 It also follows that formula_16, meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.For the special case of three numbers, formula_9, formula_10 and formula_19, the harmonic mean can be written

Three positive numbers "H", "G", and "A" are respectively the harmonic, geometric, and arithmetic means of three positive numbers if and only if the following inequality holds

If a set of weights formula_22, ..., formula_23 is associated to the dataset formula_9, ..., formula_25, the weighted harmonic mean is defined by

The unweighted harmonic mean can be regarded as the special case where all of the weights are equal.In many situations involving rates and ratios, the harmonic mean provides the truest average.