Weighted Geometric Mean

13.1 Weighted Geometric Mean

The Weighted Geometric Mean (G.M.ₓ) is used when individual values in a dataset are assigned different levels of importance or influence, represented by weights. It is analogous to the regular geometric mean, but instead of using frequencies, it incorporates specified weights (W) into the calculation.

Formula for Weighted Geometric Mean

The formula for the Weighted Geometric Mean is:

$$ G.M._{x} = \text{Antilog}\left[ \frac{\sum (W \times \log X)}{\sum W} \right] $$

Where:

• X: Represents the variable values.

• W: Represents the corresponding weights assigned to each value of X.

• log X: Denotes the logarithm of each value of X (commonly the common logarithm, base 10, or natural logarithm, base e).

• ∑W: The sum of all weights.

• ∑(W × log X): The sum of the products of each weight and the logarithm of its corresponding value.

Example: Calculate Weighted Geometric Mean

Let's calculate the Weighted Geometric Mean for the following data:

| X | W | log X | W × log X | | :-- | :-- | :------ | :-------- | | 2 | 20 | 0.3010 | 6.0200 | | 4 | 50 | 0.6020 | 30.1000 | | 6 | 40 | 0.7781 | 31.1240 | | 8 | 30 | 0.9030 | 27.0900 | | 10 | 10 | 1.0000 | 10.0000 | | 12 | 4 | 1.0791 | 4.3164 | | Total | 204 | | 108.6504 |

Step-by-Step Calculation:

1. Sum of Weights (∑W): ∑W = 20 + 50 + 40 + 30 + 10 + 4 = 204

2. Sum of Weighted Logarithms (∑(W × log X)): ∑(W × log X) = 6.0200 + 30.1000 + 31.1240 + 27.0900 + 10.0000 + 4.3164 = 108.6504

3. Calculate the average of the weighted logarithms: (∑(W × log X)) / ∑W = 108.6504 / 204 ≈ 0.5326

4. Find the Antilog of the result: G.M.ₓ = Antilog(0.5326)

   • If using common logarithm (base 10): G.M.ₓ = 100.5326 ≈ 3.41

   • If using natural logarithm (base e): G.M.ₓ = e0.5326 ≈ 1.70

   (Note: The original example's final calculation step Antilog[0.7948] = 6.23 seems to be based on a different dataset or an error in the provided ∑W log X and ∑W values for that specific calculation. The detailed breakdown above uses the provided example table's values.)

Important Notes and Limitations

• Non-Positive Values: The Geometric Mean, including its weighted version, cannot be computed if any of the variable values (X) are zero or negative. This is because the logarithm of zero or a negative number is undefined.

• Open-Ended Distributions: Open-ended distributions (e.g., "10 and above") are not suitable for Geometric Mean calculations. To apply the formula, mid-point values for each class interval are required, which cannot be determined for open-ended intervals.

• Weight Interpretation: Weights (W) reflect the relative importance or frequency of each data point. Higher weights give more influence to their corresponding values in the overall calculation.

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Interview Questions

• What is the formula for the weighted geometric mean?

• How does the weighted geometric mean differ from the regular geometric mean?

• Why are logarithms used in calculating the weighted geometric mean?

• Can you explain the step-by-step process to calculate the weighted geometric mean with an example?

• What happens if the dataset contains zero or negative values when calculating the geometric mean?

• Why is it important to assign weights in some geometric mean calculations?

• How do you interpret the weighted geometric mean in real-world data analysis?

• Can the weighted geometric mean be used with open-ended distributions? Why or why not?

• How do weights affect the final value of the weighted geometric mean?

• What are common applications of the weighted geometric mean in statistics or finance?