Index Numbers: Concepts, Construction, Tests and Uses

Overview

Index numbers are special statistical measures that show changes in a variable (or group of variables) over time, usually compared with a base period. In business and economics, index numbers help measure inflation, cost of living, changes in production, sales trends and share market movement. This topic explains index numbers, their construction methods (simple and weighted), CPI and tests like time reversal and factor reversal.

Index number: meaning

An index number is a statistical measure that shows the relative change in a variable or group of variables over time, generally expressed with base period = 100.

Uses/importance of index numbers

• measure inflation and cost of living (CPI/WPI)
• adjust wages/DA (dearness allowance) and contracts
• compare sales/production trends across periods
• support policy decisions and business planning
• evaluate purchasing power changes

Types of index numbers

• Price index (price changes)
• Quantity/volume index (quantity changes)
• Value index (price × quantity changes)

Price index: simple aggregative and simple average of relatives

• Simple aggregative price index: $P_{01} = \\frac{\\sum p_1}{\\sum p_0}\\times 100$
• Simple average of price relatives: $P_{01} = \\frac{\\sum \\left(\\frac{p_1}{p_0}\\times 100\\right)}{n}$

Weighted index numbers (Laspeyres, Paasche, Fisher)

Weighted indices use quantities as weights.

• Laspeyres price index (base-year quantity weights): $P_L = \\frac{\\sum p_1 q_0}{\\sum p_0 q_0}\\times 100$
• Paasche price index (current-year quantity weights): $P_P = \\frac{\\sum p_1 q_1}{\\sum p_0 q_1}\\times 100$
• Fisher’s ideal index: $P_F = \\sqrt{P_L\\times P_P}$

Formula table (key formulas)

Consumer Price Index (CPI): meaning and steps

CPI measures changes in the cost of a fixed basket of goods and services consumed by a group of consumers.

Basic steps:

• select basket items and base year
• collect prices for base and current period
• assign weights (quantities/expenditure shares)
• compute weighted index

Tests of index numbers (time reversal, factor reversal)

• Time reversal test: index should satisfy $P_{01}\\times P_{10} = 1$.
• Factor reversal test: $P_{01}\\times Q_{01} = V_{01}$ (value index), ideally satisfied by Fisher’s index.

Limitations of index numbers

• selection of items and weights may be subjective
• quality changes are difficult to measure
• new products and consumer preferences change over time
• data errors can distort results

Worked mini example (exam style)

Prices of two items:

Simple aggregative index: $\sum p_0 = 30$, $\sum p_1 = 37$
$P_{01} = (37/30)\\times 100 = 123.33$

Interpretation: average price level increased by about 23.33% from base to current period (in this simple example).

Common exam mistakes

• forgetting “×100” in index formula
• mixing price index and quantity index formulas
• using wrong weights (q0 vs q1) for Laspeyres/Paasche

Exam tips (how to write 3/5 marks)

• For 3 marks: definition + 3 uses + one formula.
• For 5 marks: formula table + CPI steps + tests + mini example + conclusion.

Key Terms

• Index number — measure of relative change over time.
• Base period — reference period (often = 100).
• Weights — importance given to items (quantity/expenditure).
• CPI — consumer price index (cost of living).
• Fisher index — geometric mean of Laspeyres and Paasche.

Quick Recap (1-minute revision)

• Index numbers show relative change over time (base=100).
• Weighted indices (Laspeyres/Paasche/Fisher) are more realistic.
• Fisher is “ideal” because it satisfies important tests.