Question 1

Why use the harmonic mean for averaging speeds?

Accepted Answer

If you drive equal distances at different speeds, the arithmetic mean overestimates the average speed. The harmonic mean correctly accounts for the fact that you spend more time at the slower speed. For 60 mph and 40 mph over equal distances: arithmetic mean = 50 mph, but harmonic mean = 48 mph (correct).

Question 2

How is the harmonic mean used in machine learning?

Accepted Answer

The F1 score is the harmonic mean of precision and recall: F1 = 2 × (precision × recall) / (precision + recall). The harmonic mean is used because it penalizes extreme imbalances -- if either precision or recall is very low, the F1 score will also be low.

Question 3

Why is the harmonic mean always less than or equal to the other means?

Accepted Answer

For positive values: HM ≤ GM ≤ AM (AM-GM-HM inequality). The harmonic mean gives more weight to smaller values. Equality holds only when all values are identical. This property makes HM useful when you want to penalize extreme low values.

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