Index numbers measure relative changes in variables over time, expressing one period's value as a percentage of a base period, commonly used to track inflation, production, and stock prices.
## Core concept
An index number is a statistical tool that compares a quantity, price, or value in one period (current/given period) against a reference period (base period), expressed as a ratio × 100.
Formula (Simple Index): $$I_t = \frac{P_t}{P_0} \times 100$$
Where: - $I_t$ = Index number for period t - $P_t$ = Price/value in current period - $P_0$ = Price/value in base period (always = 100)
Key characteristics: - Base period always has index = 100 - Index > 100 indicates increase from base - Index < 100 indicates decrease from base - Always expressed as a percentage relative to base
## Common applications in exams
1. Price Index (Consumer Price Index / Wholesale Price Index) - Tracks inflation by comparing current prices to base year prices - Formula: $(P_t / P_0) \times 100$
2. Quantity/Volume Index - Measures change in production or sales quantities - Formula: $(Q_t / Q_0) \times 100$
3. Value Index - Combines price and quantity changes - Formula: $(P_t Q_t) / (P_0 Q_0) \times 100$
4. Chain Index - Successive year-on-year comparisons linked together - Each period uses previous year as base - Useful for long-term trend analysis
5. Aggregate Index (Weighted average) - Used when multiple items must be combined - Weighted Aggregate Index = $\frac{\sum P_t W}{\sum P_0 W} \times 100$ - Weights ($W$) reflect importance of each item
## Worked example
Problem: A company's product sales were ₹50,000 in 2020 (base year) and ₹75,000 in 2024. Calculate the index number.
Solution: $$I_{2024} = \frac{75,000}{50,000} \times 100 = 1.5 \times 100 = 150$$
Interpretation: Sales increased 50% from 2020 to 2024 (index rose from 100 to 150).
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Weighted Index Example: | Item | 2020 Price | 2024 Price | Weight | |------|-----------|-----------|--------| | A | 10 | 12 | 2 | | B | 20 | 30 | 3 |
Weighted Aggregate Index = $\frac{(12 \times 2) + (30 \times 3)}{(10 \times 2) + (20 \times 3)} \times 100 = \frac{114}{80} \times 100 = 142.5$
## Common exam applications
- RBI/Inflation questions: Calculate CPI or WPI changes year-on-year
- Business performance: Track production, sales, or cost indices
- Real vs nominal analysis: Adjust nominal values for inflation using index numbers
- Comparative analysis: Compare growth across different periods or product lines
- Deflation calculation: Real value = Nominal value / (Index / 100)
## Common mistakes
- Forgetting base = 100: Always set base period index to exactly 100
- Incorrect formula direction: Use current/base, not base/current
- Confusing simple and weighted indices: Read problem carefully; weighted questions specify weights
- Misinterpreting index > 100: An index of 120 means a 20% increase, not 120% increase
- Not showing interpretation: State what the numerical result means contextually
- Chain index errors: Remember each link uses the previous period as base, not original base
Exam tip: Index number questions often appear in combination with inflation, real wages, or purchasing power sections. Always state your base year clearly and round to 1–2 decimals unless specified otherwise.