Written By Keerthi Kulkarni
Last Modified 24-01-2023

Construction of Index Numbers: Introduction, Methods of Construction, Examples

Construction of index numbers: There are two ways to construct an index number. They are weighted and unweighted methods of construction. It can also be calculated using both the aggregative and relative average methods. The selection of an appropriate average is required to construct index numbers.

The arithmetic mean is commonly employed in constructing index numbers since it is easier to calculate than other averages. In this article, let us learn about the different methods such as Laspeyse’s method, Paasche’s method, Dorbish & Bowley’s method, Fisher’s ideal method, Marshall Edgeworth method and Kelly’s method for constructing index numbers.

What is an Index Number?

An index number is a statistical instrument for measuring changes in the magnitude of a group of connected variables. It represents the overall trend of diverging ratios used to calculate it. It is a measure for comparing the average change in a set of related variables between two situations.

A comparison could be made between similar categories such as people, schools, and hospitals. An index number is used to track the changes such as the prices of commodities, the volume of production in industries, and the production of crops.

Methods of Construction of Index Numbers

There are two ways to generate an index number. It can be calculated using both the aggregative and relative averaging methods. In general, there are two types of index number construction:

Simple
Weighted

The sub-types of each are shown below.

Simple Aggregative Method

This method of construction is used to calculate the index price. As a result, the total cost of a commodity is expressed as a percentage of the total cost in the base year.

This approach involves adding the prices of various current-year items, dividing the total by the sum of prices for base-year items, and multiplying by \(100\).

\(P_{01}=\frac{\sum P_{1}}{\sum P_{0}} \times 100\)

\(p_{1}-\) Current year prices for various commodities
\(p_{0}-\) Base year prices for various commodities
\(P_{01}-\) Price Index number

The simple aggregative index is straightforward to comprehend. This strategy, however, has a significant flaw. In this case, the first commodity has greater influence than the other two. This is because the first commodity is more expensive than the others.

Furthermore, if we change the units in any way, the index number will also change. This is one of the method’s serious drawbacks. The use of absolute quantities reverses the tables. As a result, considering three years of independent values would be a preferable alternative.

Simple Average or Relative Price Method

First, price relatives for the various items are obtained, and then the average of these relatives is calculated using the arithmetic mean. The current year’s price expressed as a percentage of the base year’s price is known as price relative. The formula for calculating index number using the arithmetic mean is given below.

If \(N\) is the number of items, \(p_{1}\) is the current year’s commodity price, and \(p_{0}\) is the base year’s commodity price, then the average price index number is

\(P_{01}=\frac{\sum \frac{p_{1}}{p_{0}} \times 100}{N}\)

A replacement would be a better option for removing the errors and problems of a simple aggregative index. As a result, we may construct an index using a simple average of the relative’s technique.

We can reverse the actual values of any individual variable into the percentage form of the base period using this method. These percentages are referred to as relatives. One of the most convincing arguments in favour of relatives is that they are whole numbers with no absolute values, such as \(Rs. 35.60\) and \(Rs. 10.01\). As a result, the index numbers we obtain are likely to remain unchanged.

Construction of Weighted Aggregate Index Numbers

Weighing is done with an approximation factor. These variables are likely to change over time and can include anything. It can be a quantity or a volume sold off throughout the base year.

The year does not have to be the base year; it might be an average of several years or any year. The decision will be based entirely on the significance of the year in question. So, apart from the number, it is up to us to define the significance of a particular year.

The weighted aggregative index is usually expressed as a percentage. As a result, we apply a variety of formulas to accomplish the same goal. The price of each commodity is weighted by the quantity sold in the base year or the current year in this manner. There are numerous different ways to assign weights and hence many different ways to build index numbers. The following are some of the key formulas utilised in these methods.

