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Class 12 Statistics Notes · GSEB

Laspeyres, Paasche and Fisher

Index Numbers — learn the three weighted index number methods, their formulas, biases, and real-life applications. GSEB Class 12 Statistics notes.

Last updated: 21 Sep 2026

Notes

Laspeyres, Paasche and Fisher

Index Numbers — Chapter 5, GSEB Class 12 Statistics

Why Weighted Index Numbers?

In a simple average, every item gets equal importance. But in reality, rice matters more than saltin a family's budget. Weighted index numbers assign weights (quantities consumed) to reflect this importance.

Simple vs Weighted: Simple index = equal weight to all items. Weighted index = items weighted by how much is consumed. Weighted indices are more realistic.

Laspeyres Index

Laspeyres
Overstates inflation

Laspeyres Formula

IL=∑P1q0∑P0q0×100I_L = \frac{\sum P_1 q_0}{\sum P_0 q_0} \times 100
Weights: Base year quantities (q₀)

Uses base year quantities as weights. Easy to compute because q₀ stays constant. But since it uses old consumption patterns, it ignores that people may have shifted to cheaper substitutes.

Example

If 2018 basket = 100 kg rice + 50 L milk, Laspeyres always uses this same basket — even if in 2024 people buy less rice and more noodles.

Real Life

Government CPI-L uses Laspeyres approach. When inflation is reported as 6%, the actual feeling may be lower because people substitute expensive items.

Paasche Index

Paasche
Understates inflation

Paasche Formula

IP=∑P1q1∑P0q1×100I_P = \frac{\sum P_1 q_1}{\sum P_0 q_1} \times 100
Weights: Current year quantities (q₁)

Uses current year quantities as weights. Reflects current consumption but understates inflation because people naturally shift toward cheaper goods when prices rise.

Example

If rice price doubled, you buy less rice and more noodles. Paasche uses the new (lower) rice quantity, making the index appear lower than the true price increase.

Real Life

Used in GDP deflator. When GDP growth is reported, the deflator using Paasche may understate the actual price rise consumers experience.

Fisher Index

Fisher
Ideal (balances both)

Fisher Formula

IF=IL×IPI_F = \sqrt{I_L \times I_P}
Weights: Both base and current quantities

Geometric mean of Laspeyres and Paasche. Balances the overstatement of I_L with the understatement of I_P. Called the “ideal index” because it satisfies both time reversal and factor reversal tests.

Example

If I_L = 120 (overstates) and I_P = 110 (understates), I_F = √(120 × 110) = √13200 ≈ 114.89 — a balanced middle ground.

Real Life

Used by RBI for some inflation measures. International agencies prefer Fisher for cross-country comparisons.

Comparison of All Three

FeatureLaspeyres (I_L)Paasche (I_P)Fisher (I_F)
Weights UsedBase year quantities (q₀)Current year quantities (q₁)Both (geometric mean)
FormulaΣ(P₁q₀) / Σ(P₀q₀) × 100Σ(P₁q₁) / Σ(P₀q₁) × 100√(I_L × I_P)
BiasOverstates inflationUnderstates inflationIdeal (balanced)
Ease of ComputationEasy (q₀ constant)Harder (needs current data)Moderate (needs both)
Time Reversal Test✗ Fails✗ Fails✓ Satisfies
Factor Reversal Test✗ Fails✗ Fails✓ Satisfies
Remember:I_L overstates, I_P understates, I_F balances both. Fisher (I_F) is called the “ideal index” because it satisfies both time reversal and factor reversal tests.

Key Takeaways

Key Takeaways

  • Laspeyres (I_L) uses base year quantities (q₀) as weights — easy to compute but overstates inflation.
  • Paasche (I_P) uses current year quantities (q₁) as weights — reflects current patterns but understates inflation.
  • Fisher (I_F) = √(I_L × I_P) — the "ideal index" that balances both biases.
  • I_F satisfies both time reversal and factor reversal tests — I_L and I_P fail both.
  • By × bxy = r² for regression coefficients; I_F = √(I_L × I_P) for index numbers.
  • In GSEB exams, compute all three when asked — show the formulas and identify the bias direction.