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
- 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.