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Statistics

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 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.