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Index Number Construction — Laspeyres, Paasche, Fisher

🎓 Class 11 Social Science CBSE Theory Ch 7 — Index Numbers ⏱ ~28 min
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Class 11 · Statistics for Economics · Chapter 7 · Part 1

Index Numbers — Construction, Laspeyres, Paasche and Fisher Formulas

Imagine Rabi returning home from the bazaar. Tomatoes have become dearer, dal has become cheaper, soap has stayed the same and sugar has shot up. He rattles off twelve different price changes to his father; both end up confused. The industrial economy presents the same puzzle on a vastly bigger scale — coal output rising, steel falling, cement steady, sugar mills slowing down. Can a single figure summarise dozens or hundreds of changes at once? That is the job of an index number. Part 1 introduces the meaning of an index, walks through the simple aggregative method, the simple average of price relatives, and then the three weighted heavyweights — Laspeyres, Paasche and Fisher's ideal — using the worked NCERT examples step by step.

7.1 From Many Changes to One Number

You have already learned, in earlier chapters, how a single mean or median can summarise a mass of values for one variable. An index number stretches that idea to a group of related variables changing simultaneously. NCERT motivates the topic with three small cases: an industrial worker whose salary rose from Rs 1,000 in 1982 to Rs 12,000 today (has his standard of living really risen twelve times?); the BSE Sensex crossing 8,000 points and erasing Rs 1,53,690 crores of investor wealth when it dipped 600 points; and the government's claim that inflation will not accelerate even after a fuel price hike — a claim that only an inflation index can verify.

📖 Definition — Index Number
An index number? is a statistical device for measuring change in the magnitude of a group of related variables. It represents the general trend of diverging ratios from which it is calculated and acts as a measure of the average change in those variables over two different situations — typically two time periods or two places. The comparison may be between like categories (persons, schools, hospitals) or between values such as prices, output or cost of living.

7.1.1 The Base and the Current Period

Conventionally, an index number is expressed as a percentage. The two situations being compared are called the base period? and the current period?. The base is the benchmark; its index is set, by convention, to 100. The current-period index is then expressed in proportion to that benchmark. An index number of 250 therefore means that the current value is two-and-a-half times the base value. An index of 138.5 means a 38.5 per cent rise; an index of 97.86 (we shall meet it later) means a 2.14 per cent fall.

📅
Base Period
The reference year against which all comparisons are made. Always set to 100. Should be a "normal" year — no famine, no war, no boom — and routinely updated as consumption baskets evolve.
🕔 Current Period
The period being compared with the base. Its index reads as a percentage of the base: 138.5 means 38.5% above the base, 92 means 8% below.
💰
Price Index
Measures the average change in the prices of a fixed basket of goods. The most familiar — Wholesale Price Index, Consumer Price Index — fall in this family.
🏭
Quantity Index
Measures changes in the physical volume of production, construction or employment. The Index of Industrial Production is a classic quantity index.
NCERT Case 1 An industrial worker earned Rs 1,000 in 1982. Today he earns Rs 12,000. Has his standard of living really risen twelve times? Not unless prices stayed flat. If the cost of living index has also climbed sharply over those decades, much of the rupee rise is "swallowed" by inflation. The right answer needs the Consumer Price Index, not just the salary slip.

7.2 Two Families of Construction Methods

NCERT lists two ways of building a price index from the underlying numbers — the aggregative method (add up the prices, then take a ratio) and the method of averaging relatives (compute a price relative for each item, then take the average). Each can be either unweighted (every item counted equally) or weighted (more important items counted more heavily). Combine the two axes and you get four possibilities:

Index Numbers Aggregative Method Averaging Relatives Simple (P₁/P₀) Weighted Aggreg. Simple Avg of P.R. Weighted P.R. Laspeyres (q₀) Paasche (q₁) Fisher (geometric mean)
Tree of price-index formulas. The two families branch into unweighted (left, simple) and weighted (right, real-world) methods. Within "Weighted Aggregative" sit the three big-name formulae — Laspeyres, Paasche and Fisher's ideal.

7.3 Method 1 — Simple Aggregative Index

The first and easiest construction adds up the current-period prices, divides by the sum of base-period prices, and multiplies by 100.

