The geometric mean is a measure of central tendency that is calculated by multiplying together a set of numbers and then taking the nth root (where n is the number of values in the set). It is commonly used when dealing with quantities that are compounded, such as investment returns or growth rates.

The formula for the geometric mean of n numbers \(x_1, x_2, …, x_n\) is given by:

\[ GM = \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n} \]

Here, \( \sqrt[n]{} \) represents the nth root.

Key points about the geometric mean:

1. **Multiplicative Nature:** Unlike the arithmetic mean, which involves adding up values and dividing by the number of values, the geometric mean involves multiplying the values. This makes it suitable for situations where the quantities are proportional or compounded over time.

2. **Sensitivity to Small Values:** The geometric mean is sensitive to small values in the set. If there is a very small value, it will have a significant impact on the geometric mean. This property makes it useful in scenarios where small values need to be considered, such as rates of return in finance.

3. **Use in Finance:** The geometric mean is often used in finance to calculate the average rate of return on an investment over multiple periods. It provides a more accurate measure of the true return, especially when returns are compounded over time.

4. **Real-World Example:** Suppose you invest $100 in a stock that returns 10% in the first year and loses 5% in the second year. The arithmetic mean of the returns would be (10% – 5%) / 2 = 2.5%, but the geometric mean would account for the compounding effect: \( \sqrt{(1 + 0.10) \cdot (1 – 0.05)} – 1 \).

5. **Not Suitable for Negative Values:** The geometric mean is not defined for sets that contain negative values or zero because the nth root is undefined for such values.

6. **Harmonic Mean Relationship:** The geometric mean is related to the harmonic mean and the arithmetic mean. For a set of positive values, the inequality \(GM \leq AM \leq HM\) holds, where GM is the geometric mean, AM is the arithmetic mean, and HM is the harmonic mean.

The geometric mean provides a useful measure of average when dealing with multiplicative relationships or scenarios involving growth rates. It is important to choose the appropriate mean (arithmetic, geometric, or harmonic) based on the nature of the data and the mathematical characteristics of each measure.