The harmonic mean of a set of non-zero values of a variable is the reciprocal of the arithmetic mean of the reciprocals of the values.
The harmonic mean of n non-zero values of a variable x is:
For a frequency distribution:
The harmonic mean is not commonly used, but it is the appropiate average when the variable is of the form, "x per unit y", and equal amounts of x are considered. If, however, equal amounts of y are considered, arithmetic mean is the appropiate average.
- If the given values of a variable are all equal (), then the harmonic mean will be equal to their common value.
- If a variable y is related to another variable x in the form y=ax, then the harmonic mean of y is related to that of x in the similar form.
- If there are two sets of values of a variable x, consisting of n1 and n2 values, and H1 and H2 are their respective harmonic means, then the armonic mean, H, of the combined set is given by
Advantages of the harmonic mean:
- Is rigidly defined.
- Is directly based on all the values.
Disadvantages of the harmonic mean:
- It is undefined even if a single value is zero.
- It is abstract in nature.
- It involves a lot of computational labour.
- It is not amenable to algebraic treatment.