Fundamentals of Business Mathematics and Statistics · Index Numbers and Time Series

Construction of Index Numbers: Simple and Weighted Methods

Updated 10 October 2026 · Fact-checked

An index number measures the change in prices (or quantities) of a group of items against a base period, with the base set at 100. Simple methods ignore importance. Weighted methods use quantities as weights: Laspeyres uses base quantities, Paasche uses current quantities, and Fisher is the geometric mean of the two.

Understand Construction of Index Numbers: Simple and Weighted Methods

An index number compresses many price changes into one figure. The base year is set at 100. If the index for the current year is 125, prices on average are 25% higher than in the base year.

The simple methods treat every item as equally important. In the simple aggregative method, you add up current prices, add up base prices, divide and multiply by 100. In the average of price relatives method, you first find each item's price relative (P1 ÷ P0 × 100) and then average these relatives. Simple methods are weak because rice and salt count equally, even though a household buys far more rice.

The weighted methods fix this by using quantities as weights. Laspeyres keeps the base-year quantities (Q0) and asks: what would the base basket cost at today's prices? Paasche uses current-year quantities (Q1) and asks: what does today's basket cost compared with what it would have cost at base prices?

Other methods combine both quantities. Fisher takes the geometric mean of Laspeyres and Paasche. Dorbish-Bowley takes their arithmetic mean. Marshall-Edgeworth adds Q0 and Q1 and uses that sum as the weight.

In practice, when consumers shift away from items whose prices rise, Laspeyres tends to come out higher than Paasche. This is a common tendency, not a rule that holds for every data set. Fisher lies between the two and is called the ideal index because it passes both the time reversal and factor reversal tests.

Key formulas to remember

Simple aggregative index: P01 = (ΣP1 ÷ ΣP0) × 100; Unweighted. Use only when all items are in the same unit.
Average of price relatives (arithmetic mean): P01 = Σ[(P1 ÷ P0) × 100] ÷ n; Find each relative first, then divide the sum by the number of items n. If the question asks for the geometric mean, use the antilog of Σlog(relative) ÷ n.
Laspeyres index: L = (ΣP1Q0 ÷ ΣP0Q0) × 100; Base-year quantities as weights.
Paasche index: P = (ΣP1Q1 ÷ ΣP0Q1) × 100; Current-year quantities as weights.
Fisher ideal index: F = √(L × P); Geometric mean of Laspeyres and Paasche.
Dorbish-Bowley index: DB = (L + P) ÷ 2; Arithmetic mean of Laspeyres and Paasche.
Marshall-Edgeworth index: ME = [ΣP1(Q0 + Q1) ÷ ΣP0(Q0 + Q1)] × 100; Uses the sum of base and current quantities as weights.
Weighted average of price relatives: Index = ΣIV ÷ ΣV, where I = (P1 ÷ P0) × 100; With V = P0Q0 this gives the Laspeyres index.

How to solve Construction of Index Numbers: Simple and Weighted Methods questions

Use this method for any construction question, whether the data is given as prices only or as prices and quantities.

1Read which method the question names. Note the base year and the current year.
2Check what data you have: prices only (simple method) or prices and quantities (weighted method).
3Label columns clearly: P0, Q0, P1, Q1.
4Compute only the products the formula needs. Laspeyres needs P1Q0 and P0Q0. Paasche needs P1Q1 and P0Q1. Marshall-Edgeworth needs the sums of Q0 and Q1 first.
5Add each column to get the totals (Σ).
6Divide the numerator total by the denominator total and multiply by 100.
7For Fisher or Dorbish-Bowley, finish Laspeyres and Paasche first, then combine them.
8Check whether the answer is sensible: a rise in prices should give an index above 100.

Quickest way: Compute L and P once, then derive the rest

When to use it: When the question gives Laspeyres and Paasche values, or asks for Fisher or Dorbish-Bowley along with them.

Compute Laspeyres and Paasche from the table and keep both.
Dorbish-Bowley is the simple average of the two. Add and halve.
For Fisher, look for numbers whose product is a perfect square, for example 160 × 90 = 14,400, so Fisher = 120.
If the table is not given and only options are, eliminate any option for Fisher that is not between L and P.
Fisher is never above Dorbish-Bowley for positive values (geometric mean ≤ arithmetic mean). So if an option for Fisher is greater than the Dorbish-Bowley value, reject it. This does not by itself eliminate the Dorbish-Bowley option.

