Quantitative Aptitude · Index Numbers

Weighted Index Numbers: Laspeyres, Paasche and Fisher

Updated 1 October 2026 · Fact-checked

Weighted index numbers measure price change while giving each item importance through quantity weights. Laspeyres uses base-year quantities, Paasche uses current-year quantities, and Fisher is the geometric mean of the two. To solve a sum, build a table of p0q0, p1q0, p0q1 and p1q1, total each column, then apply the formula.

Understand Weighted Index Numbers: Laspeyres, Paasche, Fisher

A simple index number treats every item equally. That is unfair. A rise in the price of rice matters more to a family than a rise in the price of salt, because the family buys far more rice. Weighted index numbers fix this by multiplying each price by a quantity, which acts as the weight.

The two basic choices are the base year and the current year. In the Laspeyres index, the weights are base-year quantities (q0). It asks: what would the same basket as the base year cost today, compared with what it cost then? In the Paasche index, the weights are current-year quantities (q1). It asks: what does today's basket cost now, compared with what it would have cost in the base year?

Because people buy less of items that become costly, Laspeyres tends to be higher than Paasche when prices and quantities move in opposite directions. This is a general tendency, not a rule that holds for every data set. Laspeyres needs quantities of the base year only, so it is cheaper to compute over time. Paasche needs fresh quantity data every year.

Fisher's index is the geometric mean of Laspeyres and Paasche. It is called the ideal index because it satisfies both the time reversal test and the factor reversal test. Using the quantities of both years is a feature of its construction, not the reason for the name. The Dorbish-Bowley index is the simple arithmetic mean of Laspeyres and Paasche. The Marshall-Edgeworth index uses the sum of base and current quantities (q0 + q1) as the weight.

In every formula, the numerator holds the current-year price (p1) and the denominator holds the base-year price (p0). Only the quantity changes between methods. Learn that one idea and the formulas stop looking like five separate things.

Key formulas to remember

Laspeyres price index: P01(L) = Σp1q0 ÷ Σp0q0 × 100; Weights are base-year quantities q0.
Paasche price index: P01(P) = Σp1q1 ÷ Σp0q1 × 100; Weights are current-year quantities q1.
Fisher's ideal index: P01(F) = √(P01(L) × P01(P)); Geometric mean of Laspeyres and Paasche. It satisfies the time reversal and factor reversal tests.
Marshall-Edgeworth index: P01(ME) = Σp1(q0 + q1) ÷ Σp0(q0 + q1) × 100; Equals (Σp1q0 + Σp1q1) ÷ (Σp0q0 + Σp0q1) × 100.
Dorbish-Bowley index: P01(DB) = [P01(L) + P01(P)] ÷ 2; Arithmetic mean of Laspeyres and Paasche.
Order of means: Fisher ≤ Dorbish-Bowley; The geometric mean never exceeds the arithmetic mean of the same two positive numbers. Fisher always lies between Laspeyres and Paasche.

How to solve Weighted Index Numbers: Laspeyres, Paasche, Fisher questions

Use the same table method for every weighted index question. It keeps the sums organised and makes each formula a matter of picking the right totals.

1Read the question and note which year is the base year (0) and which is the current year (1).
2Identify the columns given: p0, q0, p1, q1. Check carefully that you have not swapped the years.
3Make four product columns: p0q0, p1q0, p0q1 and p1q1.
4Total each column to get Σp0q0, Σp1q0, Σp0q1 and Σp1q1.
5Pick the formula asked for. Laspeyres uses Σp1q0 and Σp0q0. Paasche uses Σp1q1 and Σp0q1.
6For Fisher or Dorbish-Bowley, find Laspeyres and Paasche first, then take the geometric or arithmetic mean.
7Multiply by 100 and compare with the options. Round only at the end.

Quickest way: Four totals, then option elimination

When to use it: Use this in the MCQ paper when the table has 3 to 5 items and you have about 90 seconds per question.

Compute only the totals you need. Laspeyres needs two columns, Paasche needs two, and Marshall-Edgeworth needs all four.
Work with small numbers. Compute each product mentally and add as you go.
Use the 100 check: if every price rose, every weighted index must be above 100.
Use the sandwich rule: Fisher lies between Laspeyres and Paasche. Discard any option outside that range.
If the question gives L and P directly, do not rebuild the table. Fisher is √(L × P). Try to spot a perfect square product.
Marshall-Edgeworth equals (Σp1q0 + Σp1q1) ÷ (Σp0q0 + Σp0q1) × 100, so reuse the totals from Laspeyres and Paasche.
If the calculation is long and the options are close together, skip and return at the end. Wrong answers cost 0.25 marks.

