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**Hint:**The Geometric Mean is one of the central tendencies. It is the average growth of investment calculated by taking the product of \[n\] variables and then finding the \[{n^{th}}\] root of the same. Geometric Mean is the average return of an investment over time. Geometric Mean is a metric unit which is used to calculate the performance of a single investment or portfolio.

**Complete answer:**

The Geometric Mean, abbreviated as G.M. of a series of \[n\] observation is the \[{n^{th}}\] root of the product values of the entries.

Let \[{x_1},{\text{ }}{x_2},{\text{ }}{x_3},{\text{ }}{x_4}, \ldots .{\text{ }}{x_n}\]be the observation then the geometric of the data is defined as

\[G.M. = \sqrt[n]{{{x_1} \times {\text{ }}{x_2} \times {\text{ }}{x_3} \times {x_4} \times \ldots .{\text{ }} \times {x_n}}}\]

Or,

\[G.M. = {\left( {{x_1} \times {\text{ }}{x_2} \times {\text{ }}{x_3} \times {x_4} \times \ldots .{\text{ }} \times {x_n}} \right)^{\frac{1}{n}}}\]

We have to state the advantages and disadvantages of the Geometric Mean

Advantages of Geometric Mean:

-It is rigidly defined

-It is based upon all the observations and all terms of sequences very precisely

-It gives comparatively more weight of values to small items or data

-It is not that much affected by fluctuations of sampling.

-It is suitable for further mathematical analysis and observations.

Disadvantages of Geometric Mean:

-Due to the complex numerical character and abstract behaviour it is not friendly for non-mathematical people. It will be quite difficult for them to go through the steps and calculation for the Geometric Mean.

-If any observation or data of the sequence becomes negative then the Geometric Mean becomes imaginary as the root terms take it towards complex numbers regardless of the magnitude of the other quantities.

**Note:**

The three central tendencies are mean, median and mode. Geometric Mean is the most appropriate for the series that exhibit a serial correlation, this is especially correct for the investment portfolios. Geometric Mean is technically defined as the \[{n^{th}}\] root product of \[n\] numbers or data. It must be used in the scenario where the data is in percentages, which are derived from the values. It is a complex calculation but very precise.

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