P₀₁ = ΣP₁ΣP₀ × 100

Where P₀ is the base-period price of a commodity and P₁ is its current-period price. The subscript "01" reads "from period 0 to period 1".

7.3.1 Worked Example 1 — NCERT Table 7.1

📝 NCERT Example 1 — Simple Aggregative Index
Four commodities (A, B, C, D) have base-period and current-period prices. Compute the simple aggregative price index P₀₁.
Table 7.1 — NCERT data for the simple aggregative index
CommodityBase price P₀ (Rs)Current price P₁ (Rs)Percentage change
A24+100%
B56+20%
C45+25%
D23+50%
TotalΣP₀ = 13ΣP₁ = 18
P₀₁ = 4 + 6 + 5 + 32 + 5 + 4 + 2 × 100 = 1813 × 100 = 138.5

Prices of the basket are said to have risen by 38.5 per cent. Notice that the four percentage changes (+100, +20, +25, +50) have been compressed into a single, communicable number.

⚠ Why the Simple Index is of Limited Use
Two flaws make this index a beginner's tool only. (a) The units of measurement of different commodities are not the same — adding kilograms of rice to litres of oil to bars of soap mixes apples and oranges. (b) The items are treated as equally important: a 100% jump in a low-share item like soap counts as much as a 20% jump in food, even though food eats a much bigger slice of the household budget. Real consumption baskets are not democratic; some items dominate. The remedy is weighting.

7.4 Method 2 — Weighted Aggregative Index

The cure for the equality-of-items problem is to attach a weight to each commodity that reflects its importance in real consumption. The natural choice for weights is the quantity consumed — either in the base period (q₀) or in the current period (q₁). The weighted aggregative index has the form:

P₀₁ = ΣP₁qΣP₀q × 100

where the same set of quantities q appears in both numerator and denominator. The basket of goods is fixed; only the prices change. As the total expenditure on this fixed basket changes, the change must be due to price movement alone — this is the conceptual cleverness of the weighted index. Different choices of q give different formulas, of which three are crucial.

7.4.1 Laspeyres' Price Index

The Laspeyres index?, devised by the German economist Etienne Laspeyres in the 1870s, uses the base-period quantities q₀ as weights. It answers the question: "If the expenditure on the base-period basket was Rs 100, how much would it cost in the current period?"

P₀₁L = ΣP₁q₀ΣP₀q₀ × 100

7.4.2 Paasche's Price Index

The Paasche index?, named after another German economist, Hermann Paasche (1874), uses current-period quantities q₁ as weights. It answers: "If the current-period basket had been consumed in the base period at base prices, how much would Rs 100 of expenditure on it have grown to today?"

P₀₁P = ΣP₁q₁ΣP₀q₁ × 100

7.4.3 Fisher's Ideal Index

Both Laspeyres and Paasche have a flaw — Laspeyres tends to overstate the price rise (because base-period quantities ignore the way consumers shift away from items that have become dearer) while Paasche tends to understate it (because current-period quantities already reflect the shift away). The American economist Irving Fisher proposed an "ideal" formula that takes the geometric mean of the two, balancing the two biases against each other.

P₀₁F = √( P₀₁L × P₀₁P ) = √( ΣP₁q₀ΣP₀q₀ × ΣP₁q₁ΣP₀q₁ ) × 100
📖 Definition — Fisher's Ideal Index?
Fisher's index is the geometric mean of the Laspeyres and Paasche indices. It is called "ideal" because it satisfies two formal tests of a good index — the time-reversal test (the index for period 0 to 1 equals the reciprocal of the index for period 1 to 0) and the factor-reversal test (price index times quantity index equals the value ratio). Neither Laspeyres nor Paasche satisfies both.

7.5 Worked Example 2 — Computing Laspeyres and Paasche

📝 NCERT Example 2 — Weighted Aggregative Indices
Four commodities A, B, C, D have base- and current-period prices and quantities. Compute Laspeyres' and Paasche's price indices.
Table 7.2 — Base and current period prices & quantities
CommodityP₀ (Rs)q₀P₁ (Rs)q₁
A21045
B512610
C420515
D215310

7.5.1 Building the Sums Step by Step

The four key sums are best computed in a working table. The numerator of Laspeyres is ΣP₁q₀; the denominator is ΣP₀q₀. The numerator of Paasche is ΣP₁q₁; the denominator is ΣP₀q₁.

