Class 11 Economics Chapter 8 — Index Numbers (ASSEB)
Welcome to HSLC Guru. This study guide for ASSEB Class 11 Economics (Statistics for Economics) Chapter 8 — Index Numbers covers the meaning, characteristics, types and uses of index numbers, methods of construction (Simple Aggregative, Simple Average of Price Relatives, Laspeyres, Paasche and Fisher Ideal), the major Indian indices (CPI, WPI, IIP, Sensex/Nifty), problems in construction and base year selection, along with detailed textbook style Q&A, MCQs, fill-in-the-blanks, true/false, glossary and a formula table.
Summary
An index number is a specialised statistical device designed to measure relative changes in a variable or a group of related variables across time, place, or category. It is expressed as a percentage of a chosen base value (usually taken as 100). Spiegel defined an index number as “a statistical measure designed to show changes in a variable or a group of related variables with respect to time, geographical location or other characteristics”. The chief features are: (i) it is a specialised average; (ii) it measures relative change, not absolute change; (iii) it is expressed in percentage form (without the % sign); (iv) it can compare data over time and place; and (v) it is useful where direct measurement is difficult, such as the general price level or cost of living.
Index numbers are broadly classified into Price Index (measures change in the level of prices, e.g., WPI, CPI), Quantity Index (measures change in physical volume of production or consumption, e.g., IIP) and Value Index (measures change in total monetary value = price × quantity). Methods of construction include the Unweighted methods — Simple Aggregative Method [P01 = (ΣP1 / ΣP0) × 100] and Simple Average of Price Relatives [P01 = (Σ(P1/P0)×100) / N] — and the Weighted methods — Laspeyres’ Index uses base-year quantities (q0) as weights, Paasche’s Index uses current-year quantities (q1) as weights, and Fisher’s Ideal Index is the geometric mean of Laspeyres’ and Paasche’s indices. Fisher’s Index is called “ideal” because it satisfies both the Time Reversal Test and the Factor Reversal Test and uses the geometric mean.
The principal index numbers used in India are: Consumer Price Index (CPI), which measures the change in retail prices of goods and services consumed by a defined population group and is used for measuring cost of living and granting dearness allowance; Wholesale Price Index (WPI), compiled by the Office of the Economic Adviser, which measures changes in wholesale prices of goods traded in bulk; Index of Industrial Production (IIP), a quantity index compiled by the CSO that shows growth in the industrial sector covering Mining, Manufacturing and Electricity; and stock market indices like BSE Sensex (30 stocks) and NSE Nifty (50 stocks) that capture the movement of share prices. Index numbers are widely used to study trends and tendencies, frame economic policy, measure inflation/deflation, deflate national income figures, adjust wages and pensions, and forecast future business activity.
Important problems in construction include: (a) selection of the purpose of the index; (b) selection of the base year — it should be a normal year, free from wars, famines, booms or slumps, neither too distant nor too recent; (c) selection of commodities — they should be representative, comparable and stable in quality; (d) selection of prices — wholesale or retail depending on purpose, collected from reliable sources; (e) choice of an average — the geometric mean is theoretically best but the arithmetic mean is most commonly used; (f) selection of weights — weights must reflect relative importance; and (g) choice of formula — depending on data availability and purpose. The base year may be chosen by the Fixed Base Method or the Chain Base Method.
Textbook Questions and Answers
A. Very Short Answer Type Questions (1 Mark)
Q1. What is an index number?
Answer: An index number is a statistical measure that shows the relative change in a variable or a group of variables over time, place or other characteristics, expressed as a percentage of a base value (taken as 100).
Q2. What is the value of an index number in the base year?
Answer: The value of the index number in the base year is always 100.
Q3. Name the three main types of index numbers.
Answer: The three main types are (i) Price Index Number, (ii) Quantity Index Number, and (iii) Value Index Number.
Q4. Write the full form of CPI and WPI.
Answer: CPI = Consumer Price Index; WPI = Wholesale Price Index.
Q5. Which index number is called the “Ideal Index Number”?
Answer: Fisher’s Index Number is called the Ideal Index Number because it is the geometric mean of Laspeyres’ and Paasche’s indices and satisfies both the time reversal and factor reversal tests.
Q6. What weights are used in Laspeyres’ Index?
Answer: Laspeyres’ Index uses the base-year quantities (q0) as weights.
Q7. What weights are used in Paasche’s Index?
Answer: Paasche’s Index uses the current-year quantities (q1) as weights.
Q8. What does IIP measure?
Answer: The Index of Industrial Production (IIP) is a quantity index that measures the growth in the volume of production in mining, manufacturing and electricity sectors of the economy.
