Index Numbers: ISC Class 11 Economics Notes and Numericals

What Index Numbers is about

Weightage: Index Numbers is part of the unit Statistics of the ISC Class XI Economics syllabus. CISCE gives no Class XI unit weightage for Economics: Paper I (Theory) carries 80 marks, and the school sets the paper.Source: CISCE ISC Economics syllabus, examination year 2028, Class XI paper pattern

Key concepts

Index number: meaning, features and types

An index number is a statistical device that measures the average (relative) change in a group of related variables (prices, quantities, values) over time or between places, compared with a base period whose value is taken as 100.

What it shows: a relative change, in percentage terms. A price index of 125 means prices on average are 25% higher than in the base year.

Features: (1) expressed as a pure number (no units), usually with base = 100; (2) measures average change in a group; (3) measures change relative to a base; (4) a specialised average (of relatives or aggregates).

Types:

1. Price index: change in prices (WPI, CPI).
2. Quantity index: change in physical volume of output or sales (IIP).
3. Value index: change in total value (price × quantity).

Notation: $p_0$, $q_0$ = base-year price and quantity; $p_1$, $q_1$ = current-year price and quantity; $P_{01}$ = price index of the current year with base 0.

Worked example

Suppose the CPI rises from 100 in the base year to 142. What does this indicate? [1 mark]

Try it first, then show the solutionSolution

1. 142 − 100 = 42: prices of the consumer basket have risen 42% on average since the base year.

Answer: The cost of the consumer basket has risen by 42% on average compared with the base year.

Simple (unweighted) index numbers

In a simple index, all items are given equal importance.

1. Simple aggregative method: $P_{01}=\frac{\sum p_1}{\sum p_0}\times 100$

2. Simple average of price relatives: $P_{01}=\frac{\sum \left(\frac{p_1}{p_0}\times 100\right)}{N}$

Example: prices (₹) of four items:

Item	$p_0$	$p_1$	Price relative ($p_1/p_0\times 100$)
Rice (kg)	20	25	125
Dal (kg)	40	44	110
Salt (kg)	10	15	150
Oil (litre)	50	60	120
Total	120	144	505

Aggregative: 144 ÷ 120 × 100 = 120. Average of relatives: 505 ÷ 4 = 126.25.

Limitations: the aggregative method is affected by the units in which prices are quoted (a price per quintal swamps prices per kg); both methods give equal importance to all items, though rice matters far more to a household than salt.

Worked example

Suppose prices of three goods change from ₹8, ₹12, ₹30 to ₹10, ₹15, ₹33. Find the simple aggregative index and the simple average of price relatives. [2 marks]

Try it first, then show the solutionSolution

1. Aggregative: (10 + 15 + 33) ÷ (8 + 12 + 30) × 100 = 58 ÷ 50 × 100 = 116.
2. Relatives: 125, 125, 110; average = 360 ÷ 3 = 120.

Answer: Aggregative 116; average of relatives 120.

Weighted index numbers: Laspeyres and Paasche

A weighted index gives each item a weight reflecting its importance (usually quantities bought or expenditure shares). Price-weighted quantity indices and quantity-weighted price indices both exist.

Laspeyres' price index (base-year quantities as weights):

$P_{01}^L=\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100$

Paasche's price index (current-year quantities as weights):

$P_{01}^P=\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\times 100$

Example (same four items):

Item	$p_0$	$q_0$	$p_1$	$q_1$	$p_0{q}_0$	$p_1{q}_0$	$p_0{q}_1$	$p_1{q}_1$
Rice	20	10	25	8	200	250	160	200
Dal	40	5	44	6	200	220	240	264
Salt	10	20	15	15	200	300	150	225
Oil	50	4	60	5	200	240	250	300
Total					800	1,010	800	989

Laspeyres = 1,010 ÷ 800 × 100 = 126.25. Paasche = 989 ÷ 800 × 100 = 123.63.

Why Laspeyres is usually higher: consumers buy less of goods whose prices rise most (salt: quantity fell from 20 to 15). Laspeyres keeps the old, larger quantities, so it overstates the rise in cost; Paasche uses the new, smaller quantities, so it tends to understate it.

Laspeyres: easy, needs base quantities only once; widely used (CPI-type). Paasche: needs new quantities every period, so it is costlier to compute.