Working table — products needed for Laspeyres and Paasche
CommodityP₀q₀P₁q₀P₀q₁P₁q₁
A2×10 = 204×10 = 402×5 = 104×5 = 20
B5×12 = 606×12 = 725×10 = 506×10 = 60
C4×20 = 805×20 = 1004×15 = 605×15 = 75
D2×15 = 303×15 = 452×10 = 203×10 = 30
Total190257140185

7.5.2 Laspeyres — Using Base Quantities

P₀₁L = ΣP₁q₀ΣP₀q₀ × 100 = 257190 × 100 = 135.3

Reading: holding the base-period basket fixed, the price level has risen by 35.3 per cent from period 0 to period 1. If a household used to spend Rs 100 on its 1990 basket, the same basket will cost Rs 135.3 today.

7.5.3 Paasche — Using Current Quantities

P₀₁P = ΣP₁q₁ΣP₀q₁ × 100 = 185140 × 100 = 132.1

Using the current basket as weights, the price rise is 32.1 per cent. Notice that Paasche (132.1) is a little smaller than Laspeyres (135.3) — exactly the substitution-bias gap predicted by theory. Quantities of A and B fell sharply (10→5 and 12→10) precisely because their prices doubled or rose; current weights downplay these expensive items, lowering the index.

7.5.4 Fisher's Ideal — The Geometric Mean

P₀₁F = √( 135.3 × 132.1 ) = √17,873 ≈ 133.7

Fisher's index lands between the two, at 33.7 per cent rise — a balanced compromise between the over-statement of Laspeyres and the under-statement of Paasche.

Fig. 7.2 — Side-by-side comparison of the three weighted aggregative indices on Table 7.2 data. Laspeyres tops the rise (135.3); Paasche records the smallest (132.1); Fisher's ideal balances them at 133.7.

7.6 Method 3 — Simple Average of Price Relatives

Instead of adding up rupees and rupees of different commodities (an apples-and-oranges objection), the second family of methods first standardises each price by computing a price relative — the ratio of current price to base price, multiplied by 100 — and then averages the relatives across commodities.

📖 Definition — Price Relative?
The price relative of a commodity is the ratio of its current-period price to its base-period price, expressed as a percentage: (P₁/P₀) × 100. It is a unit-free number — kilograms cancel kilograms — and so it can be safely averaged across heterogeneous commodities.

The simple (unweighted) average of price relatives is an arithmetic mean over n commodities:

P₀₁ = 1n Σ P₁P₀ × 100

7.6.1 NCERT Example — Simple Average on the Same Four Commodities

Reusing the data of Table 7.1, the price relatives are 200%, 120%, 125% and 150% for A, B, C and D respectively. The simple average is:

P₀₁ = 14 ( 42 + 65 + 54 + 32 ) × 100 = 200 + 120 + 125 + 1504 = 148.75 ≈ 149

The unweighted average reports a 49 per cent price rise — much higher than the simple aggregative figure (38.5%). The reason is that the relatives method gives the cheap-but-doubled commodity A as much voice as the expensive-but-only-mildly-rising commodity B. NCERT rounds the figure to 149.

7.7 Method 4 — Weighted Average of Price Relatives

The weighted version of the relatives method takes a weighted arithmetic mean, where each weight Wi reflects the share of expenditure on commodity i (usually in the base period). The formula is:

P₀₁ = Σ Wi ( P₁i/P₀i ) × 100Σ Wi

Weights may be expressed as percentages (in which case ΣW = 100), or as actual rupee value-shares of expenditure. NCERT prefers base-period weights because (a) computing weights every year is impractical and (b) using current weights makes successive indices strictly non-comparable, since the basket itself keeps changing.

7.7.1 NCERT Example 3 — Weighted Price Relatives

Table 7.3 — Weights and price relatives for four commodities
CommodityWeight (W) %Base price (Rs)Current price (Rs)Price relative R = (P₁/P₀)×100WR
A402420040×200 = 8000
B305612030×120 = 3600
C204512520×125 = 2500
D102315010×150 = 1500
Total10015600
P₀₁ = 15600100 = 156

The weighted price index is 156, signalling a 56 per cent rise. It is sharply higher than the unweighted figure (148.75), because the most-doubled item, A, also carries the largest weight (40 per cent) — its outsized rise pulls the average up. The contrast between unweighted and weighted figures is precisely the reason policy makers insist on weighting.

