Statistics

Geometric Mean.

The geometric mean of a set of n values of a variable is the n th root of their product. If a variable x assumes n values , then its geometric mean, denoted by is

For a frequency distribution, , where

Properties:

If the given values of a variable are all equal, then the geometric mean will be equal to their common value.
The logarithm of the geometric mean of a set of values of a variable is the aritmetic mean of their logarithms.
If y is a function of a variable x in the form y=ax, then the geometric mean of y is related to that of x in the similar form.
The geometric mean of the ratio of two variables is the ratio of their geometric means.
If there are two sets of values of a variable x, consisting of n₁ and n₂ values, and G₁ and G₂ are their respective geometric means, then the geometric mean, G, of the combined set is given by
If a variable x chages over time t exponentially, then the value of the variable at the mid-point of an interval (t₁,t₂) i.e. at is the geometric mean of its values at t₁ and t₂.
No es útil si algún valor es nulo.
No es posible su cálculo cuando hay un número par de datos y el radicando es negativo.

Advantages of the geometric mean:

The geometric mean is ridigly defined.
The geometric mean is directly based on all the observations.
Generally, the presence of a few extremely small or large values has no considerable effect on geometric mean.

Disadvantages of the geometric mean:

It is difficult to compute.
If a single value of a variable is zero, then the geometric mean becomes zero, irrespective of the magnitudes of the other values.
It may be imaginary if some values are negative (generally, use of geometric mean is restricted to positive values).

Uses of geometric mean:

It is sometimes preferred for averaging ratios of two variables: rates of population growth, rates of interest, rates of depreciation,...
It is used for finding the value of a variable at the mid-point of a time period when the variable is an exponential fuction of time.
If there are a few extreme values in a set, the geometric mean may be considered along with the median and the mode.

Ussually