Why is the harmonic mean used for averaging speeds over equal distances rather than the arithmetic mean?

Conditions

• Averaging rates (like speed)
• Fixed distance segments (equal legs)

Reasoning, step by step

1. Identify that total time is the sum of times for each leg ($t = d/v$).
2. Recognize that equal distances imply weights are proportional to inverse speeds.
3. Apply the formula $\frac{2}{\frac{1}{v_1} + \frac{1}{v_2}}$ to account for time weighting.
4. Contrast with arithmetic mean which assumes equal time weights.

Example

Driving at 40 mph and returning at 60 mph over the same route. The arithmetic mean is 50 mph, but the actual average speed is 48 mph (harmonic mean) because more time was spent driving at 40 mph.

Common misconceptions

• Using the arithmetic mean for speeds over equal distances.
• Assuming average speed is just the midpoint of min and max speeds regardless of time distribution.

Watch the explanation

Explore next

• When would the arithmetic mean be appropriate for averaging speeds?
• How does the harmonic mean compare to the geometric mean in magnitude?