Fig. 7.3 — Comparing all four index numbers on the same data context. Unweighted methods (138.5, 149) hide the dominance of high-share commodities; weighting amplifies the pull of large-budget items A and B (156).
EXPLORE — NCERT Activity (Reverse Laspeyres & Paasche)
Bloom: L3 Apply

Interchange the current-period values with the base-period values in the data given in Example 2 (Table 7.2). Calculate Laspeyres' and Paasche's indices for the reversed data. What difference do you observe from the original calculation?

✅ Sample Observation
Once we swap, the new "base" prices and quantities are 4, 6, 5, 3 and 5, 10, 15, 10. The new "current" prices and quantities are 2, 5, 4, 2 and 10, 12, 20, 15. Re-running the formulae gives a Laspeyres index for the reversed period of about 75.7 (i.e. a 24.3 per cent fall) and a Paasche index of about 73.9. Multiplied by the original Laspeyres figure of 135.3, the product is roughly 100×100 (a small rounding gap). This near-identity is the time-reversal property — Fisher's ideal index passes it exactly, while Laspeyres and Paasche only approximate it. The exercise visually shows why economists prefer Fisher's geometric mean for symmetric, two-way comparisons.

7.8 Side-by-Side — The Three Weighted Heavyweights

Laspeyres P₀₁L

  • Weight: base-period quantity q₀
  • Asks: "How much would the 1990 basket cost today?"
  • Bias: tends to overstate price rise (no substitution allowed)
  • Used by: Indian CPI, US CPI, most national statistics offices
  • Easy to update: q₀ is fixed once at base year

Paasche P₀₁P

  • Weight: current-period quantity q₁
  • Asks: "What did today's basket cost in 1990?"
  • Bias: tends to understate price rise (substitution already embedded)
  • Used by: theoretical work, GDP deflator (Paasche-style)
  • Hard to update: q₁ must be re-collected every period

Fisher's Ideal P₀₁F — The Geometric Mean

  • Definition: √(L × P), the geometric mean of Laspeyres and Paasche.
  • Why "ideal": satisfies the time-reversal test (index 0→1 = 1/index 1→0) and the factor-reversal test (price index × quantity index = value ratio).
  • Bias: averages the over- and under-statements; close to the underlying truth.
  • Drawback: requires both q₀ and q₁, so demands more data; rarely used by national agencies for routine inflation reporting.
THINK — Why is Laspeyres More Popular Than Paasche?
Bloom: L4 Analyse

If Paasche is theoretically just as legitimate as Laspeyres, why does the Reserve Bank of India and the Ministry of Statistics overwhelmingly prefer Laspeyres for the official CPI and WPI series?

✅ Sample
Three reasons. (1) Data cost. Laspeyres needs base-period quantities just once (collected via the consumer expenditure survey at the time of base-year revision). Paasche would require re-surveying household quantities every month — impossible in practice. (2) Comparability. The Laspeyres basket is fixed, so successive months compare like with like. Paasche changes its basket every period, making month-to-month comparisons strictly non-equivalent. (3) Speed. Laspeyres can be computed within weeks of the price-collection round; Paasche cannot. The price of these advantages is a small upward bias, which the statistical office tries to correct by updating the base year roughly every decade.
DISCUSS — Why Do the Three Indices Disagree?
Bloom: L5 Evaluate

For the NCERT Example 2 data, Laspeyres reports 135.3, Paasche 132.1 and Fisher 133.7. Discuss in pairs why all three are simultaneously "correct" yet give different numbers, and which one a policy-maker should quote.

✅ Sample
All three are arithmetically correct — they simply answer slightly different questions. Laspeyres asks "what would the old basket cost?", Paasche asks "what did the new basket cost in old prices?", and Fisher asks "what is the balanced average of the two?". The difference reflects substitution: as A and B became dearer, households cut back. Laspeyres ignores that adjustment and so reports a bigger pinch; Paasche embeds it and so reports a smaller pinch; Fisher splits the difference. A central bank announcing inflation usually quotes Laspeyres (CPI methodology), but a national-income statistician computing real GDP often uses a Paasche-style deflator. The "right" choice depends on the question.

