# Harmonic Mean: Characteristics, Applications and Limitations

A simple way to define a harmonic mean is to call it the reciprocal of the arithmetic mean of the reciprocals of the observations. The most important criteria for it is that none of the observations should be zero.

A harmonic mean is used in averaging of ratios. The most common examples of ratios are that of speed and time, cost and unit of material, work and time etc. The harmonic mean (H.M.) of n observations is

H.M. = 1÷ (1⁄n ∑ i= 1n (1⁄xi) )

In the case of frequency distribution, a harmonic mean is given by

H.M. = 1÷ [1⁄N (∑ i= 1n (f⁄ xi)], where N = ∑ i= 1n fi

### Properties of Harmonic Mean

• If all the observation taken by a variable are constants, say k, then the harmonic mean of the observations is also k
• The harmonic mean has the least value when compared to the geometric mean and the arithmetic mean

• A harmonic mean is rigidly defined
• It is based upon all the observations
• The fluctuations of the observations do not affect the harmonic mean
• More weight is given to smaller items