Statistics

Index Numbers

Index numbers are specialized statistical measures designed to quantify relative changes in a variable or a group of variables over time or across geographical regions.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Fisher's Ideal Index Number and Consistency Tests

Fisher's Ideal Index is defined as the geometric mean of Laspeyres' and Paasche's index numbers, expressed as $$P_{01}^F = \sqrt{P_{01}^L \times P_{01}^P} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}}$$. It uniquely satisfies both the Time Reversal Test and the Factor Reversal Test.

📐 Geometric Interpretation

It represents a balanced geometric averaging of base-weighted and current-weighted price movements, minimizing both upward and downward proportional biases inherent in unilateral index formulations.

⚙️ Algebraic & Structural Meaning

Algebraically, it satisfies the Time Reversal Test ($P_{01} \times P_{10} = 1$) and the Factor Reversal Test ($P_{01} \times Q_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0}$), making it the supreme algebraic standard among all index number formulas.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Base Period Selection	`Fixed reference epoch ($t = 0$)`	Strictly Mandatory / Necessary	If the base period is abnormal (e.g., war or famine year), all subsequent relative comparisons become economically distorted and mathematically unrepresentative.
Commodity Basket Uniformity	`Consistent set of goods ${i = 1, 2, \dots, n}$`	Strictly Mandatory	Introduction of new commodities or structural disappearance of old items invalidates direct summation across time periods, causing quality-change bias.
Homogeneity of Units	`Commensurable pricing units`	Essential for conclusion	Adding prices of items quoted in different units (e.g., per kg versus per quintal) without proper conversion leads to dimensional imbalance and numerical corruption.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Fisher from Laspeyres and Paasche

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

Rule: If Laspeyres index is given as $120$ and Paasche index as $108$, Fisher index is simply $\sqrt{120 \times 108} = \sqrt{12960} \approx 113.84$.

Time Reversal Test Condition

$$P_{01} \times P_{10} = 1$$

Rule: To check if an index satisfies the Time Reversal Test, interchange $0$ and $1$ in the formula and multiply; if the product is identically $1$, the test passes.

Application: Standard theoretical multiple-choice question in DSSSB PGT Mathematics.

Factor Reversal Test Condition

$$P_{01} \times Q_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0}$$

Rule: Multiplying the price index by the corresponding quantity index (interchanging $p$ and $q$) must equal the true value ratio.

Application: Tested in UP TGT assertions and reasons.

Marshall-Edgeworth Weighting Structure

$$P_{01}^{ME} = \frac{\sum p_1(q_0 + q_1)}{\sum p_0(q_0 + q_1)}$$

Rule: Uses arithmetic mean of base and current quantities as weights. Acts as a close approximation to Fisher's Ideal Index with simpler computation.

Cost of Living Index (Family Budget Method)

$$I = \frac{\sum R W}{\sum W}, \quad \text{where } R = \frac{p_1}{p_0} \times 100, \, W = p_0 q_0$$

Rule: When price relatives $R$ and base expenditures $W = p_0 q_0$ are provided, weighted average of relatives yields the exact Laspeyres cost of living index in fewer steps.

Application: Numerical problem-solving in UP PGT.

Circular Test Property

$$P_{01} \times P_{12} \times P_{20} = 1$$

Rule: Laspeyres, Paasche, and Fisher indices do NOT satisfy the circular test. Simple geometric mean of price relatives and Walsh index satisfy it.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Confusing Laspeyres and Paasche Weighting Baselines

Common Pitfall: Candidates mix up base quantities ($q_0$) with current quantities ($q_1$) when recalling formula denominators.

Counterexample: Laspeyres uses $\sum p_0 q_0$, whereas Paasche uses $\sum p_0 q_1$.

Why it fails: Using current weights for base index or vice versa results in completely wrong numerical values and incorrect option selection.

🛡️ Exam Defense: Mnemonic: 'L' for Laspeyres uses 'L'ast year quantities ($q_0$); 'P' for Paasche uses 'P'resent year quantities ($q_1$).

⚠️ COMMON ERROR #02

Assuming Fisher Index Satisfies Circular Test

Common Pitfall: Assuming that because Fisher's Ideal Index satisfies Time and Factor Reversal tests, it must also satisfy the Circular Test.

Counterexample: $P_{01}^F \times P_{12}^F \times P_{20}^F \neq 1$

Why it fails: Fisher index depends on fixed base periods in its component structures and fails transitiveness across multiple sequential periods.

🛡️ Exam Defense: Remember: Only the simple geometric mean of price relatives and fixed-weight index with constant weights satisfy the circular test.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Laspeyres Index	Base year quantities as weights	$P_{01}^L = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$	Upward bias in dynamic economies	Keywords: 'base period weights'
Paasche Index	Current year quantities as weights	$P_{01}^P = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$	Downward bias due to substitution effect	Keywords: 'current period weights'
Fisher's Ideal Index	Geometric mean of Laspeyres & Paasche	$P_{01}^F = \sqrt{L \times P}$	Balanced mean eliminating biases	Keywords: 'ideal', 'satisfies both reversal tests'
Marshall-Edgeworth Index	Sum of base and current quantities	$P_{01} = \frac{\sum p_1(q_0 + q_1)}{\sum p_0(q_0 + q_1)}$	Arithmetic mean quantity weighting	Keywords: 'quantities combined'

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If the Laspeyres index number is $110$ and the Paasche index number is $121$, find Fisher's Ideal Index number.

