Class 11 Statistics Notes Chapter 8 (Index numbers) – Statistics For Economics Book

Alright class, let's get straight into Chapter 8: Index Numbers. This is a very important topic, not just for your Class 11 syllabus, but also frequently tested in various government exams. Index numbers are essential tools used by economists and policymakers, so understanding them well is crucial.

Chapter 8: Index Numbers - Detailed Notes for Government Exam Preparation

1. What are Index Numbers?

• Definition: An index number is a statistical measure designed to show changes in a variable or a group of related variables with respect to time, geographic location, or other characteristics (like income, profession, etc.).
• Purpose: They are used to measure the relative change in the level of a phenomenon (like price, quantity produced/consumed, cost of living) over a period.
• Nature: Index numbers are expressed as percentages, allowing for easy comparison, but the percentage sign (%) is usually omitted. They are specialized averages.
• Base Period: The period chosen as the standard point of comparison is called the base period (or base year). The index number for the base period is always taken as 100.
• Current Period: The period for which the change is being measured is called the current period (or current year).

2. Key Characteristics of Index Numbers:

• Specialized Averages: Capable of averaging variables expressed in different units.
• Measure Relative Changes: They show changes over time or location, not absolute values.
• Expressed as Percentages: Facilitates comparison.
• Basis for Comparison: Provide a common platform (base period) for comparing different time periods or locations.

3. Basic Terms:

• P0: Price in the base period.
• P1: Price in the current period.
• Q0: Quantity in the base period.
• Q1: Quantity in the current period.
• P01: Price index number for the current period (1) with respect to the base period (0).
• Q01: Quantity index number for the current period (1) with respect to the base period (0).
• V01: Value index number for the current period (1) with respect to the base period (0). (Value = Price × Quantity)

4. Types of Index Numbers:

• Price Index Numbers (Most Common): Measure changes in the general price level of goods and services over time. (e.g., WPI, CPI).
• Quantity Index Numbers: Measure changes in the volume or quantity of goods produced, consumed, or sold. (e.g., Index of Industrial Production - IIP).
• Value Index Numbers: Measure changes in the total monetary value of a set of items. Value = Price × Quantity. V01 = (∑P1Q1 / ∑P0Q0) × 100.

5. Methods of Constructing Index Numbers:

(A) Simple (Unweighted) Index Numbers: Assume equal importance for all items.

• i) Simple Aggregative Method:
  • Formula: P01 = (∑P1 / ∑P0) × 100
  • Pros: Simple to calculate.
  • Cons:
    • Affected by the units in which prices are quoted (e.g., kg vs. quintal).
    • Gives equal weightage to all items, regardless of their relative importance.
    • Highly influenced by items with large prices.
• ii) Simple Average of Price Relatives Method:
  • Calculate Price Relative for each item: P = (P1 / P0) × 100
  • Formula (using Arithmetic Mean): P01 = [∑((P1 / P0) × 100)] / N (where N = number of items)
  • Formula (using Geometric Mean): P01 = Antilog [ (∑ log P) / N ]
  • Pros:
    • Not affected by the units of measurement.
    • Gives equal weight to relative changes of each item.
  • Cons: Still gives equal importance to all items, which might not be realistic. Using AM can give an upward bias compared to GM.

(B) Weighted Index Numbers: Assign weights to items according to their relative importance.

• i) Weighted Aggregative Method: Weights are usually quantities (Q) or values (P×Q).