Worked example

Suppose Σp0q0 = 118, Σp1q0 = 145, Σp0q1 = 124 and Σp1q1 = 152. Find Laspeyres' and Paasche's price indices. [2 marks]

Try it first, then show the solutionSolution

1. Laspeyres = 145 ÷ 118 × 100 = 122.88.
2. Paasche = 152 ÷ 124 × 100 = 122.58.

Answer: Laspeyres 122.88; Paasche 122.58.

Fisher's ideal index

Fisher's index is the geometric mean of Laspeyres' and Paasche's indices:

$P_{01}^F=\sqrt{\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times \frac{\sum p_1{q}_1}{\sum p_0{q}_1}}\times 100$

Example: $\sqrt{126.25\times 123.625}$ = 124.93.

Why called 'ideal':

1. Uses both base-year and current-year quantities, so it avoids the upward bias of Laspeyres and the downward bias of Paasche.
2. Uses the geometric mean, which is suitable for averaging ratios.
3. Satisfies the time reversal test ($P_{01}\times P_{10}=1$, in ratio form) and the factor reversal test ($P_{01}\times Q_{01}=\frac{\sum p_1{q}_1}{\sum p_0{q}_0}$).

Limitation: needs data on both years' quantities and involves more calculation, so official indices often use Laspeyres-type formulas.

Worked example

If Laspeyres' index is 122.88 and Paasche's index is 122.58, find Fisher's index. [1 mark]

Try it first, then show the solutionSolution

1. Fisher = √(122.88 × 122.58) = √15,062.6 = 122.73.

Answer: 122.73.

WPI, CPI and IIP

Wholesale Price Index (WPI): measures the average change in prices of goods traded in bulk (wholesale) by producers and traders: primary articles, fuel and power, and manufactured products. Services are not included. Compiled by the Office of the Economic Adviser (Ministry of Commerce and Industry). Used to measure inflation at the producer and wholesale level and to deflate output.

Consumer Price Index (CPI): measures the average change in retail prices of a fixed basket of goods and services bought by a particular group of consumers (food, housing, clothing, fuel, health, education, transport). Series include CPI (Rural, Urban, Combined) by the National Statistics Office, and CPI for Industrial Workers by the Labour Bureau. The CPI (Combined) is used by the RBI for its inflation target; CPI-IW is used to revise dearness allowance of government employees.

Index of Industrial Production (IIP): a quantity index measuring short-term changes in the volume of industrial output: mining, manufacturing and electricity. Compiled monthly by the National Statistics Office. It shows industrial growth or decline.

WPI vs CPI: WPI uses wholesale prices and excludes services; CPI uses retail prices and includes services; weights differ; CPI reflects the cost of living for households.

Inflation rate from an index = (Index this year − Index last year) ÷ Index last year × 100.

(Base years of these series are revised periodically; check the latest official release before quoting one.)

Worked example

Suppose the CPI was 180 last year and 189 this year. Find the rate of inflation. [1 mark]

Try it first, then show the solutionSolution

1. Inflation = (189 − 180) ÷ 180 × 100 = 5%.

Answer: 5%.

Uses of index numbers and problems in constructing them

Uses:

1. Measure inflation (WPI, CPI) and guide monetary policy.
2. Deflate money values into real values: real income = money income × 100 ÷ price index.
3. Wage adjustment: dearness allowance linked to CPI-IW.
4. Policy formulation: industrial policy from IIP; agricultural price policy from farm price indices.
5. Measure changes in the standard of living and purchasing power of money (purchasing power = 1 ÷ price index).
6. Business decisions: sales and cost trends; share-price indices (stock market indices).

Problems in constructing a price index:

1. Purpose: the purpose decides every other choice (whose prices, which goods).
2. Choice of base year: it must be a normal year (no war, drought, boom) and not too distant; otherwise the index exaggerates or understates change.
3. Coverage of commodities: which and how many items; they must be representative of the group's consumption, and their quality must stay comparable over time.
4. Choice of prices: wholesale or retail; which markets; average of which period; controlled or open-market prices.
5. Choice of weights: quantity or value weights; base or current year.
6. Choice of average and method: arithmetic or geometric mean; Laspeyres, Paasche or Fisher.
7. Changes in tastes, quality and new products over time make the basket outdated, so indices are periodically revised.

