When do we use reciprocal values in Harmonic Mean?

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David Eugene Booth

See the Wikipedia article on harmonic means and it's references. Best, David

Salvatore S. Mangiafico

Imagine you have a car that goes 3 km in total. For the first kilometer it goes 2 km/hr., for the second kilometer it goes 4 km/hr., and for the third kilometer it goes 5 km/hr.

So the first kilometer took 0.50 hours, the second kilometer took 0.25 hours, and the third kilometer took 0.20 hours.

The overall average speed is 3 km in 0.95 hours, or 3.16 km/hr.

This is the same as the harmonic mean of (2, 4, 5).

Ette Etuk

The harmonic mean (HM) is defined as the reciprocal of the mean of the reciprocals of the data. That is also its explanation in simple language without much technical stuff. It is therefore used when there is an inverse relationship. Salvatore's example is very appropriate. Also, in weighted least square modelling the weights of the variables are reciprocals of their variances. The mean weight is their HM. The use of reciprocals in the HM makes it relatively not sensitive to outliers as the arithmetic or geometric mean.