Index Numbers in Statistics: Laspeyres, Paasche, Fisher Formulas, CPI, WPI and Solved Examples for SSC CGL Statistics
Rahul Kumar
SSC Exam Expert & Content Editor
In this note
- 1.Basics
- 2.Types of Index Numbers
- 3.Formulas
- 4.Solved Example
- 5.Tests of an Ideal Index Number
- 6.Base Shifting, Splicing and Deflating
- 7.Important Index Numbers in India
- 8.More Examples
- 9.Quick Tips
- 10.Quick Revision Points
- 11.Practice Questions With Answers
- 12.How to Practise This Chapter
- 13.Frequently Asked Questions
PDF Notes: Index Numbers in Statistics: Laspeyres, Paasche, Fisher Formulas, CPI, WPI and Solved Examples for SSC CGL Statistics
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An index number measures the relative change in a variable, such as price or quantity, over time or between places. The Consumer Price Index and the Wholesale Price Index are the best-known examples. SSC CGL Statistics asks formula-based questions on Laspeyres, Paasche and Fisher index numbers, the tests of an ideal index and base shifting. This guide explains each with solved examples, practice questions and FAQs.
Basics
Let P0 and Q0 be the price and quantity in the base year, and P1 and Q1 in the current year. The index number for the base year is always taken as 100.
Types of Index Numbers
| Type | Meaning |
|---|---|
| Price index | Measures changes in the price level |
| Quantity index | Measures changes in the quantity (volume) of production or consumption |
| Value index | Measures changes in the total value (price × quantity) |
Formulas
| Index | Formula |
|---|---|
| Simple aggregative price index | (ΣP1 / ΣP0) × 100 |
| Price relative method | (1/n) Σ(P1/P0 × 100) |
| Laspeyres price index (base-year weights) | (ΣP1Q0 / ΣP0Q0) × 100 |
| Paasche price index (current-year weights) | (ΣP1Q1 / ΣP0Q1) × 100 |
| Fisher ideal index | √(Laspeyres × Paasche) |
| Marshall–Edgeworth index | (ΣP1(Q0 + Q1) / ΣP0(Q0 + Q1)) × 100 |
| Dorbish–Bowley index | (Laspeyres + Paasche)/2 |
| Value index | (ΣP1Q1 / ΣP0Q0) × 100 |
Solved Example
| Commodity | P0 | Q0 | P1 | Q1 |
|---|---|---|---|---|
| A | 10 | 5 | 12 | 6 |
| B | 20 | 2 | 25 | 3 |
Simple aggregative index = (12 + 25)/(10 + 20) × 100 = 37/30 × 100 = 123.3.
Laspeyres = (12×5 + 25×2)/(10×5 + 20×2) × 100 = 110/90 × 100 = 122.2.
Paasche = (12×6 + 25×3)/(10×6 + 20×3) × 100 = 147/120 × 100 = 122.5.
Fisher = √(122.2 × 122.5) = about 122.4.
Tests of an Ideal Index Number
| Test | Condition |
|---|---|
| Time reversal test | P01 × P10 = 1 (the index for the current year on base year and the reverse are reciprocals) |
| Factor reversal test | P01 × Q01 = value index (ΣP1Q1 / ΣP0Q0) |
| Circular test | P01 × P12 × P20 = 1 |
| Unit test | The index should not depend on the unit of measurement |
- The Fisher index satisfies both the time reversal and the factor reversal tests, and is called the ideal index number.
- Laspeyres and Paasche satisfy neither test.
- Laspeyres usually overstates the price rise; Paasche usually understates it.
- Fisher is the geometric mean of Laspeyres and Paasche.
Base Shifting, Splicing and Deflating
- Base shifting: New index = (Old index / Index of the new base year in the old series) × 100.
- Splicing: joining two index series with different base years into one series.
- Deflating: Real value = (Nominal value / Price index) × 100.
- Real wages: (Money wages / Consumer price index) × 100.
Important Index Numbers in India
| Index | Compiled by | Remarks |
|---|---|---|
| Consumer Price Index (CPI) | National Statistical Office (MoSPI) | Measures retail inflation; used by RBI for monetary policy |
| Wholesale Price Index (WPI) | Office of the Economic Adviser, Ministry of Commerce and Industry | Measures wholesale inflation; the current base year is 2011–12 |
| Index of Industrial Production (IIP) | National Statistical Office | Measures the industrial output; base year 2011–12 |
| CPI for Industrial Workers (CPI-IW) | Labour Bureau | Used for the dearness allowance |
More Examples
Example 1. The price index of a commodity in 2020 was 150 with 2010 as the base. If the price in 2010 was 40 rupees, what was the price in 2020?
40 × 150/100 = 60 rupees.
Example 2. The money wage of a worker is 12,000 rupees and the CPI is 120. Find the real wage.
12000/120 × 100 = 10,000 rupees.
Example 3. If the Laspeyres index is 125 and the Paasche index is 80, find the Fisher index.
√(125 × 80) = √10000 = 100.
Quick Tips
- Memorise the Laspeyres and Paasche formulas, with their weights.
- Remember that Fisher is the geometric mean of the two.
- Remember that Fisher satisfies both reversal tests.
- Use real value = nominal value / index × 100 for deflating.
Quick Revision Points
- Laspeyres uses base-year quantities; Paasche uses current-year quantities.
- Fisher = √(L × P).
- The Fisher index is the ideal index number.
- Real wage = money wage / CPI × 100.
- WPI base year is 2011–12.
Practice Questions With Answers
- What is the base year index always taken as?
Answer: 100 - Which index uses the base-year quantities as weights?
Answer: Laspeyres - Which index uses the current-year quantities as weights?
Answer: Paasche - Which index is called the ideal index number?
Answer: Fisher - If the Laspeyres index is 144 and the Paasche index is 100, what is the Fisher index?
Answer: 120 - The money wage is 18,000 and the CPI is 150. Find the real wage.
Answer: 12,000 - Which test requires P01 × P10 = 1?
Answer: Time reversal test - Which institution compiles the CPI in India?
Answer: National Statistical Office (MoSPI) - What is the formula for the simple aggregative index?
Answer: ΣP1/ΣP0 × 100 - What does a CPI of 120 with base 100 mean?
Answer: Prices have risen by 20%
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Statistics notes page, read the previous topic Data Collection and Presentation and the next topic Time Series Analysis.
Frequently Asked Questions
What is an index number?
A statistical measure showing the relative change in a variable over time or place.
What is the Laspeyres index?
A price index that uses base-year quantities as weights.
What is the Paasche index?
A price index that uses current-year quantities as weights.
What is the Fisher index?
The geometric mean of the Laspeyres and Paasche indices; it is called the ideal index.
What is the time reversal test?
A test which says that the index for year 1 on base year 0 multiplied by the index for year 0 on base year 1 should be 1.
What is the factor reversal test?
A test which says that the product of the price index and the quantity index should equal the value index.
What is base shifting?
Changing the base year of an index series to another year.
What is deflating?
Converting nominal values into real values by dividing by the price index.
What is the CPI?
The Consumer Price Index, which measures the retail price changes faced by consumers.
What is the WPI?
The Wholesale Price Index, which measures changes in wholesale prices.
Is this topic asked in SSC?
Yes. It is an important topic in the SSC CGL Statistics paper.
Which index overestimates the price change?
The Laspeyres index tends to overestimate.
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Time Series Analysis: Trend, Seasonal, Cyclical and Irregular Components, Moving Average and Least Squares for SSC Statistics
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.