  • Laspeyres' Price Index (Uses Base Year Quantities - Q0 - as weights):
    • Formula: P01(L) = (∑P1Q0 / ∑P0Q0) × 100
    • Interpretation: Measures the change in cost of purchasing the base year's basket of goods at current year prices.
    • Pros: Easy to calculate (only need base year quantities), allows comparison over time as the basket is fixed.
    • Cons: Tends to overestimate price changes (upward bias) as it ignores consumption pattern shifts away from relatively expensive items. Uses outdated quantities.
  • Paasche's Price Index (Uses Current Year Quantities - Q1 - as weights):
    • Formula: P01(P) = (∑P1Q1 / ∑P0Q1) × 100
    • Interpretation: Measures the change in cost of purchasing the current year's basket of goods at current year prices relative to base year prices.
    • Pros: Uses current consumption patterns.
    • Cons: Requires quantity data for every year, making it computationally intensive and difficult for year-to-year comparison (basket changes). Tends to underestimate price changes (downward bias).
  • Fisher's Ideal Price Index (Geometric Mean of Laspeyres' and Paasche's):
    • Formula: P01(F) = √[ P01(L) × P01(P) ] = √[ (∑P1Q0 / ∑P0Q0) × (∑P1Q1 / ∑P0Q1) ] × 100
    • Why 'Ideal'? It satisfies certain mathematical tests (Time Reversal Test and Factor Reversal Test - important for competitive exams). It moderates the biases of Laspeyres' and Paasche's.
    • Cons: More complex to calculate, requires both Q0 and Q1. Interpretation is less direct.
  • Dorbish-Bowley Price Index (Arithmetic Mean of Laspeyres' and Paasche's):
    • Formula: P01(DB) = [ P01(L) + P01(P) ] / 2
  • Marshall-Edgeworth Price Index (Uses average of base and current year quantities):
    • Formula: P01(ME) = [ ∑P1(Q0+Q1) / ∑P0(Q0+Q1) ] × 100

• ii) Weighted Average of Price Relatives Method: Weights are typically base year values (P0Q0).

  • Calculate Price Relatives: P = (P1 / P0) × 100
  • Assign Weights: W = P0Q0 (Base year value)
  • Formula: P01 = (∑PW / ∑W)

6. Quantity Index Numbers:

• Similar formulas as price indices, but P and Q are interchanged.
• Laspeyres' Quantity Index: Q01(L) = (∑Q1P0 / ∑Q0P0) × 100
• Paasche's Quantity Index: Q01(P) = (∑Q1P1 / ∑Q0P1) × 100
• Fisher's Quantity Index: Q01(F) = √[ Q01(L) × Q01(P) ]

7. Tests of Adequacy (for Weighted Aggregative Indices):

• Time Reversal Test: An index number formula should work both forwards and backwards in time. P01 × P10 = 1 (where P10 is the index for period 0 based on period 1). Fisher's index satisfies this test. Laspeyres' and Paasche's do not.
• Factor Reversal Test: The product of a price index and the corresponding quantity index should equal the value index. P01 × Q01 = V01 = (∑P1Q1 / ∑P0Q0). Fisher's index satisfies this test. Laspeyres' and Paasche's do not.

8. Consumer Price Index (CPI) or Cost of Living Index (COLI):

• Definition: Measures the average change over time in the prices paid by urban consumers for a market basket of consumer goods and services. It reflects the cost of living for a specific group of consumers.
• Construction Methods:
  • Aggregate Expenditure Method: Identical to Laspeyres' Price Index. CPI = (∑P1Q0 / ∑P0Q0) × 100. Here, Q0 represents the quantities consumed by the specific consumer group in the base year.
  • Family Budget Method: Identical to the Weighted Average of Price Relatives method, using base year values (P0Q0) as weights. CPI = (∑PW / ∑W), where P = (P1/P0)×100 and W = P0Q0.
• Steps in Constructing CPI:
  1. Define the target consumer group.
  2. Conduct a Family Budget Survey to determine the basket of goods and their weights (importance).
  3. Select a Base Year.
  4. Collect current and base year prices for items in the basket.
  5. Choose a formula (usually Laspeyres' or Weighted Average of Relatives).
  6. Calculate the index.
• Uses: Wage negotiations (DA calculation), measuring purchasing power of money, deflation of income/value series, policy formulation.
• Different CPIs in India: CPI-IW (Industrial Workers), CPI-AL (Agricultural Labourers), CPI-RL (Rural Labourers), CPI-Urban, CPI-Rural, CPI-Combined.

9. Wholesale Price Index (WPI):

• Definition: Measures the average change in prices of goods traded in bulk in the wholesale market. It does not include services.
• Base Year in India: Currently 2011-12.
• Publisher: Office of Economic Adviser, Ministry of Commerce and Industry.
• Use: Often considered a headline inflation measure, used for policy making, forecasting demand/supply, deflating national accounts.
• Difference from CPI: WPI tracks wholesale prices (bulk), CPI tracks retail prices (consumer level). WPI excludes services, CPI includes them. Weights differ based on wholesale transactions vs. consumer expenditure.

