Harmonic Mean.
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.
Properties:
- 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 n_{1} and n_{2} values, and H_{1} and H_{2} 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.