Common mistakes in Construction of Index Numbers: Simple and Weighted Methods

Using current-year quantities in the Laspeyres formula (or base quantities in Paasche).
The two formulas look alike and students mix the Q subscript.
Fix: Remember: Laspeyres = Q0 (base-year basket). Paasche = Q1 (current-year basket). Mnemonic: 'L for Legacy basket'.
Forgetting to multiply by 100.
The division gives a ratio such as 1.40, and students stop there.
Fix: Index numbers are always shown with base = 100. Multiply the final ratio by 100 every time.
Using the arithmetic mean for Fisher.
Students confuse Fisher with Dorbish-Bowley.
Fix: Fisher uses a square root of L × P. Dorbish-Bowley uses (L + P) ÷ 2.
Dividing the price relatives total by 100 or by the wrong n.
Students are unsure of the final step in the relatives method.
Fix: Add the relatives, which already include ×100, and divide only by the number of items.
Treating the simple aggregative method as weighted.
The word aggregative suggests totals of value, so students bring in quantities.
Fix: Simple aggregative uses only prices: ΣP1 ÷ ΣP0 × 100. Quantities appear only in weighted methods.
Swapping base and current prices.
Students read the table columns carelessly when years are listed in a different order.
Fix: Mark P0 and P1 against the years before calculating. The current year's price always goes in the numerator.

Worked examples

Example 1

For two items, the data is: Item A: base price ₹2, base quantity 10, current price ₹4, current quantity 5. Item B: base price ₹5, base quantity 12, current price ₹6, current quantity 10. Find the Laspeyres, Paasche and Dorbish-Bowley price indices.

Show the solution

Laspeyres numerator ΣP1Q0 = 4 × 10 + 6 × 12 = 40 + 72 = 112.
Laspeyres denominator ΣP0Q0 = 2 × 10 + 5 × 12 = 20 + 60 = 80.
L = 112 ÷ 80 × 100 = 140.
Paasche numerator ΣP1Q1 = 4 × 5 + 6 × 10 = 20 + 60 = 80.
Paasche denominator ΣP0Q1 = 2 × 5 + 5 × 10 = 10 + 50 = 60.
P = 80 ÷ 60 × 100 = 133.33 (approximately).
Dorbish-Bowley = (140 + 133.33) ÷ 2 = 136.67 (approximately).

Answer: Laspeyres = 140, Paasche ≈ 133.33, Dorbish-Bowley ≈ 136.67.

Example 2

The Laspeyres price index for a group of goods is 160 and the Paasche price index is 90. What is the Fisher ideal index? (a) 110 (b) 120 (c) 125 (d) 135

Show the solution

Fisher = √(L × P).
L × P = 160 × 90 = 14,400.
√14,400 = 120.
Option (c), 125, is the Dorbish-Bowley value (160 + 90) ÷ 2, a common trap.
Fisher must lie between 90 and 160, which does not remove any option. But Fisher cannot exceed the Dorbish-Bowley value of 125, so (d) 135 can be rejected. Calculation is still needed to choose among the rest.

Answer: (b) 120

Exam tips

Write the formula name and its Q subscript at the top of your rough work before filling the table.
Questions often say only 'price index using weights'. Check whether the weights given are base-year (Laspeyres) or current-year (Paasche) values.
If both L and P are given, Fisher and Dorbish-Bowley need no table work at all. Do these first to save time.
When asked for the average of price relatives, find every relative to two decimals but round only at the end.
There is no negative marking, so always mark an option. Eliminate those outside the range between L and P for Fisher and Dorbish-Bowley.

Practice questions from Index Numbers and Time Series

Construction of Index Numbers: Simple and Weighted Methods in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Construction of Index Numbers: Simple and Weighted Methods: frequently asked questions

What is the difference between Laspeyres and Paasche index?

Laspeyres uses base-year quantities as weights, so it tracks the cost of the old basket. Paasche uses current-year quantities, so it tracks the cost of the present basket. Laspeyres needs quantity data only for the base year, which makes it easier to compute.

Why is the Fisher index called the ideal index?

It is the geometric mean of Laspeyres and Paasche and it satisfies both the time reversal test and the factor reversal test. These tests are covered in a separate topic.

How is Marshall-Edgeworth different from Laspeyres and Paasche?

It uses the sum of base and current quantities (Q0 + Q1) as the weight. So one formula uses both years' quantities without needing a separate Laspeyres and Paasche calculation.

Which method should I use if only prices are given?

Use a simple method. If the question says aggregative, use ΣP1 ÷ ΣP0 × 100. If it says price relatives, find each relative and average them. Weighted methods need quantities or values as weights.