# MTH5P2A: Index Number

##### This unit explains Index Number

An index number is the measure of change in a variable (or group of variables) over time. It is typically used in economics to measure trends in a wide variety of areas including: stock market prices, cost of living, industrial or agricultural production, and imports. Index numbers are one of the most used statistical tools in economics.

Index numbers are not directly measurable, but represent general, relative changes. They are typically expressed as percents.

Example
In the table, the year 2015 has been chosen as the base year.
The value of 2015 is therefore the base value and is 100%.
The index numbers can be found in the last column.

 Year Value In- or decrease compared to 2015 in % Index number 2012 53 412 –9.1 1 90.9 2013 53 726 –8.5 1 91.5 2014 56 243 –4.3 1 95.7 2015 58 743 –0 100 2016 61 467 –4.6 104.6 2017 59 551 –1.4 101.4

### Calculating with index numbers

Because an index number is a percentage, you can calculate these the same way.

Example 1
See the table above.
The value that corresponds to 2018 is 60 342.
Calculate the index number for the year 2018.

Answer:
60 34258 743 × 100 ≈ 102.7

Example 2
See the table above.
It is known that the index number of the year 2011 is 89.7.
Calculate the value for the year 2011.

Answer:
89.7 : 100 × 58 743 ≈ 52 692

### Types of index numbers

Simple Index Number

A simple index number is a number that measures a relative change in a single variable with respect to a base.

Composite Index Number

A composite index number is a number that measures an average relative changes in a group of relative variables with respect to a base.

The following types of index numbers are usually used: price index numbers and quantity index numbers.

Price Index Numbers

Price index numbers measure the relative changes in the price of a commodity between two periods. Prices can be either retail or wholesale.

Quantity Index Numbers

These index numbers are considered to measure changes in the physical quantity of goods produced, consumed or sold for an item or a group of items.

## Construction of Index Numbers

• Current year: It refers to the year for which we aim to find index number for.
• Base year: It acts as the reference about which we wish to find the change in the value of the variable.

There are two methods of deducing formulae for each of the two types of index numbers.

• Average or Price Relatives Method
• Aggregative Method

### Construction of Simple Index Numbers

#### 1] Simple Average or Price Relatives Method

In this method, we find out the price relative of individual items and average out the individual values. Price relative refers to the percentage ratio of the value of a variable in the current year to its value in the year chosen as the base.

Price relative (R) = (P1÷P2) × 100

Here, P1= Current year value of item with respect to the variable and P2= Base year value of the item with respect to the variable. Effectively, the formula for index number according to this method is:

P = ∑[(P1÷P2) × 100] ÷N

Here, N= Number of goods and P= Index number.

#### 2] Simple Aggregative Method

It calculates the percentage ratio between the aggregate of the prices of all commodities in the current year and aggregate prices of all commodities in the base year.

P= (∑P1÷∑P2)×100

Here, ∑P1= Summation of the prices of all commodities in current year and ∑P2= Summation of prices of all commodities in base year.

### Construction of Weighted Index Number

#### 1] Weighted Average or Price Relatives Method

Here we calculate the ratio between the summation of the product of weights with price relatives and summation of the weights.

P=∑RW÷∑W

Here, R= Price relative and W= weight.

#### 2] Weighted Aggregate Method

Here different goods are assigned weight according to the quantity bought. There are three well-known sub-methods based on the different views of economists as mentioned below:

#### A] Laspeyre’s Method

Laspeyre was of the view that base year quantities must be chosen as weights. Therefore the formula is :

P=(∑P1Q0÷∑P0Q0)×100

Here,  ∑P1Q0= Summation of prices of current year multiplied by quantities of the base year taken as weights and ∑P0Q0= Summation of, prices of base year multiplied by quantities of the base year taken as weights.

#### B] Paasche’s Method

Unlike the above mentioned, Paasche believed that the quantities of the current year must be taken as weights. Hence the formula:

P=(∑P1Q1÷∑P0Q1) ×100

Here, ∑P1Q1= Summation of, prices of current year multiplied by quantities of the current year taken as weights and ∑P0Q1= Summation of, prices of base year multiplied with quantities of the current year taken as weights.

#### C] Fisher’s Method

Fisher combined the best of both above-mentioned formulas which resulted in an ideal method. This method uses both current and base year quantities as weights as follows:

P =  √[ (∑P1Q0÷∑P0Q0) × (∑P1Q1÷∑P0Q1) ]  ×100

NOTE: Index number of base year is generally assumed to be 100 if not given

### Fisher’s Method is an Ideal Measure

As noted Fisher’s method uses views of both Laspeyres and Paasche. Hence it takes into account the prices and quantities of both years. Moreover, it is based on the concept of the geometric mean, which is considered as the best mean method.

However, the most important evidence for the above affirmation is that it satisfies both time reversal and factor reversal tests. Time reversal test checks that when we reverse the current year to base year and vice-versa, the product of indexes should be equal to unity. This confirms the working of a formula in both directions. Also, factor reversal test implies that interchanging the piece and quantities do not give varying results. This proves the consistency of the formula.

LEARNING MORE ABOUT INDEX NUMBERS

### Common Problems with Construction of Index Numbers

Due to the availability of a wide range of index numbers we have to select an index number that matches the objective we want to fulfill. For example, to study the impact of a change in the government’s budget on people, one should refer to the price index number.

It must be noted that the selected base year should be a normal one. In other words, there should be no reforms in that year which can influence the economy in a drastic manner. If such is chosen as the base year there will be a big variation in the index numbers, which would not reflect the accurate changes over the years.

Also, it is not possible to include all the goods and services along with their prices in our calculations. This means we need to select various goods and services that can effectively represent all of them. In a word, a sample size has to be selected. Larger the sample size more is the accuracy. And we need to select the method of calculation that suits best with the objective in hand.

## A Solved Example For You

Q: Construct index numbers of prices of items in the year 2012 from the following data by:

1. Laspeyres method
2.  Paasche’s method
3.  Fisher’s method
 Items Price (2004) Quantity(2004) Price(2012) Quantity (2012) A 10 10 5 25 B 35 4 35 10 C 30 3 15 15 D 10 25 20 20 E 40 3 40 5

Ans:

 Items P0 Q0 P1 Q1 P0Q0 P0Q1 P1Q0 P1Q1 A 10 10 5 25 100 250 50 125 B 35 4 35 10 140 350 140 350 C 30 3 15 15 90 450 45 225 D 10 25 4 20 250 200 100 80 E 40 3 40 5 120 200 120 200 ∑=700 ∑=1450 ∑=455 ∑=980
1. Laspeyre’s method= (455/700) × 100 = 65
2. Paasche’s method= (980/1450) × 100= 67.58
3. Fisher’s method= √0.43927  × 100 = 66.27

## STEM Elearning

We at FAWE have built this platform to aid learners, trainers and mentors get practical help with content, an interactive platform and tools to power their teaching and learning of STEM subjects, more

#### How to find your voice as a woman in Africa

© FAWE, Powered by: Yaaka DN.