Explanation

As long as there is variability in the data, the arithmetic mean is greater than geometric mean, which is greater than the harmonic mean. This relationship holds true for any set of positive numbers with at least some variation.

Mathematical Relationship: For any set of positive numbers with variation: Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean

The equality holds only when all numbers in the dataset are identical (no variation).

Why this relationship exists:

1. Arithmetic Mean gives equal weight to all values
2. Geometric Mean is more affected by smaller values due to the multiplicative nature
3. Harmonic Mean gives more weight to smaller values and is most sensitive to extreme low values

This relationship is important in finance for calculating returns, averages of ratios, and other applications where different averaging methods are appropriate.