Worked example

Suppose a worker's money wage rises from ₹15,000 to ₹19,500 while the CPI-IW rises from 100 to 125. Find her real wage and its change. [2 marks]

Try it first, then show the solutionSolution

1. Real wage = 19,500 × 100 ÷ 125 = ₹15,600.
2. Change = (15,600 − 15,000) ÷ 15,000 × 100 = 4%.

Answer: Real wage ₹15,600 at base prices: a real rise of 4% (money wage rose 30%).

Index Numbers in real life

Each concept, met in an everyday situation. Read the situation, answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Topics in this chapter

• Simple Index Numbers: Aggregative and Price Relatives Methods, Base Shifting, Real Wages
• Weighted Index Numbers: Laspeyres, Paasche, Fisher, and WPI, CPI and IIP

Index Numbers: key facts

Simple aggregative

$P_{01}=\frac{\sum p_1}{\sum p_0}\times 100$

Simple average of price relatives

$P_{01}=\frac{\sum (p_1/p_0\times 100)}{N}$

Laspeyres

$P_{01}^L=\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100$

Paasche

$P_{01}^P=\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\times 100$

Fisher

$P_{01}^F=\sqrt{P^L\times P^P}$

Satisfies time reversal and factor reversal tests.

Weighted average of price relatives

$P_{01}=\frac{\sum RW}{\sum W}$, where $R=\frac{p_1}{p_0}\times 100$

The method behind CPI-type indices.

Real value and inflation

Real value = Money value × 100 ÷ Index; Inflation = (I1 − I0) ÷ I0 × 100

Index Numbers: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the school exam asks Index Numbers

The question patterns to expect, and the method that works for each.

Tricks to remember Index Numbers

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Laspeyres looks back, Paasche looks at the present

  L uses q0 (last time's quantities); P uses q1 (present quantities).

  Use it when: Choosing weights.

• Fisher sits in the middle

  Geometric mean of L and P, so it lies between them.

  Use it when: Fisher's index.

• WPI: Wholesale, no services; CPI: Consumer, with services

  W for warehouse, C for cart.

  Use it when: WPI vs CPI.

• Problems: B-C-P-W-M

  Base year, Coverage, Prices, Weights, Method.

  Use it when: Problems in construction.

Index Numbers: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section C style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 2 + 1 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Index Numbers

The slips examiners see most often, and what to write instead.

• ✗Mistake: Swapping the weights of Laspeyres and Paasche

  Laspeyres uses q0; Paasche uses q1.

  ✓Do this: L = last (base) quantities; P = present quantities.

  Usually costs: 2 marks.

• ✗Mistake: Forgetting to multiply by 100

  Index numbers are expressed with base = 100.

  ✓Do this: Write '× 100' in every formula.

  Usually costs: 1 mark.

• ✗Mistake: Reading an index of 130 as '130% rise'

  130 means a 30% rise over the base.

  ✓Do this: Change = index − 100 (from the base year).

  Usually costs: 1 mark.

• ✗Mistake: Saying WPI includes services

  WPI covers goods traded wholesale; CPI includes services.

  ✓Do this: W for warehouse goods.

  Usually costs: 1 mark.

Practice questions on Index Numbers

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Fisher's index is the:
   (a) arithmetic mean of Laspeyres' and Paasche's indices
   (b) geometric mean of Laspeyres' and Paasche's indices
   (c) harmonic mean of the price relatives
   (d) simple aggregative index

   Hint

   Square root of the product.

   Answer

   (b) geometric mean of Laspeyres' and Paasche's indices.

2. Q2 · 1 mark · Very short answerExam level

   What does a price index of 85 indicate?

   Hint

   Below 100.

   Answer

   Prices are on average 15% lower than in the base year.

3. Q3 · 2 marks · Short answerExam level

   State two differences between WPI and CPI.

   Hint

   Prices and coverage.

   Answer

   WPI uses wholesale prices, CPI retail prices. WPI covers only goods; CPI also includes services such as housing, education and health (weights also differ).

4. Q4 · 4 marks · Short answerHarder exam level

   Suppose Σp1q0 = 690, Σp0q0 = 600, Σp1q1 = 780 and Σp0q1 = 650. Calculate Laspeyres', Paasche's and Fisher's indices.

   Hint

   Fisher = √(L × P).

