Understanding Geometric Mean and Ratios

This question asks us to find the geometric mean of a set of ratios, given the geometric means of the numerators and the denominators separately. Let's first recall the definition of the geometric mean.

The geometric mean (GM) of \(n\) positive observations \(z_1, z_2, \ldots, z_n\) is defined as the \(n\)-th root of their product:

\(\text{GM} = \left( z_1 \cdot z_2 \cdot \ldots \cdot z_n \right)^{1/n}\)

Applying the Definition to the Given Information

We are given two sets of observations and their geometric means:

1. Observations: \(x_1, x_2, \ldots, x_n\). Their geometric mean is \(G_1\).
2. Observations: \(y_1, y_2, \ldots, y_n\). Their geometric mean is \(G_2\).

Using the definition of the geometric mean, we can write:

• For the first set: \(G_1 = \left( x_1 \cdot x_2 \cdot \ldots \cdot x_n \right)^{1/n}\)
• Raising both sides to the power of \(n\): \(G_1^n = x_1 \cdot x_2 \cdot \ldots \cdot x_n\)

• For the second set: \(G_2 = \left( y_1 \cdot y_2 \cdot \ldots \cdot y_n \right)^{1/n}\)
• Raising both sides to the power of \(n\): \(G_2^n = y_1 \cdot y_2 \cdot \ldots \cdot y_n\)

Calculating Geometric Mean of Ratios

We need to find the geometric mean of the observations \(\frac{{{{\rm{x}}_1}}}{{{{\rm{y}}_1}}},\frac{{{{\rm{x}}_2}}}{{{{\rm{y}}_2}}},\frac{{{{\rm{x}}_3}}}{{{{\rm{y}}_3}}}, \ldots \frac{{{{\rm{x}}_{\rm{n}}}}}{{{{\rm{y}}_{\rm{n}}}}}\). Let's call this geometric mean \(G\).

Using the definition of geometric mean for this new set of observations:

\(G = \left( \frac{{{{\rm{x}}_1}}}{{{{\rm{y}}_1}}} \cdot \frac{{{{\rm{x}}_2}}}{{{{\rm{y}}_2}}} \cdot \ldots \cdot \frac{{{{\rm{x}}_{\rm{n}}}}}{{{{\rm{y}}_{\rm{n}}}}} \right)^{1/n}\)

We can rewrite the product inside the parenthesis as the product of the numerators divided by the product of the denominators:

\(G = \left( \frac{{{{\rm{x}}_1} \cdot {{\rm{x}}_2} \cdot \ldots \cdot {{\rm{x}}_{\rm{n}}}}}{{{{\rm{y}}_1} \cdot {{\rm{y}}_2} \cdot \ldots \cdot {{\rm{y}}_{\rm{n}}}}} \right)^{1/n}\)

Now, we can substitute the expressions we found for the product of x's and the product of y's:

\(G = \left( \frac{{G_1^n}}{{G_2^n}} \right)^{1/n}\)

Using the property of exponents \(\left( \frac{a^n}{b^n} \right) = \left( \frac{a}{b} \right)^n\), we get:

\(G = \left( \left( \frac{{G_1}}{{G_2}} \right)^n \right)^{1/n}\)

Using the property of exponents \((a^n)^{1/n} = a\), we get:

\(G = \frac{{G_1}}{{G_2}}\)

Therefore, the geometric mean of the ratios is the ratio of the individual geometric means.

Conclusion

The geometric mean of the observations \(\frac{{{{\rm{x}}_1}}}{{{{\rm{y}}_1}}},\frac{{{{\rm{x}}_2}}}{{{{\rm{y}}_2}}},\frac{{{{\rm{x}}_3}}}{{{{\rm{y}}_3}}}, \ldots \frac{{{{\rm{x}}_{\rm{n}}}}}{{{{\rm{y}}_{\rm{n}}}}}\) is \(\frac{{{{\rm{G}}_1}}}{{{{\rm{G}}_2}}}\).

Revision Table: Key Concepts in Geometric Mean

Additional Information: Why Geometric Mean for Ratios?

The geometric mean is particularly useful when dealing with percentages, ratios, or growth rates, because it's less affected by extreme values than the arithmetic mean and better reflects multiplicative relationships. In this problem, we are dealing with ratios \(\frac{x_i}{y_i}\), which are essentially comparisons through division. The property \(GM\left(\frac{x_i}{y_i}\right) = \frac{GM(x_i)}{GM(y_i)}\) shows how the geometric mean behaves consistently with division, making it the appropriate average for this scenario.

Comparing with other means:

• Arithmetic Mean (AM): \(AM(z_i) = \frac{1}{n} \sum z_i\). \(AM\left(\frac{x_i}{y_i}\right) \neq \frac{AM(x_i)}{AM(y_i)}\) generally.
• Harmonic Mean (HM): \(HM(z_i) = \left(\frac{1}{n}\sum \frac{1}{z_i}\right)^{-1}\). \(HM\left(\frac{x_i}{y_i}\right) \neq \frac{HM(x_i)}{HM(y_i)}\) generally.

The multiplicative nature of the geometric mean aligns perfectly with operations involving products and ratios.