Method	Description	Formula
Laspeyres Method	The values of weights are the quantities in the base year for this type of index	\(P_{01}^{L}=\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times 100\)
Paasche’s Method	As a weight, the current year’s quantities are used. We employ continuously revised weights in this method, so it’s not usually used when there are a lot of goods.	\(P_{01}^{P}=\frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}} \times 100\)
Dorbish and Bowley’s Method	We utilise the simple arithmetic mean of Laspeyres and Paasche’s formulas to account for the impact of both the base and current year.	\(P_{01}^{D B}=\frac{P_{01}^{L}+P_{01}^{P}}{2}=\frac{\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}}+\frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}}}{2} \times 100\)
Fisher’s Ideal Index	It’s the geometric mean of Laspeyres and Paasche’s indices	\(P_{01}^{F}=\sqrt{P_{01}^{L} \times P_{01}^{P}}=\sqrt{\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times \frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}}} \times 100\)
Marshall-Edgeorth Method	Current and base year pricing and quantities are considered in this strategy.	\(P_{01}^{M E}=\frac{\sum p_{1}\left(q_{0}+q_{1}\right)}{p_{0}\left(q_{0}+q_{1}\right)} \times 100=\frac{\sum p_{1} q_{0}+\sum p_{1} q_{1}}{\sum p_{0} q_{0}+\sum p_{0} q_{1}} \times 100\)
Kelly’s Method	In the Kelly’s Index, where \(q\) denotes the quantity of a certain time, not necessarily the mean of the base and current years. Weights can also be calculated using the average quantity of two or more years. Fixed weight aggregative index is the name of this method.	\(P_{01}^{K}=\frac{\sum p_{1} q}{\sum p_{0} q} \times 100, q=\frac{q_{0}+q_{1}}{2}\)

Weighted Average of Price Relative Index

The weighted average of price relatives may be calculated by adding weights to the unweighted price relatives. To average weighted price relatives, we can use the arithmetic mean.

If \(P=\frac{p_{1}}{p_{0}} \times 100\) is the price relative index and \(w=p_{0} q_{0}\) is attached to the commodity, then

\(P_{01}=\frac{\sum\left[\frac{p_{1}}{p_{0}} \times 100\right] \times p_{0} q_{0}}{\sum p_{0} q_{0}}=\frac{\sum w P}{\sum w}\)

Practice Exam Questions

Solved Examples – Construction of Index Numbers

1. Find the simple aggregate index of the years \(1999\) and \(2000\) over the years \(1998\).

*Commodity (Item)*	\(1998\)	\(1999\)	\(2000\)
*Potato (Per kg)*	\(5\)	\(6\)	\(5.70\)
*Cheese (\(100\) gm)*	\(12\)	\(15\)	\(15.60\)
*Egg (per piece)*	\(3\)	\(3.60\)	\(3.30\)

Ans:
Given:

Commodity (Item)	\(1998\)	\(1999\)	\(2000\)
Potato (Per kg)	\(5\)	\(6\)	\(5.70\)
Cheese (\(100\) gm)	\(12\)	\(15\)	\(15.60\)
Egg (per piece)	\(3\)	\(3.60\)	\(3.30\)
Aggregate	\(20\)	\(24.60\)	\(24.60\)

By simple aggregate index of the year \(1999\)over the year \(1998\) is given by \(\frac{\sum P_{n}}{\sum P_{0}} \times 100\)
\(=\frac{24.60}{20} \times 100=123\)
By simple aggregate index of the year \(2000\) over the year \(1998\) is given by \(\frac{\sum P_{n}}{\sum P_{0}} \times 100\)
\(=\frac{24.60}{20} \times 100=123\)

2. Calculate the weighted aggregative price index for the given data.

Commodity	Base Period		Current Period
Commodity	Price	Quantity	Price	Quantity
\(A\)	\(2\)	\(10\)	\(4\)	\(5\)
\(B\)	\(5\)	\(12\)	\(6\)	\(10\)
\(C\)	\(4\)	\(20\)	\(5\)	\(15\)
\(D\)	\(2\)	\(15\)	\(3\)	\(10\)

Ans:
By the method of weighted aggregative, we know that the index number is
\(P_{01}=\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times 100\)
\(P_{01}=\frac{(4 \times 5+6 \times 10+5 \times 15+3 \times 10)}{(2 \times 10+5 \times 12+4 \times 20+2 \times 15)} \times 100\)
\(P_{01}=\frac{257}{190} \times 100\)
\(P_{01}=135.3\)
Hence, the weighted aggregative price index is \(135.3\).