7.9 A Worked Case-Based Question

📋 Case-Based Question — Building the Mini-CPI for a Hostel Mess

A college mess buys three commodities every week — rice (R), dal (D) and oil (O). In the base week of January 2024 the price-quantity figures were: R = Rs 40/kg, 50 kg; D = Rs 100/kg, 20 kg; O = Rs 120/litre, 10 litres. In the current week of January 2025 the figures are: R = Rs 50/kg, 45 kg; D = Rs 130/kg, 18 kg; O = Rs 140/litre, 12 litres. The mess accountant wants to build a price index to negotiate the next month's grant from the college.
Q1. Define an "index number" and state, in one sentence, the role of the base period.
L1 Remember
Answer: An index number is a statistical device that measures the average change in a group of related variables (such as prices) across two situations, expressed conventionally as a percentage. The base period is the benchmark whose value is set to 100 and against which all other values are compared.
Q2. Compute the simple aggregative price index for the mess.
L3 Apply
Answer: ΣP₀ = 40 + 100 + 120 = 260. ΣP₁ = 50 + 130 + 140 = 320. P₀₁ = (320/260) × 100 = 123.1. The simple aggregative index reports a 23.1 per cent rise.
Q3. Compute Laspeyres' price index using base-period quantities, and explain what its value means for the mess accountant.
L4 Analyse
Answer: ΣP₀q₀ = 40×50 + 100×20 + 120×10 = 2000 + 2000 + 1200 = 5200. ΣP₁q₀ = 50×50 + 130×20 + 140×10 = 2500 + 2600 + 1400 = 6500. P₀₁L = (6500/5200) × 100 = 125.0. The mess basket of January 2024 would cost 25 per cent more in January 2025. The accountant should ask for at least a 25 per cent grant hike to keep buying the same basket.
Q4. Compute Paasche's index and Fisher's ideal index. Explain why Fisher's value lies between Laspeyres and Paasche.
L5 Evaluate
Answer: ΣP₀q₁ = 40×45 + 100×18 + 120×12 = 1800 + 1800 + 1440 = 5040. ΣP₁q₁ = 50×45 + 130×18 + 140×12 = 2250 + 2340 + 1680 = 6270. P₀₁P = (6270/5040) × 100 = 124.4. Fisher's index = √(125.0 × 124.4) ≈ 124.7. Fisher's value lies between Laspeyres and Paasche by definition — it is their geometric mean. Whenever L and P differ, the geometric mean is squeezed between them; this averaging property is precisely why Fisher's index is called "ideal".

7.10 Assertion–Reason Questions

⚖ Assertion–Reason Questions (Class 11)

Choose: (A) Both A and R are true and R is the correct explanation of A. (B) Both A and R are true but R is not the correct explanation of A. (C) A is true, R is false. (D) A is false, R is true.

Assertion (A): The simple aggregative price index is the most reliable measure of inflation because it adds up all prices directly.
Reason (R): A simple aggregative index treats every commodity as equally important and does not adjust for the units of measurement, so it is of limited use in real economies.
Correct: (D) — Assertion is false. The simple aggregative index is in fact the least reliable of the methods studied because it ignores both the units of measurement (kilograms cannot be added to litres without absurdity) and the relative importance of items in expenditure. Reason is true and is itself the standard textbook critique that pushes statisticians toward weighted formulae.
Assertion (A): Fisher's ideal index satisfies both the time-reversal and the factor-reversal tests.
Reason (R): Fisher's index is defined as the geometric mean of Laspeyres' and Paasche's indices, and the geometric mean structure mathematically forces both reversal tests to hold exactly.
Correct: (A) — Both statements are true and R is the correct explanation of A. The very name "ideal" was given by Irving Fisher precisely because the geometric mean of Laspeyres and Paasche is symmetric in periods 0 and 1 and so passes the formal axiomatic tests of an index, while the arithmetic Laspeyres and Paasche each fail at least one of them.
Assertion (A): Indian government statistical agencies use the Laspeyres formula rather than the Paasche formula for routine CPI and WPI publication.
Reason (R): Laspeyres requires a fresh quantity survey every month, which is logistically simpler than Paasche's once-in-a-decade base-year survey.
Correct: (C) — Assertion is true (Laspeyres is indeed the workhorse of Indian official statistics, as it is for most countries), but Reason is false — the practical convenience runs the opposite way. Laspeyres needs the base-period basket only once (collected at the base-year revision), whereas Paasche would need fresh quantity data every period. The data-cost argument is a major reason for Laspeyres' dominance, but the Reason statement has the logic inverted.

