Which one of the following measures of central tendency is used in construction of index numbers?
Geometric mean
Understanding Measures of Central Tendency in Index Numbers
Index numbers are statistical measures used to track changes in a variable or group of variables over time or across different locations. They are particularly useful for comparing relative changes rather than absolute values. When constructing index numbers, especially those that aggregate price or quantity changes of multiple items, a suitable measure of central tendency is required to represent the overall change.
Why Measures of Central Tendency are Needed for Index Numbers
An index number often summarizes the price or quantity changes of several commodities. To get a single representative figure for the group's change, we need to average the individual changes. The choice of the measure of central tendency significantly impacts the resulting index number.
Analyzing Options for Index Number Construction
Let's look at the common measures of central tendency and their suitability for constructing index numbers:
- Harmonic Mean: The harmonic mean is typically used for averaging rates or ratios when the data are expressed as reciprocals, such as averaging speeds over fixed distances or calculating average price when the number of items bought for a fixed amount is given. While useful for specific types of averages, it is not the standard or preferred method for the general construction of index numbers involving price or quantity relatives.
- Geometric Mean: The geometric mean is defined as the nth root of the product of n values. It is particularly appropriate for averaging rates of change, ratios, and percentages. When calculating an index number, we often deal with price relatives (current price / base price) or quantity relatives. Averaging these relatives using the geometric mean is preferred because it gives equal weight to equal proportional changes and is not unduly influenced by extreme values in the same way the arithmetic mean can be when dealing with ratios. It also satisfies the time reversal test and factor reversal test under certain conditions, which are desirable properties for index numbers.
- Median: The median is the middle value in a dataset that is ordered from least to greatest. It is a measure of position and is not suitable for averaging ratios or rates of change, which is fundamental to index number construction. The median doesn't account for the magnitude of all changes, only the central one.
- Mode: The mode is the value that appears most frequently in a dataset. Like the median, it is a measure of position/frequency and is not appropriate for calculating a representative average of price or quantity relatives needed for index numbers.
Geometric Mean: The Preferred Measure for Index Numbers
Based on the characteristics required for averaging price and quantity relatives, the geometric mean is widely recognized and used in the construction of many standard index numbers, such as the Fisher's Ideal Index and certain types of average-of-relatives indices. Its ability to handle proportional changes makes it superior to arithmetic mean for this purpose and certainly more suitable than median or mode.
For example, if the price of item A increases by 100% (relative = 2) and item B decreases by 50% (relative = 0.5), the arithmetic mean of the relatives is $(2 + 0.5) / 2 = 1.25$, suggesting a 25% average increase. The geometric mean is $\sqrt{2 \times 0.5} = \sqrt{1} = 1$, suggesting no average change, which is a more balanced representation of opposing proportional changes.
Thus, the measure of central tendency most commonly used in the construction of index numbers is the geometric mean.
Revision Table: Central Tendencies in Index Numbers
| Measure of Central Tendency | Suitability for Index Numbers | Reason |
|---|---|---|
| Harmonic Mean | Less common for general index construction | More suitable for specific types of rates/ratios (e.g., speed, price per unit amount). |
| Geometric Mean | Most Suitable and Widely Used | Appropriate for averaging ratios/relatives; accounts for proportional changes; satisfies certain index number tests. |
| Median | Not suitable | Measure of position, not average of ratios/rates. |
| Mode | Not suitable | Measure of frequency, not average of ratios/rates. |
Additional Information on Index Numbers and Averages
While geometric mean is preferred for averaging relatives, some index numbers (like the Laspeyres or Paasche price indices) implicitly use a form of weighted average of prices or quantities, which relates more to arithmetic means of values rather than geometric means of relatives directly. However, when explicitly averaging price/quantity relatives to construct an index, the geometric mean is the theoretically sounder choice for many applications because it handles multiplicative relationships inherent in ratios better.
Understanding index numbers is crucial in economics and statistics for measuring inflation, changes in industrial production, stock market movements, etc.
Important Questions from Measures of Central Tendency
If X̅ = 20 is the mean of 10 observations x1, x2, ... x10; then what is the value of \(\displaystyle \sum_{i=1}^{10}\left(\frac{3 x_i-4}{5}\right) ?\) ?
What is the mean of the numbers 1, 2, 3, ... 10 with frequencies 9C0, 9C1, 9C2 ..., 9C9, respectively?
The mean of five numbers is 30. If one number is excluded, their mean becomes 28. The excluded number is
The ‘less than’ ogive curve and the ‘more than’ ogive curve intersect at
The observations 4, 1, 4, 3, 6, 2, 1, 3, 4, 5, 1, 6 are outputs of 12 dices thrown simultaneously. If m and M are means of lowest 8 observations and highest 4 observations respectively, then what is (2m + M) equal to ?