10. Index of Industrial Production (IIP):

• Definition: Measures the changes in the level of physical volume of production in the industrial sector (manufacturing, mining, electricity).
• Type: It's a Quantity Index Number.
• Base Year in India: Currently 2011-12.
• Publisher: National Statistical Office (NSO), Ministry of Statistics and Programme Implementation.
• Use: Tracks industrial growth, policy making related to industry.

11. Uses of Index Numbers:

• Measure changes in price levels/value of money.
• Help in policy formulation (monetary, fiscal).
• Used for adjusting wages and allowances (DA).
• Act as economic barometers, indicating overall economic trends.
• Facilitate comparison of living standards over time or between places.
• Used for deflating – adjusting original data for price changes to understand 'real' changes. (e.g., Real Income = (Nominal Income / Price Index) × 100).

12. Problems in Constructing Index Numbers:

• Defining the Purpose: The purpose dictates the type of index, items, weights, etc.
• Selecting the Base Period: Should be a normal period, not too distant.
• Selecting Items (Basket): Should be representative, standardized, and stable in quality.
• Obtaining Price Quotations: Need representative and reliable price data from relevant markets.
• Choosing the Formula: Different formulas give different results; choice depends on purpose and data availability.
• Selecting Weights: Assigning appropriate importance is crucial but difficult. Weights may become outdated.
• Choosing the Average: AM or GM for simple average of relatives?

13. Index Numbers and Inflation:

• Inflation refers to a sustained increase in the general price level.
• Price indices like CPI and WPI are the primary tools used to measure the rate of inflation.
• Rate of Inflation = [(Index(Current) - Index(Previous)) / Index(Previous)] × 100

Multiple Choice Questions (MCQs):

1. An index number which accounts for the relative importance of items is known as:
   a) Simple Index Number
   b) Weighted Index Number
   c) Value Index Number
   d) Quantity Index Number

2. Laspeyres' price index uses weights based on:
   a) Current year quantities
   b) Base year quantities
   c) Average of base and current year quantities
   d) Base year prices

3. If the price index for a year is 150, it means that prices have increased by ______ compared to the base year.
   a) 150%
   b) 100%
   c) 50%
   d) 250%

4. Which index number satisfies both the Time Reversal Test and the Factor Reversal Test?
   a) Laspeyres' Index
   b) Paasche's Index
   c) Fisher's Ideal Index
   d) Simple Aggregative Index

5. The Consumer Price Index (CPI) measures changes in:
   a) Wholesale prices of commodities
   b) Retail prices of consumer goods and services
   c) Volume of industrial production
   d) Prices of agricultural inputs

6. The formula ∑P1Q0 / ∑P0Q0 × 100 represents:
   a) Paasche's Price Index
   b) Fisher's Price Index
   c) Laspeyres' Price Index
   d) Simple Aggregative Price Index

7. The base period for an index number should ideally be:
   a) A period of very high prices
   b) A period of very low prices
   c) A normal period without major fluctuations
   d) The most recent period available

8. Which method of constructing index numbers is affected by the units in which prices are quoted?
   a) Simple Average of Price Relatives
   b) Weighted Average of Price Relatives
   c) Simple Aggregative Method
   d) Laspeyres' Method

9. If nominal income is ₹50,000 and the CPI is 125, what is the real income?
   a) ₹40,000
   b) ₹62,500
   c) ₹50,000
   d) ₹37,500

10. The Index of Industrial Production (IIP) is an example of a:
    a) Price Index
    b) Value Index
    c) Quantity Index
    d) Cost of Living Index

Answers to MCQs:

1. b) Weighted Index Number
2. b) Base year quantities
3. c) 50% (Index 150 means 150 - 100 = 50% increase over base 100)
4. c) Fisher's Ideal Index
5. b) Retail prices of consumer goods and services
6. c) Laspeyres' Price Index
7. c) A normal period without major fluctuations
8. c) Simple Aggregative Method
9. a) ₹40,000 (Real Income = (50000 / 125) * 100 = 40000)
10. c) Quantity Index

Make sure you understand the formulas, the difference between the types of indices, and their specific applications like CPI, WPI, and IIP. Remember the limitations and problems associated with constructing these indices as well. Good luck with your preparation!