3.3:
Geometric Mean
The geometric mean is used for the analysis of data related to economics or biology, where the values change exponentially. If n number of data values are given, their geometric mean is expressed as the nth root of the product.
For example, consider the following set of numbers. Since these numbers are changing exponentially, their arithmetic mean would be skewed towards larger values. So, calculating the geometric mean can help find the mean of such exponentially changing values.
Begin by multiplying all the given numbers. Since there are four numbers in the data set, take the 4th root of the product. The resulting value is the geometric mean of the data.
Alternatively, convert the data values into corresponding logarithmic numbers. Then, add up all the log numbers and divide them by the total number of values in the data set. Finally, take antilog to arrive at the geometric mean.
It is important to note that the geometric mean cannot be used if given data contains zero or negative value.
3.3:
Geometric Mean
The mean is a measure of the central tendency of a data set. In some data sets, the data is inherently multiplicative, and the arithmetic mean is not useful. For example, the human population multiplies with time, and so does the credit amount of financial investment, as the interest compounds over successive time intervals.
In cases of multiplicative data, the geometric mean is used for statistical analysis. First, the product of all the elements is taken. Then, if there are n elements in the dataset, the nth root of the products is defined as the geometric mean of the data set. It can also be expressed via the use of the natural logarithmic function.
For example, suppose money compounds at annual interest rates of 10%, 5%, and 2%. In that case, the average growth factor can be calculated by computing the geometric mean of 1.10, 1.05, and 1.02. Its value comes out to be 1.056, which means that the average growth rate is 5.6% per annum.
It can be shown that the geometric mean of a sample data set is always quantitatively less than or at most equal to the arithmetic mean of the sample.