13.2 Properties of the Geometric Mean

The Geometric Mean (G.M.) possesses several valuable algebraic properties that make it particularly well-suited for specific types of data analysis, especially when dealing with multiplicative relationships or data that exhibits exponential growth.

Key Properties

1. Product of Observations Equals (G.M.)ⁿ

This property states that the product of all n observations in a dataset is equal to the geometric mean raised to the power of n.

Formula: $X_1 \times X_2 \times X_3 \times \dots \times X_n = (\text{G.M.})^n$

Explanation: This property is a direct consequence of the definition of the geometric mean: $\text{G.M.} = (X_1 \times X_2 \times \dots \times X_n)^{1/n}$ Raising both sides to the power of n isolates the product of the observations.

Example: Consider the dataset: {4, 8, 4, 2}

First, calculate the Geometric Mean (G.M.): $\text{G.M.} = (4 \times 8 \times 4 \times 2)^{1/4}$ $\text{G.M.} = (256)^{1/4}$ $\text{G.M.} = 4$

Now, verify the property: Product of the values = $4 \times 8 \times 4 \times 2 = 256$ $(\text{G.M.})^n = 4^4 = 256$

Thus, the product of the observations equals $(\text{G.M.})^n$.

2. Proportion Property Around the Geometric Mean

This property describes a balance around the geometric mean. The product of the ratios of the geometric mean to all values less than or equal to the geometric mean is equal to the product of the ratios of all values greater than the geometric mean to the geometric mean.

Formula: $\prod_{X_i \le \text{G.M.}} \left(\frac{\text{G.M.}}{X_i}\right) = \prod_{X_j > \text{G.M.}} \left(\frac{X_j}{\text{G.M.}}\right)$

Explanation: This property highlights how the geometric mean acts as a central point in a multiplicative sense. For every observation below or equal to the G.M., there's a corresponding "gap" above the G.M. that balances it out multiplicatively.

Example (using values 4, 8, 4, 2): From the previous example, we know G.M. = 4.

• Values $\le$ G.M.: {4, 4, 2}
• Values $>$ G.M.: {8}

Calculate the Left-Hand Side (LHS): LHS = $\left(\frac{\text{G.M.}}{4}\right) \times \left(\frac{\text{G.M.}}{4}\right) \times \left(\frac{\text{G.M.}}{2}\right)$ LHS = $\left(\frac{4}{4}\right) \times \left(\frac{4}{4}\right) \times \left(\frac{4}{2}\right)$ LHS = $1 \times 1 \times 2 = 2$

Calculate the Right-Hand Side (RHS): RHS = $\left(\frac{8}{\text{G.M.}}\right)$ RHS = $\left(\frac{8}{4}\right)$ RHS = $2$

Since LHS = RHS = 2, the proportion property holds true.

3. Combined Geometric Mean for Two Groups

When you have two datasets with different sample sizes and their respective geometric means, you can calculate the combined geometric mean for the merged dataset. This is often done using logarithms due to the nature of the geometric mean.

Formula: $\text{Combined G.M.} = \text{Antilog}\left(\frac{n_1 \times \log(\text{G.M.}_1) + n_2 \times \log(\text{G.M.}_2)}{n_1 + n_2}\right)$

Where:

• $n_1$ = size of the first dataset
• G.M.$_1$ = geometric mean of the first dataset
• $n_2$ = size of the second dataset
• G.M.$_2$ = geometric mean of the second dataset
• $\log$ refers to the logarithm (commonly base 10 or natural logarithm)
• Antilog is the inverse of the logarithm (e.g., $10^x$ if using log base 10, or $e^x$ if using natural log)

Explanation: This formula essentially calculates a weighted average of the logarithms of the geometric means, where the weights are the sample sizes. The antilog of this weighted average gives the geometric mean of the combined data. This method is used because the logarithm transforms the multiplicative property of the G.M. into an additive property, allowing for easier averaging.

Example: Suppose we have two datasets:

• Dataset 1: $n_1 = 10$, G.M.$_1 = 18.5$
• Dataset 2: $n_2 = 15$, G.M.$_2 = 7.8$

Step-by-step calculation (using log base 10):

1. Calculate the logarithms of the individual geometric means: $\log(18.5) \approx 1.267$ $\log(7.8) \approx 0.892$

2. Substitute these values into the combined G.M. formula: Combined G.M. = $\text{Antilog}\left(\frac{(10 \times 1.267) + (15 \times 0.892)}{10 + 15}\right)$

3. Perform the calculations inside the antilog: Combined G.M. = $\text{Antilog}\left(\frac{12.67 + 13.38}{25}\right)$ Combined G.M. = $\text{Antilog}\left(\frac{26.05}{25}\right)$ Combined G.M. = $\text{Antilog}(1.042)$

4. Calculate the antilog (10 raised to the power of 1.042): Combined G.M. $\approx 10.94$

Therefore, the combined geometric mean for the two datasets is approximately 10.94.

Summary of Key Formulas

• Product Property: $X_1 \times X_2 \times \dots \times X_n = (\text{G.M.})^n$
• Proportional Balancing Property: $\prod_{X_i \le \text{G.M.}} \left(\frac{\text{G.M.}}{X_i}\right) = \prod_{X_j > \text{G.M.}} \left(\frac{X_j}{\text{G.M.}}\right)$
• Combined Geometric Mean: $\text{G.M.} = \text{Antilog}\left(\frac{n_1 \times \log(\text{G.M.}_1) + n_2 \times \log(\text{G.M.}_2)}{n_1 + n_2}\right)$