13. Geometric Mean in Business Statistics: Concept, Properties, and Uses

The Geometric Mean is a type of average that is particularly useful for calculating the average rate of change or growth over time, especially when dealing with percentages or ratios. It is calculated by multiplying all the values together and then taking the nth root, where n is the number of values.

13.1 Weighted Geometric Mean

The weighted geometric mean is used when different values have different levels of importance or frequency. Instead of simply multiplying the values, each value is raised to the power of its weight before multiplication, and then the nth root is taken, where n is the sum of the weights.

The formula for the weighted geometric mean ($G_w$) is:

$G_w = (\prod_{i=1}^{n} x_i^{w_i})^{1/\sum_{i=1}^{n} w_i}$

Where:

• $x_i$ is the individual value.
• $w_i$ is the weight assigned to the value $x_i$.
• $n$ is the number of values.

Example:

Suppose you have the annual growth rates of an investment over three years: 10%, 20%, and 15%. If the investment amounts were different for each year, say $1000, $2000, and $1500 respectively, you would use the weighted geometric mean. However, if the question implies weights related to the value itself and not an external frequency, consider the following:

Let's say we have two sets of growth rates:

• Year 1: 5% growth (weighted by 1)
• Year 2: 10% growth (weighted by 2)

$G_w = ((1.05)^1 \times (1.10)^2)^{1/(1+2)}$ $G_w = (1.05 \times 1.21)^{1/3}$ $G_w = (1.2705)^{1/3}$ $G_w \approx 1.083$

This indicates an average growth rate of approximately 8.3%.

13.2 Properties of Geometric Mean

The geometric mean possesses several key properties that make it suitable for specific types of data analysis:

• Positivity: The geometric mean is only defined for positive values. If any value is zero or negative, the geometric mean cannot be calculated in the standard way.
• Sensitivity to Small Values: The geometric mean is heavily influenced by small values in the dataset. A single small value can significantly lower the overall mean.
• Uniqueness: For a given set of positive numbers, there is only one geometric mean.
• Relationship with Arithmetic Mean: The geometric mean is always less than or equal to the arithmetic mean ($\text{GM} \le \text{AM}$). They are equal only when all values in the dataset are identical.
• Logarithmic Property: The logarithm of the geometric mean is equal to the arithmetic mean of the logarithms of the individual values. This property is often used for calculation: $\log(\text{GM}) = \frac{\sum_{i=1}^{n} \log(x_i)}{n}$

13.3 Uses of Geometric Mean

The geometric mean finds its primary applications in business statistics for analyzing data that involves rates of change, growth, or proportional relationships:

• Calculating Average Growth Rates: This is the most common use. For instance, calculating the average annual growth rate of an investment over several years, or the average rate of increase in sales.
• Index Numbers: It is used in the construction of certain index numbers, such as price indices or economic indicators, where the compounding effect of price changes is important.
• Finance and Investment: Evaluating the performance of portfolios or the average return on investments, especially when returns are compounded over time.
• Measuring Relative Changes: When comparing relative changes across different entities or over different periods, the geometric mean provides a more accurate average than the arithmetic mean, as it accounts for compounding.
• Averaging Ratios and Proportions: For datasets consisting of ratios or proportions, the geometric mean is the appropriate measure for central tendency.

Example: Investment Returns

Consider an investment that grew by 10% in Year 1, 20% in Year 2, and 5% in Year 3.

• Year 1 factor: $1 + 0.10 = 1.10$
• Year 2 factor: $1 + 0.20 = 1.20$
• Year 3 factor: $1 + 0.05 = 1.05$

To find the average annual growth rate using the geometric mean:

$G = (1.10 \times 1.20 \times 1.05)^{(1/3)}$ $G = (1.386)^{(1/3)}$ $G \approx 1.1148$

The average annual growth rate is approximately $1.1148 - 1 = 0.1148$, or 11.48%. If we used the arithmetic mean (10% + 20% + 5%) / 3 = 11.67%, it would overstate the true average compounded growth.