A careful reading of Harmonic Mean
The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. It gives relatively more influence to smaller values, which is appropriate when rates apply to equal distances, quantities, or workloads.
Math calculator
Harmonic Mean Calculator
Combine rates or ratios when equal quantities are traveled, processed, or purchased. The displayed harmonic mean includes enough working to inspect signs and scale.
Harmonic Mean inputs
Numerical setup
Equal distance versus equal time
For two positive rates a and b, the harmonic mean simplifies to 2ab/(a + b). This form makes the influence of the slower rate visible and provides a convenient hand calculation when only two equal-distance legs are involved.
Unequal distances can still be combined, but they require distance weights: total distance divided by total time. Feeding the rates directly into an unweighted harmonic mean would silently assume that every rate covers the same distance.
The equal-distance condition matters. If the time spent at each speed is equal, use the arithmetic mean instead. Zero rates make a reciprocal undefined, while mixed positive and negative entries can cancel reciprocals. If the Harmonic Mean assumptions do not fit, consider multiplicative mean.
Take the reciprocal of each value, add those reciprocals, and divide the number of values by the sum. Keep units consistent before starting. A hand-worked extension of Harmonic Mean is weighted observations.
Testing the Harmonic Mean result
Start the Harmonic Mean cross-check with Nonzero values. Apply the original the fixed condition and recompute Harmonic mean. Treat a different the fixed condition as new Harmonic Mean data. That prevents its Harmonic mean from being attributed to the earlier Harmonic Mean setup.
Reverse the Harmonic Mean reasoning once: begin with the shown Harmonic mean and ask whether Nonzero values could produce it under the fixed condition. When that Harmonic Mean relationship fails, the contradiction narrows the error to an entry, order choice, or convention.
Cross-checking Harmonic Mean
Work backward from the displayed Harmonic mean once. Ask whether Nonzero values can produce that Harmonic mean under the fixed condition. If the reverse Harmonic Mean relationship fails, recheck the entry order and any convention attached to the fixed condition.
A one-input trial is useful when Harmonic Mean behaves unexpectedly. Change the fixed condition alone, retain Nonzero values and the fixed condition, and observe Harmonic mean. Multiple simultaneous edits would make the cause of the changed Harmonic mean ambiguous within the Harmonic Mean setup.
Useful Harmonic Mean clarifications
Why is the harmonic mean pulled toward small values?
Reciprocals make smaller positive inputs contribute more heavily to the denominator.
Can it average speeds?
Yes when the distances covered at each speed are equal.
What if travel times are equal?
Then ordinary time-weighting leads to the arithmetic mean of the speeds.