How is the geometric mean affected when one of the data points is zero? GM becomes zero GM becomes infinite GM remains unchanged GM becomes negativ

Analysis

The geometric mean (GM) is calculated as the nth root of the product of n data points. If one of the data points in the set is zero, the product of all the data points will also be zero.

Step-by-step Explanation

1. Definition of Geometric Mean: The geometric mean of a set of n positive numbers $x_1, x_2, ..., x_n$ is given by: $GM = \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n}$

2. Impact of Zero in Data Points: If $x_k = 0$ (where $x_k$ is any data point in the set): $Product = x_1 \times x_2 \times \cdots \times x_k \times \cdots \times x_n = 0$

3. Calculation of GM: Then the geometric mean is calculated as: $GM = \sqrt[n]{0} = 0$

Verification and Conclusion

Since the geometric mean involves the product of all data points, having at least one data point equal to zero means that the product will always be zero, and therefore the geometric mean will also be zero.

Final Answer

The geometric mean becomes zero when one of the data points is zero.