Both the geometric mean and harmonic mean are measures of central tendency, but they have different formulas and applications.
Get the full solved assignment PDF of MCO-022 of 2023-24 session now.
Let’s explore each:
Geometric Mean:
The geometric mean is calculated by multiplying all the values together and then taking the nth root, where n is the number of values.
[ \text{Geometric Mean} = \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n} ]
- Properties:
- Suitable for data with multiplicative relationships.
- Sensitive to both small and large values.
- The product of the geometric mean with itself (n) times equals the product of the values.
- Example:
- For the numbers 2, 4, and 8: [ \text{Geometric Mean} = \sqrt[3]{2 \cdot 4 \cdot 8} = \sqrt[3]{64} = 4 ]
Harmonic Mean:
The harmonic mean is calculated by dividing the number of values by the reciprocal of each value, and then taking the reciprocal of the result.
[ \text{Harmonic Mean} = \frac{n}{\left(\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}\right)} ]
- Properties:
- Suitable for data with rates or ratios.
- Sensitive to small values.
- The harmonic mean is always less than or equal to the arithmetic mean.
- Example:
- For the numbers 2, 4, and 8: [ \text{Harmonic Mean} = \frac{3}{\left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8}\right)} = \frac{3}{\left(\frac{4}{8} + \frac{2}{8} + \frac{1}{8}\right)} = \frac{3}{\frac{7}{8}} = \frac{24}{7} ]
Comparison:
- The geometric mean is suitable for values with multiplicative relationships, like growth rates or investment returns.
- The harmonic mean is useful for values with rates or ratios, such as speed or efficiency.
In summary, while the arithmetic mean is more commonly used, the geometric and harmonic means offer valuable insights in specific contexts, addressing scenarios where relationships are multiplicative or rates are involved.