(A) $110$(B) $115$(C) $121$(D) $132$

📚 Method A: Formal Textbook Derivation

Step 1: Write Fisher's formula $P_F = \sqrt{P_L \times P_P}$. Step 2: Substitute given values $P_L = 110$ and $P_P = 121$. Step 3: Compute $P_F = \sqrt{110 \times 121} = \sqrt{13310} \approx 115.37$, closest to $115$.

⚡ Method B: 25-Second Exam Shortcut

Recognize that Fisher is the geometric mean lying between 110 and 121; check square roots: $115^2 = 13225$, which aligns closely with $13310$.

💡 Tip: Direct formula application accounts for 40% of index number questions in UP PGT.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Which of the following index numbers satisfies both the Time Reversal Test and the Factor Reversal Test?

(A) Laspeyres' Index(B) Paasche's Index(C) Fisher's Ideal Index(D) Marshall-Edgeworth Index

📚 Method A: Formal Textbook Derivation

Evaluate each index against test criteria. Laspeyres and Paasche fail both reversal tests due to weight asymmetry. Fisher's index satisfies both identically.

⚡ Method B: 25-Second Exam Shortcut

Direct theoretical recall: Fisher = Ideal = Reversal Test satisfaction.

💡 Tip: Frequent theoretical question in DSSSB and KVS PGT exams.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: Fisher's Ideal Index is given by the formula $P_{01} = $

A $\frac{\sum p_1 q_0}{\sum p_0 q_0}$

B $\sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}}$

C $\frac{\sum p_1 q_1}{\sum p_0 q_0}$

D $\frac{\sum p_1(q_0+q_1)}{\sum p_0(q_0+q_1)}$

Derivation & Analysis:

Fisher's Ideal Index is defined as the geometric mean of Laspeyres and Paasche index numbers.

MCQ 2: If $P_{01} \times P_{10} = 1$, then the index number satisfies:

A Circular Test

B Factor Reversal Test

C Time Reversal Test

D Unit Test

Derivation & Analysis:

The condition $P_{01} \times P_{10} = 1$ is the definition of the Time Reversal Test formulated by Irving Fisher.

MCQ 3: Laspeyres' price index uses weights corresponding to:

A Current year quantities ($q_1$)

B Base year quantities ($q_0$)

C Average quantities $\frac{q_0 + q_1}{2}$

D Current year prices ($p_1$)

Derivation & Analysis:

Laspeyres index uses base year quantities $q_0$ as weights for both numerator and denominator.

MCQ 4: Paasche's index number formula uses weights of:

A Base year ($q_0$)

B Current year ($q_1$)

C Both base and current year

D Arithmetic mean of years

Derivation & Analysis:

Paasche's index utilizes current period quantities ($q_1$) as weights.

MCQ 5: Which test requires that the index for period $c$ with respect to $a$ multiplied by index for $b$ with respect to $c$ equals index for $b$ with respect to $a$?

A Time Reversal Test

B Factor Reversal Test

C Circular Test

D Unit Test

Derivation & Analysis:

The Circular Test is an extension of the time reversal test for more than two periods: $P_{01} \times P_{12} \times P_{20} = 1$.

MCQ 6: If Laspeyres index is $140$ and Paasche index is $90$, Fisher's Ideal Index is approximately:

A $112.25$

B $115.00$

C $116.19$

D $120.00$

Derivation & Analysis:

$P_F = \sqrt{140 \times 90} = \sqrt{12600} \approx 112.25$ wait, $\sqrt{12600} \approx 112.25$. Let's recalculate: $112.25^2 = 12600$. Correct option is (A).

MCQ 7: The factor reversal test requires that $P_{01} \times Q_{01}$ must equal:

A $1$

B $\frac{\sum p_1 q_1}{\sum p_0 q_0}$

C $\frac{\sum p_0 q_1}{\sum p_1 q_0}$

D $\sum p_1 q_1$

Derivation & Analysis:

Factor reversal test states that product of price index and quantity index must equal the total value ratio $\frac{\sum p_1 q_1}{\sum p_0 q_0}$.

MCQ 8: Cost of living index numbers are generally constructed using:

A Laspeyres' Method

B Paasche's Method

C Family Budget Method / Aggregative Expenditure

D Simple Average of Relatives

Derivation & Analysis:

Cost of living indices typically employ the Family Budget Method or Aggregated Expenditure method with base quantities.

MCQ 9: Which of the following indices satisfies the circular test?

A Fisher's Index

B Simple geometric mean of price relatives

C Paasche's Index

D Laspeyres' Index

Derivation & Analysis:

Simple geometric mean of price relatives satisfies the circular test, unlike Laspeyres or Paasche.

Q10

MCQ 10: An index number that is free from unit of measurement bias is said to satisfy:

A Time Reversal Test

B Circular Test

C Unit Test

D Factor Reversal Test

Derivation & Analysis:

The Unit Test ensures that the index number is independent of the units in which prices and quantities are quoted.

Common questions

It is termed ideal because it satisfies both of Fisher's rigorous consistency tests (Time Reversal and Factor Reversal) and accounts for changes in both price and quantity structures symmetrically.

Laspeyres index tends to have an upward bias because consumers substitute away from goods whose prices rise rapidly. Paasche index tends to have a downward bias because it uses current weights.

Remember that any index formula with asymmetric weights (like Laspeyres using $q_0$ exclusively or Paasche using $q_1$ exclusively) automatically fails the time reversal test.

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