3. Calculate the weighted price relative index for the data:

*Commodity*	*Weight in* \(%\)	*Base year price (in Rs)*	*Current year price (in Rs.)*	*Price relative*
\(A\)	\(40\)	\(2\)	\(4\)	\(200\)
\(B\)	\(30\)	\(5\)	\(6\)	\(120\)
\(C\)	\(20\)	\(4\)	\(5\)	\(125\)
\(D\)	\(10\)	\(2\)	\(3\)	\(150\)

Ans:
The weighted price index is given by
\(P_{01}=\frac{\sum_{i=1}^{n} W_{i}\left(\frac{P_{1 i}}{P_{0 i}} \times 100\right)}{\sum_{i=1}^{n} W_{i}}\)
\(P_{01}=\frac{40 \times\left(\frac{4}{2} \times 100\right)+30 \times\left(\frac{6}{5} \times 100\right)+20 \times\left(\frac{5}{4} \times 100\right)+10 \times\left(\frac{3}{2} \times 100\right)}{40+30+20+10}\)
\(P_{01}=\frac{40 \times 200+30 \times 120+20 \times 125+10 \times 150}{100}\)
\(P_{01}=156\)
Hence, the weighted price relative index is \(156\)

4. Construct the price index number of the given data by using Paasche’s method.

Items	Base Year		Current Year
Items	Quantity	Price	Quantity	Price
\(A\)	\(3\)	\(5\)	\(2\)	\(8\)
\(B\)	\(7\)	\(4\)	\(5\)	\(6\)
\(C\)	\(4\)	\(7\)	\(3\)	\(10\)
\(D\)	\(6\)	\(6\)	\(5\)	\(7\)

Ans:
We know that Paasche’s price index is given by
\(P_{01}=\frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}} \times 100\)
\(P_{01}=\frac{2 \times 8+5 \times 6+3 \times 10+5 \times 7}{5 \times 2+4 \times 5+7 \times 3+6 \times 5} \times 100\)
\(P_{01}=\frac{111}{81} \times 100=137.04\)
Hence, Paschees index number is \(137.40\)

5. Find the price index number by Fisher’s method of the given data for the year \(2014\) over the year \(2004\)

Goods	2004		2014
Goods	Quantity	Price	Quantity	Price
Goods I	\(5\)	\(10\)	\(4\)	\(12\)
Goods II	\(8\)	\(6\)	\(7\)	\(7\)
Goods III	\(6\)	\(3\)	\(5\)	\(4\)

Ans:

We know that Fisher’s price index is given by
\(P_{01}=\sqrt{\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times \frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}}} \times 100\)
\(P_{01}=\sqrt{\frac{(12 \times 5+7 \times 8+4 \times 6)}{(5 \times 10+8 \times 6+\times 6 \times 3)} \times \frac{(4 \times 12+7 \times 7+5 \times 4)}{(10 \times 4+6 \times 7+3 \times 5)}} \times 100\)
\(P_{01}=\sqrt{\frac{97}{116} \times \frac{117}{140}} \times 100\)
\(P_{01}=83.59\)
Hence, the Fisher’s price index is \(83.59\)

Summary

You can construct a single measure of change for many objects by estimating index numbers. Price, quantity, volume, and other index numbers can all be determined. The aggregate and average or relative methods are the two main methods used to construct the index numbers. The formulas also show that the index numbers must be interpreted with caution.

It is crucial to consider the elements to include and the time span to use. Their diverse applications show that index numbers are essential in policymaking. Laspeyse’s method, Paasche’s method, Dorbish & Bowley’s method, Fisher’s ideal method, Marshall Edgeworth method and Kelly’s method are some of the weighted aggregate methods of index numbers.

Frequently Asked Questions (FAQs)

Students must have many questions with respect to the Construction of Index Numbers. Here are a few commonly asked questions and answers.

Q.1. What is used in the construction of index numbers?
Ans: The construction of index numbers requires the selection of an appropriate average. The arithmetic mean is commonly used in the construction of index numbers.

Q.2. What are the steps for the construction of index numbers?
Ans: Steps to be followed:

The first step or problem in preparing index numbers is determining the base year.
The second issue in index number construction is the selection of commodities.
The fourth problem is to select a suitable average because the index numbers are a specialised average.
In general, not all of the commodities included in the construction of index numbers are equally important. As a result, proper weights should be assigned to the commodities based on their relative importance if the index numbers are to be representative.
The objective of the index numbers is the most important consideration in their construction. All other problems or steps must be viewed in light of the purpose for which a particular problem or step was created.

Q.3. What is the formula for constructing simple index number?
Ans: This approach involves adding the prices of various current-year items, dividing the total by the sum of prices for base-year items, and multiplying by \(100\).
\(P_{01}=\frac{\sum P_{n}}{\sum P_{0}} \times 100\)

Q.4. What is the aim of index numbers?
Ans: The primary functions of an index number are to provide a value useful for comparing the magnitudes of aggregates of related variables, and to measure changes in their magnitudes.

Q.5. What is Fisher’s price index number?
Ans: Fisher’s price index is the geometric mean of Laspeyres and Paasche’s indices.

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