Q9. What is Sensex?
Answer: Sensex is the stock market index of the Bombay Stock Exchange (BSE) that reflects the price movement of 30 financially sound and actively traded company shares.
Q10. What is meant by base year?
Answer: The base year is the reference year against which prices, quantities or values of the current year are compared. The index value of the base year is taken as 100.
B. Short Answer Type Questions (2-3 Marks)
Q1. Calculate the Simple Aggregative Price Index from the following data: Prices in 2010 (P0): Rice 20, Wheat 15, Pulses 30, Sugar 25. Prices in 2024 (P1): Rice 50, Wheat 40, Pulses 90, Sugar 45.
Answer: ΣP0 = 20 + 15 + 30 + 25 = 90; ΣP1 = 50 + 40 + 90 + 45 = 225. P01 = (ΣP1 / ΣP0) × 100 = (225 / 90) × 100 = 250. Hence prices have risen by 150% over the base year.
Q2. From the following, construct the Index Number using the Simple Average of Price Relatives Method. P0: A=10, B=20, C=40, D=50. P1: A=15, B=25, C=60, D=75.
Answer: Price Relatives (P1/P0)×100 = A: 150, B: 125, C: 150, D: 150. Sum = 575. N = 4. P01 = 575/4 = 143.75. Prices have increased by 43.75% on the average.
Q3. Distinguish between Price Index and Quantity Index.
Answer: A Price Index measures the relative change in the level of prices of a group of commodities between two time periods (e.g., CPI, WPI). A Quantity Index measures the relative change in the physical volume of production, consumption or stocks of commodities (e.g., IIP). Price index uses prices as the variable, whereas quantity index uses quantities.
Q4. Mention any three uses of Index Numbers.
Answer: (i) They measure the change in the value of money (inflation/deflation) and the cost of living. (ii) They help the government in framing economic policies and revising wages and dearness allowances. (iii) They serve as economic barometers for measuring trends and forecasting future business conditions.
Q5. Differentiate between Fixed Base Method and Chain Base Method.
Answer: In the Fixed Base Method, prices of all years are compared with the prices of one fixed base year. In the Chain Base Method, the price of each year is compared with the immediately preceding year (link relatives), and these are then chained together to obtain chain indices. The chain method is more flexible and useful when commodities or weights change frequently.
Q6. What are the qualities of a good base year?
Answer: A good base year should be: (i) a normal year free from natural calamities, wars, booms or depressions; (ii) not too distant from the current year, so that comparisons remain meaningful; (iii) one for which reliable and complete data are available; and (iv) not affected by abnormal economic events.
C. Long Answer Type Questions (5-6 Marks)
Q1. Construct (a) Laspeyres’ (b) Paasche’s and (c) Fisher’s Ideal Price Index from the following data:
| Commodity | P0 | q0 | P1 | q1 |
|---|---|---|---|---|
| A | 2 | 10 | 4 | 5 |
| B | 5 | 12 | 6 | 10 |
| C | 4 | 20 | 5 | 15 |
| D | 2 | 15 | 3 | 10 |
Answer: Compute the four sums.
| Commodity | p0q0 | p1q0 | p0q1 | p1q1 |
|---|---|---|---|---|
| A | 20 | 40 | 10 | 20 |
| B | 60 | 72 | 50 | 60 |
| C | 80 | 100 | 60 | 75 |
| D | 30 | 45 | 20 | 30 |
| Total | 190 | 257 | 140 | 185 |
(a) Laspeyres’ Index = (Σp1q0 / Σp0q0) × 100 = (257/190) × 100 = 135.26.
(b) Paasche’s Index = (Σp1q1 / Σp0q1) × 100 = (185/140) × 100 = 132.14.
(c) Fisher’s Ideal Index = √(Laspeyres × Paasche) = √(135.26 × 132.14) = √17873.25 = 133.69.
Q2. Calculate Laspeyres’ and Paasche’s Price Index Numbers from the following data, and verify that Laspeyres > Fisher > Paasche.
| Item | p0 | q0 | p1 | q1 |
|---|---|---|---|---|
| X | 10 | 5 | 20 | 4 |
| Y | 20 | 10 | 30 | 8 |
| Z | 40 | 2 | 50 | 2 |
Answer: p0q0 = 50+200+80 = 330; p1q0 = 100+300+100 = 500; p0q1 = 40+160+80 = 280; p1q1 = 80+240+100 = 420. Laspeyres = (500/330)×100 = 151.52; Paasche = (420/280)×100 = 150.00; Fisher = √(151.52×150) = √22728 = 150.76. Hence Laspeyres (151.52) > Fisher (150.76) > Paasche (150.00).
