10.1 Index Numbers
An index number is a statistical measure used to calculate the ratio or percentage change of a variable in a specific year compared to a chosen base year. It helps in measuring economic, environmental, demographic, and social trends without using units.
Learning Objectives
- Define and describe index numbers based on real-life situations.
- Calculate the index number using the standard formula \(I = \dfrac{Q_1}{Q_0} \times 100\).
- Interpret the index number obtained in terms of percentage increase or decrease relative to a base year.
- Determine values of variables (\(Q_1\) or \(Q_0\)) when the index number is given.
- Calculate index numbers involving shifts or changes in base years.
- Apply a structured four-step problem-solving strategy to solve real-world problems involving index numbers.
Relationship Between Base Year and Current Year Index
See how \(Q_0\) and \(Q_1\) relate to the index number \(I\), and how to interpret values greater than, equal to, or less than 100.
Base Year \(Q_0\)
The reference year is always assigned an index value of \(100\). It is the denominator in the formula \(I = \dfrac{Q_1}{Q_0} \times 100\).
Current Year \(Q_1\)
The particular year under study. Its value is compared against the base year to measure the percentage change.
General formula for index number
\(I = \dfrac{Q_1}{Q_0} \times 100\)\(Q_0\) = base year value • \(Q_1\) = current year value
\(I > 100\)
Indicates an increase of \((I - 100)\%\) compared to the base year. Example: \(I = 110\) means a \(10\%\) increase.
\(I < 100\)
Indicates a decrease of \((100 - I)\%\) compared to the base year. Example: \(I = 89.4\) means a \(10.6\%\) decrease.
\(I = 100\)
Indicates no change between the compared year and the base year.
Short Explanation
Index numbers are dimensionless indicators that express how much a price, quantity, or statistical count has changed compared to a baseline period.
Definition and Formula
An index number is a statistical measure used to measure the change of a variable of a particular year as compared to another year considered to be the base year. The base year is always assigned an index value of \(100\).
Interpretation
An index number represents the percentage ratio without writing the percentage sign (\(\%\)). \(I > 100\) means an increase of \((I - 100)\%\), while \(I < 100\) means a decrease of \((100 - I)\%\).
Variety of Index Numbers
Common examples include the Consumer Price Index (CPI), Death Index of Road Accidents, Air Quality Index, and Body Mass Index (BMI).
Changing the Base Year
When connecting three time periods \(a\), \(b\), and \(c\) where \(a < b < c\): \(I_{c/b} = \dfrac{I_{c/a}}{I_{b/a}} \times 100\).
| Index Type | Formula | Notes |
|---|---|---|
| Consumer Price Index (CPI / IHP) | \(\text{IHP} = \dfrac{\text{Market price at current year}}{\text{Market price at base year}} \times 100\) | Measures changes in the price level of consumer goods and services. |
| Death Index of Road Accidents | \(I = \dfrac{a}{\sum \text{vehicles}} \times 10\,000\) | \(a\) = total death at current year; \(\sum \text{vehicles}\) = total registered vehicles till current year. |
| Air Quality Index | \(I = \dfrac{I_{\text{high}} - I_{\text{low}}}{C_{\text{high}} - C_{\text{low}}} (C - C_{\text{low}}) + I_{\text{low}}\) | \(C\) = concentration of pollutants. |
| Body Mass Index (BMI) | \(\text{BMI} = \dfrac{\text{Weight (kg)}}{\text{Height (cm)} \times \text{Height (cm)}} \times 100\) | Used as a general health indicator. |
Four-Step Problem-Solving Model
Understanding the problem: Identify known values, base years, and required unknowns.
Planning the strategy: Set up the appropriate index formula relating given periods.
Implementing the strategy: Substitute values into \(I = \dfrac{Q_1}{Q_0} \times 100\) and solve algebraically.
Making a conclusion: State the final value or interpret the index number within the context.
Index number formula
\(I = \dfrac{Q_1}{Q_0} \times 100\)\(Q_0\) = base year value, \(Q_1\) = current year value
Changing base year
\(I_{c/b} = \dfrac{I_{c/a}}{I_{b/a}} \times 100\)For three periods \(a < b < c\)
Try to Answer First
Answer in your mind, then press "Check Answer".
The price of a school bag in the year 2018 was RM80. In the year 2020, its price was RM96. Calculate the price index for the year 2020 based on the year 2018 and interpret the result.
The production index of a factory in the year 2019 based on the year 2015 was 125. If the factory produced 50 000 units in 2019, calculate its production in 2015.
Given that the price index of a laptop in 2019 based on 2016 was 115, and the price index in 2021 based on 2016 was 138, find the price index of the laptop in 2021 based on 2019.
Common Mistakes
- Placing the current year quantity (\(Q_1\)) in the denominator and the base year quantity (\(Q_0\)) in the numerator, reversing the ratio.
- Writing a percentage sign (\(\%\)) at the end of an index number, for example writing \(I = 120\%\) instead of \(I = 120\).
- Confusing the index value with the percentage change directly when \(I < 100\), for example claiming an index of \(85\) represents an \(85\%\) decrease instead of a \(15\%\) decrease.
- Subtracting indices directly to change base years instead of using the ratio formula \(I_{c/b} = \dfrac{I_{c/a}}{I_{b/a}} \times 100\).
Concept Misunderstandings
An index number of 0 means there is no change.
An index number of 100 represents no change because the base year value is assigned 100.
The base year must always be the first calendar year mentioned in the question.
The base year is strictly the reference year specified by the phrase "based on the year..." (\(Q_0\)), which can be any designated reference year.
An index number has physical units such as RM, kg, or tonnes.
Index numbers are dimensionless ratio measurements and do not have units.
Summary
- Formula: \(I = \dfrac{Q_1}{Q_0} \times 100\), where \(Q_0\) is the base quantity and \(Q_1\) is the current quantity.
- Base year always equals \(100\).
- \(I > 100\) means an increase of \((I - 100)\%\).
- \(I < 100\) means a decrease of \((100 - I)\%\).
- For three periods: \(I_{c/b} = \dfrac{I_{c/a}}{I_{b/a}} \times 100\).
Short Activity
Objective: Test your understanding of index numbers, base years, and percentage changes.
A. Objective Quiz
1 The price of an item in the year 2017 was RM500 and its price in the year 2018 was RM550. What is the price index of the item in the year 2018 based on the year 2017?
2 The price index of a bicycle in the year 2018 based on the year 2010 was 176, and based on the year 2015 was 110. What was the price index of the bicycle in the year 2015 based on the year 2010?
B. Fill in the Blanks
3 In calculating index numbers, the base year is always assigned an index value of ________.
4 If the price index of a commodity in 2020 based on 2016 is 89.4, this indicates a percentage decrease of ________% compared to the year 2016.
C. Matching / Drag and Drop
Match each index number situation or formula component to its correct description or interpretation. On a phone, tap a term first, then tap the matching box.
