All three means, the same two numbers
For the numbers 4 and 9: the arithmetic mean is (4 + 9) ÷ 2 = 6.5. The geometric mean is √(4 × 9) = 6. The harmonic mean is 2 ÷ (1/4 + 1/9) ≈ 5.538462. Ordered from largest to smallest: arithmetic (6.5) ≥ geometric (6) ≥ harmonic (5.538462).
This ordering always holds
This isn't specific to 4 and 9 — for any list of positive numbers, arithmetic mean is always greater than or equal to geometric mean, which is always greater than or equal to harmonic mean. The three are only ever equal when every number in the list is identical.
Why the gap exists at all
Arithmetic mean treats every value at face value. Geometric mean, by multiplying values together, is pulled down by any relatively small value in the list. Harmonic mean, by averaging reciprocals, is pulled down even further — small values produce large reciprocals, which dominate the sum. The more spread out the numbers are, the wider the gap between all three means becomes.
Why this matters when choosing which mean to use
Since arithmetic mean is always the largest of the three, using it where geometric or harmonic mean is actually appropriate — like averaging investment returns or speeds — will systematically overstate the true result. The size of the overstatement depends on how volatile or spread out the underlying numbers are.