Continue to Part 2 — Some Important Indian Index Numbers (CPI, WPI, IIP, Sensex, HDI), Issues in Construction, NCERT Exercises and Summary.

Frequently Asked Questions — Index Numbers — Construction, Laspeyres, Paasche and Fisher Formulas

What is an index number in NCERT Class 11 Statistics Chapter 7?

An index number is a statistical device that measures relative changes in a variable or a group of related variables across time, geographic locations or other characteristics, with a chosen reference period (base year) set to 100. NCERT Class 11 Statistics Chapter 7 Part 1 explains that index numbers help compare changes in prices, quantities, wages or production over time. Common examples include the Consumer Price Index (CPI), Wholesale Price Index (WPI), Index of Industrial Production (IIP) and Sensex. They are called economic barometers because they summarise complex movements into a single number that can be tracked easily.

What is the formula for Laspeyres price index in Class 11 Statistics?

The Laspeyres price index uses the base-year quantities as weights and is calculated as P01(L) = (Σp1·q0 / Σp0·q0) × 100, where p1 and p0 are current-year and base-year prices respectively, and q0 is the base-year quantity. NCERT Class 11 Statistics Chapter 7 Part 1 explains that Laspeyres is the most widely used formula because it requires only base-year quantity data, which is fixed over time. However, it overstates inflation because it ignores how consumers substitute away from goods whose prices rise — the substitution bias.

What is the difference between Laspeyres and Paasche price index?

Laspeyres uses base-year quantities (q0) as weights, giving P01(L) = (Σp1·q0/Σp0·q0)×100, while Paasche uses current-year quantities (q1), giving P01(P) = (Σp1·q1/Σp0·q1)×100. NCERT Class 11 Statistics Chapter 7 Part 1 notes that Laspeyres tends to overstate price increases (upward bias) because it ignores substitution toward cheaper goods, while Paasche tends to understate them (downward bias) because it gives less weight to goods whose prices have risen most. Fisher's ideal index averages the two using a geometric mean to balance these biases and satisfies more tests of a good index number.

Why is Fisher's index called the ideal index in NCERT Class 11 Statistics?

Fisher's index is called the ideal index in NCERT Class 11 Statistics Chapter 7 Part 1 because it is the geometric mean of Laspeyres and Paasche indexes — P01(F) = √(L × P) — and balances out the upward bias of Laspeyres and the downward bias of Paasche. It also satisfies two important tests of a good index number: the time reversal test (P01 × P10 = 1) and the factor reversal test (price index × quantity index = value index). Despite these theoretical advantages, Fisher's index requires both base-year and current-year quantity data, making it harder to compute regularly than Laspeyres.

What is the difference between simple and weighted index numbers?

Simple index numbers treat all items as equally important and are calculated either by simple aggregative method P01 = (Σp1/Σp0)×100 or by simple average of price relatives. Weighted index numbers assign different weights to different items based on their economic importance — typically quantities consumed, expenditure shares or production volumes. NCERT Class 11 Statistics Chapter 7 Part 1 stresses that weighted indexes (Laspeyres, Paasche, Fisher) are far more reliable because they reflect real economic significance: a price rise in rice matters more to a household than a price rise in saffron, so weights are essential for meaningful CPI or WPI calculations.

How do you choose a base year for an index number in NCERT Class 11?

NCERT Class 11 Statistics Chapter 7 Part 1 lists four rules for selecting a base year: (1) it should be a normal year, free from war, famine, natural disaster or unusual economic events; (2) it should not be too distant from the current year, so the basket of goods and economic structure remains broadly comparable; (3) reliable and complete data must be available for the base year; and (4) it should be reviewed and updated periodically — India's WPI base year was revised from 2004-05 to 2011-12, for example. Older base years lose relevance as consumption patterns and technology change.

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