Geometric Mean (GM)

The geometric mean is a type of average that indicates the central tendency or typical value of a set of numbers by using the product of their values. It is especially useful when comparing different items with very different range values, or when the values are not distributed normally (e.g., when the data is log-normally distributed).

Definition

The geometric mean of n non-negative numbers a1, a2, ..., an is given by the nth root of the product of the numbers:

$$ GM = \left( \prod_{i=1}^{n} a_i \right)^{\frac{1}{n}} = \sqrt[n]{a_1 \cdot a_2 \cdot \ldots \cdot a_n} $$

Properties of Geometric Mean

• All the numbers in the set must be non-negative because we cannot calculate the root of a negative number in the real number system.
• The geometric mean is less sensitive to extreme values (outliers) compared to the arithmetic mean.
• It is used in various financial calculations, such as compound interest, stock index, and in many other scientific contexts.
• The geometric mean of a data set is always less than or equal to the arithmetic mean of the same data set.

Comparison with Arithmetic Mean

Here is a table comparing the geometric mean with the arithmetic mean:

Feature	Geometric Mean (GM)	Arithmetic Mean (AM)
Definition	The nth root of the product of n numbers	The sum of n numbers divided by n
Applicability	Best for multiplicative processes and percentages	Best for additive processes
Sensitivity to Extremes	Less sensitive to extreme values	More sensitive to extreme values
When to Use	When dealing with rates of growth, ratios, or index numbers	When dealing with sums or totals

Formulas

Calculation of GM for Two Numbers

For two numbers a and b, the geometric mean is:

$$ GM = \sqrt{a \cdot b} $$

Calculation of GM for n Numbers

For n numbers a1, a2, ..., an, the geometric mean is:

$$ GM = \left( \prod_{i=1}^{n} a_i \right)^{\frac{1}{n}} $$

Examples

Example 1: GM for Two Numbers

Calculate the geometric mean of 4 and 9.

$$ GM = \sqrt{4 \cdot 9} = \sqrt{36} = 6 $$

Example 2: GM for a Set of Numbers

Calculate the geometric mean of 1, 3, 9, 27, and 81.

$$ GM = \sqrt[5]{1 \cdot 3 \cdot 9 \cdot 27 \cdot 81} = \sqrt[5]{59049} = 9 $$

Example 3: GM in Growth Rates

Suppose an investment grows by 10% in the first year, 20% in the second year, and 15% in the third year. To find the average growth rate per year, we use the geometric mean:

$$ GM = \sqrt[3]{1.10 \cdot 1.20 \cdot 1.15} \approx 1.1467 $$

The average growth rate per year is approximately 14.67%.

Example 4: GM in Finance

If a stock index goes from 1000 to 1100 in one year, then to 1210 the next year, the geometric mean of the growth rate is:

$$ GM = \sqrt{1.10 \cdot 1.10} = 1.10 $$

This means the average annual growth rate is 10%.

Conclusion

The geometric mean is a powerful tool for finding the central tendency of a set of numbers, especially when dealing with multiplicative processes or growth rates. It is less affected by extreme values and provides a more accurate measure of the average in certain situations compared to the arithmetic mean. Understanding when and how to use the geometric mean is essential for various applications in finance, economics, and the sciences.