Harmonic Mean Calculator

The harmonic mean is the number of values divided by the sum of their reciprocals. It gives more weight to smaller values and is always less than or equal to the geometric and arithmetic means of the same data.

Use it to average rates and ratios that share a common base, such as speeds over equal distances, price-to-earnings ratios, or rates of work. In those cases the arithmetic mean would overstate the true average.

If you travel equal distances at different speeds, the time spent at each speed differs, so a simple average is wrong. The harmonic mean weights by time correctly. Driving 60 then 40 over equal distances gives an average of 48, not 50.

The harmonic mean adds the reciprocals of the values. A zero has no reciprocal and would make the result undefined, while mixing signs can produce a meaningless figure, so the calculator requires every value to be greater than zero.

For any data set with positive values that are not all equal, the harmonic mean is smaller than the arithmetic mean. The two are equal only when every value is the same. The harmonic mean is the most conservative of the common averages.