10.2 Composite Index
In everyday life, products or services are rarely made from a single component. A composite index combines multiple index numbers to give an overall statistical measure of performance, price changes, or trends across a sector or group of items.
Learning Objectives
- Determine and calculate the composite index using the formula \(\bar{I} = \dfrac{\sum I_i w_i}{\sum w_i}\).
- Interpret the value of the composite index in terms of overall price increase, decrease, or stability.
- Understand the role of weightages presented as percentages, ratios, frequencies, or charts.
- Calculate the composite index when no weightages are given by assigning equal weightage to each item.
- Solve real-world problems involving index numbers and composite numbers.
Summary of Composite Index Calculation and Weightage Forms
See how individual indices and weightages combine into one overall measure, and how weightage can be represented in different ways.
Composite Index Formula
\[\bar{I} = \dfrac{\sum I_i w_i}{\sum w_i}\]
Multiply each index by its weightage, add all products, then divide by the total weightage.
Interpreting \(\bar{I}\)
\(\bar{I} = 100\) → no change.
\(\bar{I} > 100\) → increase of \((\bar{I} - 100)\%\).
\(\bar{I} < 100\) → decrease of \((100 - \bar{I})\%\).
Weightage Forms
Percentages: \(\sum w_i = 100\)
Ratios: e.g. \(60 : 40\)
Frequencies or raw numbers
Bar chart or pie chart readings
Example: Utility Expenditure
Water \(17\%\), Electricity \(58\%\), Fuel \(25\%\).
These percentages are the weightages used in the composite index.
Example: Credit Points
Chemistry: 3, Economy: 2, Science: 3.
These frequencies are the weightages used in the composite index.
Short Explanation
A composite index is a weighted average of individual index numbers.
Definition of Composite Index
A composite index is a statistical measure that combines several individual indices to oversee market or sector performance over time, taking into account the relative importance of each item.
Understanding Weightage
Weightage represents the relative importance, quantity, frequency, or proportion of usage of each item.
When No Weightage Is Given
Each item is assumed to have the same weightage, \(w = 1\). The composite index is then the simple arithmetic mean of the individual index numbers.
Composite Costs
Use \(\bar{I} = \dfrac{Q_1}{Q_0} \times 100\), where \(Q_1\) is the current total cost and \(Q_0\) is the base-year total cost.
Composite Index Formula
\[\bar{I} = \dfrac{I_1 w_1 + I_2 w_2 + I_3 w_3 + \dots + I_n w_n}{w_1 + w_2 + w_3 + \dots + w_n} = \dfrac{\sum I_i w_i}{\sum w_i}\]\(I_i\) is the index number of the \(i\)-th item and \(w_i\) is its corresponding weightage.
Steps to Calculate the Composite Index
Multiply each individual index number \(I_i\) by its weightage \(w_i\).
Sum all the products: \(\sum I_i w_i\).
Sum all the weightages: \(\sum w_i\).
Divide the total weighted sum by the total weightage.
| Value of \(\bar{I}\) | Interpretation |
|---|---|
| \(\bar{I} = 100\) | No overall change compared to the base year. |
| \(\bar{I} > 100\) | Overall increase of \((\bar{I} - 100)\%\). |
| \(\bar{I} < 100\) | Overall decrease of \((100 - \bar{I})\%\). |
Try to Answer First
Answer in your mind, then press "Check Answer".
The price indices of three items \(X\), \(Y\), and \(Z\) in 2020 based on 2018 are \(110\), \(125\), and \(140\) respectively. If the weightages of \(X\), \(Y\), and \(Z\) are in the ratio \(2 : 3 : 5\), find the composite index for the three items in 2020 based on 2018.
The composite index for four components in the year 2021 based on 2019 is \(120\). If the total cost of production in 2019 was RM\(45\text{ million}\), determine the total cost of production in 2021.
Three types of fruits have price indices of \(175\), \(120\), and \(160\) in 2018 based on 2010. If no weightage is given, calculate the composite index and state the percentage change.
Common Mistakes
- Forgetting to divide by the sum of weightages \(\sum w_i\) and dividing by the number of items \(n\) instead when weightages are not all equal to \(1\).
- Incorrectly assuming \(\sum w_i\) is always \(100\) when weightages are given as ratios, raw numbers, or frequencies rather than percentages.
- Mixing up the base year and current year when using the composite index to find actual total costs: \(\bar{I} = \dfrac{Q_1}{Q_0} \times 100\).
- Misinterpreting the final composite index: stating that \(\bar{I} = 135.4\) means an increase of \(135.4\%\), rather than an increase of \(35.4\%\) (\(135.4 - 100\)).
Concept Misunderstandings
If one component in a composite index decreases in price, the composite index must also decrease.
The overall composite index depends on the weightages. If the component that decreased has a small weightage while other components with large weightages increased significantly, the overall composite index will still increase.
Weightage must always be expressed as a percentage.
Weightages can be expressed as percentages, ratios, frequencies, degrees in a pie chart, or scale readings on a bar chart. Any consistent set of proportional values works in the formula.
You cannot calculate a composite index if weightages are not stated in the question.
When no weightage is given, each item is assumed to carry equal weightage (\(w = 1\)), and the composite index is calculated as the simple average of the individual indices.
Summary
- Composite index (\(\bar{I}\)) measures the combined relative change of several items using the formula: \(\bar{I} = \dfrac{\sum I_i w_i}{\sum w_i}\).
- Weightage (\(w\)) reflects the relative importance of each item and can be represented as percentages, ratios, frequencies, or chart readings.
- If no weightage is assigned, equal weightage (\(w = 1\)) is assumed for each item.
- An overall composite index can be interpreted as a percentage increase (\(> 100\)), no change (\(= 100\)), or percentage decrease (\(< 100\)).
- Total costs or values between two years can be determined using: \(\bar{I} = \dfrac{Q_{\text{current}}}{Q_{\text{base}}} \times 100\).
Short Activity
Objective: Test your understanding of composite index calculations, interpretation, and key terms.
A. Objective Quiz
1 The price indices of two items \(P\) and \(Q\) are \(120\) and \(150\) with weightages of \(3\) and \(2\) respectively. What is the composite index?
2 If the composite index for the production cost of an item in the year 2020 based on 2016 is \(135\), what is the percentage change in the production cost?
B. Fill in the Blanks
3 In the formula for composite index, \(\bar{I} = \dfrac{\sum I_i w_i}{\sum w_i}\), the symbol \(w_i\) represents the ________ of the item.
4 When no weightage is given for the items in calculating a composite index, the weightage for each item is assumed to be ________.
C. Matching / Drag and Drop
Drag each concept to the correct meaning. On a phone, tap a concept first, then tap the matching box.
