To obtain a new series of index numbers, the old index numbers are multiplied with which of the following common factor?
Understanding Index Numbers and Base Year
Index numbers are statistical tools used to measure the average change in price, quantity, or value of a group of related items over time. They are typically expressed as percentages, relative to a base period (often a specific year) which is assigned an index value of 100.
The base year serves as a reference point. When we say the index number for a particular year is 120, it means the value (price, quantity, etc.) in that year is 120% of the value in the base year, representing a 20% increase.
Changing the Base Year of Index Numbers
Sometimes, it becomes necessary to shift the base year of an existing series of index numbers. This might be done for several reasons:
- To make comparisons easier with other series that use a different base year.
- To update the reference point if the original base year is too old or no longer relevant.
- To reflect changes in the pattern of consumption or production over time.
The process of changing the base year involves recalculating each index number in the series so that the index number for the new base year becomes 100.
The Formula for Changing Base Year
To convert an old series of index numbers to a new series with a different base year, we use the following formula:
\(\text{New Index Number for a given year} = \frac{\text{Old Index Number for that year}}{\text{Old Index Number for the new base year}} \times 100\)
Identifying the Common Factor
Let's look at the formula again:
\(\text{New Index Number} = \text{Old Index Number} \times \left( \frac{100}{\text{Old Index Number for the new base year}} \right)\)
In this formula, the term \(\left( \frac{100}{\text{Old Index Number for the new base year}} \right)\) is a constant value for the entire series. It does not change depending on the specific year for which you are calculating the new index number, because the "Old Index Number for the new base year" is fixed once the new base year is chosen.
This constant value is the "common factor" by which every old index number is multiplied to get the new index number.
Therefore, the common factor used to multiply the old index numbers to obtain a new series based on a new base year is \(\frac{100}{\text{Index number of new base year}}\).
Example of Changing Base Year
Let's consider a simple example. Suppose we have the following index numbers with Year 2000 as the base (Index = 100):
| Year | Old Index (Base: 2000=100) |
|---|---|
| 2000 | 100 |
| 2005 | 150 |
| 2010 | 180 |
| 2015 | 200 |
Now, let's change the base year to 2005. The old index number for the new base year (2005) is 150.
The common factor is \(\frac{100}{\text{Old Index Number for 2005}} = \frac{100}{150}\).
Let's calculate the new index numbers:
- For Year 2000: Old Index = 100. New Index = \(100 \times \frac{100}{150} = 66.67\)
- For Year 2005: Old Index = 150. New Index = \(150 \times \frac{100}{150} = 100\) (As expected, the new base year index is 100)
- For Year 2010: Old Index = 180. New Index = \(180 \times \frac{100}{150} = 120\)
- For Year 2015: Old Index = 200. New Index = \(200 \times \frac{100}{150} = 133.33\)
The new series of index numbers with 2005 as the base would be approximately 66.67, 100, 120, 133.33.
| Year | New Index (Base: 2005=100) |
|---|---|
| 2000 | 66.67 |
| 2005 | 100 |
| 2010 | 120 |
| 2015 | 133.33 |
This example clearly shows that each old index number is multiplied by the common factor \(\frac{100}{150}\), which is \(\frac{100}{\text{Old Index Number for the new base year}}\).
Conclusion on Common Factor
Based on the formula and the explanation, the common factor required to convert an old series of index numbers to a new series with a different base year is \(\frac{100}{\text{Index number of new base year}}\).
Revision Table: Changing Index Number Base
| Concept | Description | Formula/Method |
|---|---|---|
| Index Number | Measures relative change over time compared to a base period. | \(\frac{\text{Value in Current Period}}{\text{Value in Base Period}} \times 100\) |
| Base Year | The reference period assigned an index value of 100. | Chosen period for comparison. |
| Changing Base Year | Shifting the reference point to a different year in the series. | Multiply old indices by a common factor. |
| Common Factor | The constant multiplier used for base shifting. | \(\frac{100}{\text{Old Index Number of the new base year}}\) |
Additional Information on Index Numbers
Understanding index numbers involves knowing a few related concepts:
- Simple Index Numbers: These measure the change in a single variable.
- Composite Index Numbers: These measure the average change in a group of related variables (like price index, quantity index).
- Price Index Numbers: Measure changes in the price level of goods and services. Examples include the Consumer Price Index (CPI) and Wholesale Price Index (WPI).
- Quantity Index Numbers: Measure changes in the volume of production or consumption.
- Value Index Numbers: Measure changes in the total value (Price \(\times\) Quantity).
- Methods of Construction: Index numbers can be simple aggregative, simple average of relatives, weighted aggregative (like Laspeyres' and Paasche's), or weighted average of relatives.
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