   Answer

   Laspeyres = 690 ÷ 600 × 100 = 115; Paasche = 780 ÷ 650 × 100 = 120; Fisher = √(115 × 120) = √13,800 = 117.47.

5. Q5 · 2 marks · Short answerHarder exam level

   Suppose prices of 4 items rise from ₹10, ₹25, ₹40, ₹5 to ₹12, ₹30, ₹44, ₹8. Find the simple average of price relatives.

   Hint

   Relatives first.

   Answer

   Relatives: 120, 120, 110, 160; average = 510 ÷ 4 = 127.5.

6. Q6 · 2 marks · Short answerHarder exam level

   Explain why the base year should be a normal year and not too distant.

   Hint

   Abnormal prices; outdated basket.

   Answer

   An abnormal base (war, drought, boom) has unusual prices, so every later index is distorted. A distant base has outdated consumption patterns and quality, so the basket and weights no longer represent current spending.

7. Q7 · 4 marks · Short answerHOTS / competency

   Suppose a household's spending in the base year was: food ₹6,000, rent ₹3,000, education ₹1,000. Current price relatives are 125, 140 and 160. Calculate the cost of living index using expenditure weights, and the rise in the cost of living.

   Hint

   ΣRW ÷ ΣW with W = spending.

   Answer

   ΣRW = 125 × 6,000 + 140 × 3,000 + 160 × 1,000 = 7,50,000 + 4,20,000 + 1,60,000 = 13,30,000; ΣW = 10,000. Index = 133: the cost of living has risen 33%.

8. Q8 · 2 marks · Short answerHOTS / competency

   Suppose a firm's sales rose from ₹50 crore to ₹66 crore while the relevant price index rose from 100 to 120. Did the volume of sales rise? By how much?

   Hint

   Deflate.

   Answer

   Real sales = 66 × 100 ÷ 120 = ₹55 crore at base prices: volume rose by 10% (money sales rose 32%).

9. Q9 · 4 marks · Short answerHarder exam level

   Explain the uses of index numbers in economics.

   Hint

   Inflation, deflating, wages, policy.

   Answer

   Measuring inflation (WPI, CPI) to guide monetary policy; deflating money values to real values (real income, real GDP); adjusting wages and dearness allowance (CPI-IW); guiding industrial and agricultural policy (IIP, farm price indices); measuring the purchasing power of money and changes in the standard of living; and helping businesses track costs and sales.

10. Q10 · 6 marks · Long answerHOTS / competency

    'Laspeyres overstates and Paasche understates; Fisher gets it right.' Examine this statement with reference to a situation where consumers switch away from goods that become dearer.

    Hint

    Substitution bias.

    Answer

    When the price of a good rises, consumers buy less of it and more of substitutes. Laspeyres weights prices by base-year quantities, so it gives too much weight to the goods that became dearer and overstates the rise in the cost of maintaining living standards. Paasche weights by current quantities, which already reflect switching away, and so tends to understate the rise. Fisher, the geometric mean of the two, lies between them (in the example, 126.25 > 124.93 > 123.63) and satisfies the time reversal and factor reversal tests, so it is called 'ideal'. However, it is not perfect: it needs current quantities every period, and no index fully captures quality changes or new products. In practice, official indices often use Laspeyres-type formulas with periodically revised weights.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Base-year index value?Show answer

   100.

2. 2.Laspeyres' weights?Show answer

   Base-year quantities (q0).

3. 3.Paasche's weights?Show answer

   Current-year quantities (q1).

4. 4.Fisher's index?Show answer

   √(Laspeyres × Paasche).

5. 5.Who compiles CPI-IW?Show answer

   The Labour Bureau.

6. 6.IIP measures?Show answer

   Volume of industrial output (quantity index).

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Index number: average relative change over a base = 100.
• ☐Simple: Σp1/Σp0 × 100 or average of p1/p0 × 100.
• ☐Laspeyres Σp1q0/Σp0q0 × 100; Paasche Σp1q1/Σp0q1 × 100.
• ☐Fisher = √(L × P): ideal; satisfies time and factor reversal.
• ☐WPI (wholesale goods), CPI (retail goods and services), IIP (industrial volume).
• ☐Problems: base year, coverage, prices, weights, method.

Index Numbers: frequently asked questions