Q3. Discuss the main problems faced in the construction of an Index Number.
Answer: The construction of an index number involves the following problems:
- Purpose of the Index: The objective should be clearly defined as it determines the choice of items, weights and formula.
- Selection of Base Year: The base year should be a normal year, neither too distant nor too recent, free from abnormalities.
- Selection of Commodities: Items must be representative of consumption habits of the target group, and stable in quality.
- Selection of Prices: Wholesale or retail prices should be chosen depending on the purpose, and collected from reliable sources at the same point of the chain.
- Choice of Average: The geometric mean is theoretically best, but the arithmetic mean is widely used due to its simplicity.
- Selection of Weights: Weights must reflect the relative importance of items in consumption or production.
- Choice of Formula: The formula (Laspeyres, Paasche, Fisher, etc.) is chosen on the basis of accuracy required and data availability.
Q4. Explain the meaning, construction and uses of Consumer Price Index (CPI).
Answer: The Consumer Price Index (CPI), also known as the Cost of Living Index, measures the average change over time in the retail prices of a fixed basket of goods and services purchased by a defined consumer group (industrial workers, agricultural labourers, urban or rural population). Its construction involves: (i) defining the target population and conducting a family budget enquiry to determine the consumption pattern; (ii) selecting representative commodities under heads such as Food, Fuel & Light, Housing, Clothing, and Miscellaneous; (iii) collecting retail prices regularly from selected markets; (iv) assigning weights based on the proportion of expenditure on each group; and (v) computing the index, usually by the Family Budget Method [P01 = ΣRW / ΣW, where R = price relative and W = weight]. CPI is used to (a) measure changes in the cost of living and the purchasing power of money, (b) grant dearness allowance to employees, (c) frame wage policy and welfare schemes, (d) deflate national income to obtain real income, and (e) study the impact of inflation on different income groups.
Q5. Calculate the Cost of Living Index Number using the Family Budget Method from the following data:
| Item | Weight (W) | Price Relative (R = P1/P0×100) |
|---|---|---|
| Food | 50 | 250 |
| Fuel & Light | 10 | 200 |
| Clothing | 15 | 180 |
| Housing | 15 | 160 |
| Miscellaneous | 10 | 140 |
Answer: ΣW = 50+10+15+15+10 = 100. ΣRW = (50×250)+(10×200)+(15×180)+(15×160)+(10×140) = 12500+2000+2700+2400+1400 = 21000. Cost of Living Index = ΣRW / ΣW = 21000 / 100 = 210. The cost of living has increased by 110% over the base year.
Additional Multiple Choice Questions
Q1. The index number for the base year is always:
(a) 0 (b) 1 (c) 100 (d) 1000
Answer: (c) 100.
Q2. Index numbers are expressed in:
(a) Rupees (b) Percentage (c) Ratio (d) Tonnes
Answer: (b) Percentage.
Q3. Laspeyres’ Index uses:
(a) Current-year quantities as weights (b) Base-year quantities as weights (c) Average of both (d) None
Answer: (b) Base-year quantities as weights.
Q4. Fisher’s Ideal Index is the:
(a) Arithmetic mean of Laspeyres and Paasche (b) Geometric mean of Laspeyres and Paasche (c) Median (d) Mode
Answer: (b) Geometric mean of Laspeyres and Paasche.
Q5. Which one of the following is a Quantity Index?
(a) CPI (b) WPI (c) IIP (d) Sensex
Answer: (c) IIP.
Q6. Sensex is associated with:
(a) NSE (b) BSE (c) RBI (d) SEBI
Answer: (b) BSE.
Q7. Nifty consists of how many companies?
(a) 20 (b) 30 (c) 50 (d) 100
Answer: (c) 50.
Q8. CPI is also called:
(a) Cost of Living Index (b) Wholesale Index (c) Quantity Index (d) Value Index
Answer: (a) Cost of Living Index.
Q9. Wholesale Price Index in India is compiled by:
(a) RBI (b) NSO (c) Office of the Economic Adviser (d) SEBI
Answer: (c) Office of the Economic Adviser.
Q10. Which average is theoretically considered the best for index numbers?
(a) Arithmetic Mean (b) Median (c) Mode (d) Geometric Mean
Answer: (d) Geometric Mean.
Fill in the Blanks
1. The index number of the base year is taken as ________.
Answer: 100.
2. Fisher’s Ideal Index is the ________ mean of Laspeyres’ and Paasche’s indices.
Answer: Geometric.
3. CPI is used to measure changes in the ________ of money.
Answer: Purchasing power.
4. The IIP is a ________ index number.
Answer: Quantity.
