
1. Base prices ₹40, ₹50, ₹110; current prices ₹50, ₹60, ₹140. Find the simple aggregative index.
ΣP0 = 200, ΣP1 = 250. Index = 250 ÷ 200 × 100 = 125. Prices rose 25%.
2. Same prices; base quantities 10, 8, 2. Find Laspeyres' index.
Σp0q0 = 400 + 400 + 220 = 1020. Σp1q0 = 500 + 480 + 280 = 1260. Index = 1260 ÷ 1020 × 100 ≈ 123.5.
3. Price relatives are 125, 120 and 127.3. Find the simple average of relatives.
(125 + 120 + 127.3) ÷ 3 = 372.3 ÷ 3 ≈ 124.1.
4. CPI was 150 last year and 162 this year. Find the inflation rate.
(162 − 150) ÷ 150 × 100 = 12 ÷ 150 × 100 = 8%.
5. A worker's salary rose from ₹20,000 to ₹24,000 while CPI rose from 100 to 125. Did his real income rise?
Real income now = 24,000 ÷ 125 × 100 = ₹19,200. It fell from ₹20,000. Prices rose faster than his salary.
6. Weights 4, 3, 1 on price relatives 110, 120, 150. Find the weighted index.
ΣRW = 440 + 360 + 150 = 950; ΣW = 8. Index = 950 ÷ 8 = 118.75.
Common mistakes
- Forgetting to multiply by 100.
- Dividing ΣP0 by ΣP1 (upside down). Current over base.
- Calling the index itself the inflation rate. An index of 108 means 8% above the base, not 8% inflation this year unless last year was the base.
- Mixing up WPI (wholesale, no services) with CPI (retail, includes services).