Common mistakes in Weighted Index Numbers: Laspeyres, Paasche, Fisher

Using q1 in Laspeyres or q0 in Paasche.
Students remember that one uses base quantities and one uses current quantities, then swap them under pressure.
Fix: Remember: Laspeyres = Last-year (base) quantities. Paasche = Present-year quantities.
Using the wrong denominator in Paasche, such as Σp0q0.
The denominator is copied from the Laspeyres formula.
Fix: In Paasche, both sums use q1. The numerator is Σp1q1 and the denominator is Σp0q1.
Calling the average of Laspeyres and Paasche the Fisher index.
Both are means of the same two numbers, so they are easily confused.
Fix: The arithmetic mean is Dorbish-Bowley. Fisher is the square root of the product. Read the name in the question carefully.
Forgetting to multiply by 100.
The ratio is computed correctly and the student stops there.
Fix: Index numbers are expressed with base = 100. Always multiply the ratio by 100 and check the options are on that scale.
Applying the Marshall-Edgeworth formula with q0 or q1 alone.
Students forget that the weight is the sum of both years' quantities.
Fix: Write (q0 + q1) next to each price in the formula, or add the Laspeyres and Paasche totals.
Rounding Laspeyres and Paasche early before finding Fisher.
Students round to a whole number to simplify the square root.
Fix: Keep at least two decimals until the end, or use the exact fraction, then round the final answer.

Worked examples

Example 1

Use the data below for the first two questions. Items A, B, C. Base-year prices p0 = 10, 20, 5. Base-year quantities q0 = 4, 3, 10. Current-year prices p1 = 12, 25, 6. Current-year quantities q1 = 5, 2, 12. Q1. The Laspeyres price index is: (a) 118 (b) 121.33 (c) 122 (d) 125

Show the solution

Laspeyres needs Σp1q0 and Σp0q0.
Σp0q0 = 10×4 + 20×3 + 5×10 = 40 + 60 + 50 = 150.
Σp1q0 = 12×4 + 25×3 + 6×10 = 48 + 75 + 60 = 183.
Index = 183 ÷ 150 × 100 = 122.
Option (b), 121.33, is the Paasche index for this data, so it is the trap option.

Answer: (c) 122

Example 2

Using the same data (p0 = 10, 20, 5; q0 = 4, 3, 10; p1 = 12, 25, 6; q1 = 5, 2, 12), the Marshall-Edgeworth price index is closest to: (a) 121.33 (b) 121.67 (c) 122 (d) 122.67

Show the solution

Formula: Σp1(q0 + q1) ÷ Σp0(q0 + q1) × 100.
Σp1q0 = 183 (from the previous example).
Σp1q1 = 12×5 + 25×2 + 6×12 = 60 + 50 + 72 = 182.
Numerator = 183 + 182 = 365.
Σp0q0 = 150. Σp0q1 = 10×5 + 20×2 + 5×12 = 50 + 40 + 60 = 150.
Denominator = 150 + 150 = 300.
Index = 365 ÷ 300 × 100 = 121.67 (rounded).

Answer: (b) 121.67

Example 3

For a set of commodities the Laspeyres price index is 162 and the Paasche price index is 128. The Fisher ideal index is: (a) 140 (b) 144 (c) 145 (d) 150

Show the solution

Fisher = √(L × P).
L × P = 162 × 128 = 20,736.
√20,736 = 144, because 144 × 144 = 20,736.
Check the sandwich rule: 144 lies between 128 and 162.
Option (c), 145, is the arithmetic mean (162 + 128) ÷ 2, which is the Dorbish-Bowley index. It is the trap option.

Answer: (b) 144

Exam tips

Write the four column headings p0q0, p1q0, p0q1, p1q1 on rough paper at the start. Most errors come from picking the wrong column.
Expect trap options built from the other methods. A wrong Laspeyres answer will often appear as the Paasche or Dorbish-Bowley value.
When L and P are given, use Fisher = √(L × P) directly. Look for pairs whose product is a perfect square.
Remember that Fisher satisfies both the time reversal and factor reversal tests. Simple theory MCQs often ask this, or ask which index is called ideal.
Check the base year in the question. If the base is not 100, or if two years are compared in reverse, read the data carefully before computing.

Practice questions from Index Numbers

Weighted Index Numbers: Laspeyres, Paasche, Fisher: frequently asked questions

What is the difference between Laspeyres and Paasche index?

Laspeyres uses base-year quantities as weights, so it compares the cost of the old basket at new and old prices. Paasche uses current-year quantities, so it compares the cost of the new basket at new and old prices. Laspeyres needs only base-year quantity data, while Paasche needs fresh quantity data each year.

Why is Fisher's index called the ideal index number?

It is called ideal because it satisfies both the time reversal test and the factor reversal test. Using the quantities of both years, through Laspeyres and Paasche, is a feature of how it is built, not the reason for the name. It is harder to compute in practice.

How is Marshall-Edgeworth different from Dorbish-Bowley?

Marshall-Edgeworth uses (q0 + q1) as the weight in a single ratio of sums. Dorbish-Bowley is the simple average of the Laspeyres and Paasche indices. They can give the same or different values depending on the data.

Do I always multiply by 100 in weighted index number sums?

Yes, for price index numbers with the base year taken as 100. The ratio of the sums is multiplied by 100. If your answer is near 1.2 instead of near 120, you have missed this step.