5. Sensex is the index of ________ Stock Exchange.
Answer: Bombay (BSE).
True or False
1. Index numbers are specialised averages. — True.
2. Laspeyres’ Index uses current-year quantities as weights. — False (it uses base-year quantities).
3. Fisher’s Index satisfies both the Time Reversal and Factor Reversal tests. — True.
4. WPI is based on retail prices. — False (it is based on wholesale prices).
5. Index numbers are expressed in absolute units. — False (they are expressed in percentage form).
Glossary
| Term | Meaning |
|---|---|
| Index Number | A statistical measure of relative change in a variable or group of variables over time, expressed as a percentage of the base value. |
| Base Year | The reference year against which comparisons are made; its index value is 100. |
| Current Year | The year for which the index value is to be computed. |
| Price Relative | (P1/P0)×100 — the ratio of current-year price to base-year price expressed as percentage. |
| Weighted Index | An index in which different commodities are given different weights according to their importance. |
| Laspeyres’ Index | Weighted price index using base-year quantities as weights. |
| Paasche’s Index | Weighted price index using current-year quantities as weights. |
| Fisher’s Ideal Index | Geometric mean of Laspeyres’ and Paasche’s indices; satisfies Time and Factor Reversal Tests. |
| CPI | Consumer Price Index — measures change in retail prices for a defined consumer group; cost of living index. |
| WPI | Wholesale Price Index — measures change in wholesale prices of bulk-traded goods. |
| IIP | Index of Industrial Production — quantity index for mining, manufacturing and electricity. |
| Sensex / Nifty | BSE 30-share / NSE 50-share stock market indices. |
| Inflation | A sustained rise in the general price level of goods and services. |
| Deflator | A price index used to convert nominal values to real values. |
Important Formulae
| Method | Formula |
|---|---|
| Simple Aggregative Method | P01 = (ΣP1 / ΣP0) × 100 |
| Simple Average of Price Relatives | P01 = [Σ(P1/P0)×100] / N |
| Weighted Aggregative — Laspeyres | P01(L) = (Σp1q0 / Σp0q0) × 100 |
| Weighted Aggregative — Paasche | P01(P) = (Σp1q1 / Σp0q1) × 100 |
| Fisher’s Ideal Index | P01(F) = √(P01(L) × P01(P)) |
| Marshall–Edgeworth Index | P01(ME) = [Σp1(q0+q1) / Σp0(q0+q1)] × 100 |
| Family Budget / Weighted Average of Relatives | P01 = ΣRW / ΣW, where R = (P1/P0)×100 |
| Quantity Index (Laspeyres) | Q01(L) = (Σq1p0 / Σq0p0) × 100 |
| Value Index | V01 = (Σp1q1 / Σp0q0) × 100 |
| Inflation Rate (using CPI) | {(CPI_current − CPI_previous) / CPI_previous} × 100 |
Key Differences Between Laspeyres and Paasche Indices
| Basis | Laspeyres’ Index | Paasche’s Index |
|---|---|---|
| Weights | Base-year quantities (q0) | Current-year quantities (q1) |
| Formula | (Σp1q0 / Σp0q0) × 100 | (Σp1q1 / Σp0q1) × 100 |
| Bias | Has an upward bias | Has a downward bias |
| Data Requirement | Only base-year quantities collected once | Current-year quantities collected each period |
| Practical Use | Widely used (e.g., CPI) | Less common due to data costs |
Tests of Adequacy of an Index Number
Three important tests are applied to test the consistency of an index number formula:
- Unit Test: The formula should give the same result irrespective of the units in which prices are quoted. All weighted aggregative formulae satisfy this test except the Simple Aggregative Method.
- Time Reversal Test: P01 × P10 = 1. The formula should produce consistent results when the time periods are reversed. Fisher’s Ideal Index satisfies this test.
- Factor Reversal Test: P01 × Q01 = (Σp1q1 / Σp0q0) — the product of the price index and the quantity index should equal the value ratio. Only Fisher’s Index satisfies this test, hence it is called the Ideal Index.
Limitations of Index Numbers
- They are based on samples and hence have a margin of error.
- They are approximate indicators and cannot show absolute changes.
- Different methods may yield different results, leading to confusion.
- They ignore changes in quality of commodities.
- They may be manipulated by faulty selection of items, weights or base year.
- The choice of an unrepresentative base year distorts comparisons.
This completes the ASSEB Class 11 Economics Chapter 8 — Index Numbers study notes from HSLC Guru, covering theory, numerical methods, the major Indian index numbers, MCQs, fill-in-the-blanks, true/false, glossary and a complete formula table for examination revision.