Harmonic mean

The harmonic mean of nonzero values is the reciprocal of the arithmetic mean of their reciprocals. It gives greater weight to smaller values than the arithmetic mean does.

Arithmetic mean: The sum of a set of values divided by the number of values. The harmonic mean is the reciprocal of the arithmetic mean applied to reciprocals.
Average speed: Total distance traveled divided by total elapsed time. For equal-distance legs, average speed is the harmonic mean of the leg speeds.
Mean: A single value summarizing a set of numbers according to a specified averaging rule. The harmonic mean is one member of the broader family of means.
Equal values: A dataset in which every entry has the same numerical value. The harmonic mean equals that shared value when all inputs are equal and nonzero.
Geometric mean: The positive nth root of the product of n values. It is another multiplicative average, with a different formula and range of use.
Parallel resistance: The equivalent resistance of resistors connected across the same two electrical nodes. For identical resistors in parallel, equivalent resistance is their value divided by their count.
Zero: The additive identity, whose reciprocal is undefined. A zero input makes the reciprocal formula undefined.
Two-speed journey: A trip split into two equal-distance legs traveled at different constant speeds. Its overall speed demonstrates why equal distances require a harmonic rather than arithmetic average.
Power mean: A family of means defined by averaging powered values and then taking a root. The harmonic mean is the power mean of order −1.
F-number: A dimensionless measure of a camera lens aperture, equal to focal length divided by entrance-pupil diameter. Combining equal-duration exposures at different f-numbers involves averaging reciprocal squared values.

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