Geometric Mean: Properties, Application, Example

In this lesson, we will examine the relationship between geometric mean, arithmetic mean, and heuristic mean, as well as the definition, formula, properties, and applications of the geometric mean.

The Geometric Mean (GM) is a mathematical term that denotes the central tendency of a group of integers by determining the product of their values. Essentially, we multiply all of the numbers and then take the nth root of the multiplied numbers, where n is the total number of data values. For example, the geometric mean of a pair of numbers such as 3 and 1 is √(3 ×1) = √3 = 1.732.

The geometric mean is the nth root of the product of n numbers, in other words. The geometric mean differs from the arithmetic mean, as shown below. Because we add the data values and then divide them by the entire number of values in arithmetic mean. However, in the geometric mean, we multiply the given data values and then take the root of the total number of data values using the radical index. Take the square root if you have two data, the cube root if you have three data, and so on. If you have four data values, take the fourth root, and so on.

Relation Between AM, GM and HM

It is necessary to first understand the formulas for each of these three forms of meaning in order to comprehend the nature of the connection that exists between the geometric mean, the arithmetic mean, and the geometric mean. Let’s say “x” and “y” are two numbers, and let’s further assume that the number of values is two.

AM = (a+b)/2

⇒ 1/AM = 2/(a+b) ……. (1)

GM = √(ab)

⇒GM² = ab ……. (2)

HM= 2/[(1/a) + (1/b)]

⇒HM = 2/[(a+b)/ab

⇒ HM = 2ab/(a+b) ….. (3)

By making these two changes in (3), we arrive at the following:

HM = GM² /AM

⇒GM² = AM × HM

Or else,

GM = √[ AM × HM]

Hence, the relation between AM, GM and HM is GM² = AM × HM

Geometric Mean Properties

The following is a list of important qualities that the G.M. possesses:

• The geometric mean for the data set in question is invariably lower than the arithmetic mean for the data set in question.

• The product of the objects does not change if the G.M. is used to replace each object in the data set; alternatively, the product of the objects does not change.

• The ratio of the geometric means of two series is equal to the ratio of the corresponding observations of the geometric mean in two different series.

• The product of the geometric mean of two series is equal to the product of the products of the items that correspond to the geometric mean in two different series.

Application of Geometric Mean

The most fundamental assumption made by the G.M. is that data can actually be interpreted in the form of a scaling factor. Before that, we need to have a good understanding of when to use the G.M. The correct response is that it must be used exclusively with positive numbers, and it is typically applied to a set of numbers whose values are exponential in nature and are meant to be multiplied together. Because of this, there won’t be any values that are zero or negative, meaning we won’t be able to use them. The geometric mean is useful in a wide variety of contexts and offers a number of benefits to those who employ it. The following are examples of possible applications:

• It is utilised in the calculation of stock indexes. G.M. is utilised in a significant number of the value line indexes that are utilised by financial departments.

• The annual return on the portfolio can be computed with the help of this factor.

• It is utilised in the field of finance for the purpose of determining average growth rates, which are also known as the compounded annual growth rate.

• It is also put to use in scientific research on topics such as the growth of bacteria and the division of cells.

CONCLUSION

• For any given set of positive numbers, the geometric mean will be lower than the arithmetic mean. However, when all of the values in a series are the same, the geometric mean and the arithmetic mean will be equal.

• The geometric mean is not appropriate to use if it is infinity, which it is if any value in a series is 0.

• The calculation cannot be performed if there are an odd number of negative values. This is due to the fact that the product of the values will become negative, and we won’t be able to determine what the cause of a negative product is because it won’t have a root.

• The geometric mean multiplied by the number of values in the product is equal to the nth power of the geometric mean.

• The geometric mean of any set of numbers with the same N and product is the same. This holds true for any set of numbers.

• The product of the geometric mean’s side ratios will always result in both sides having the same value.

• Even if each number in a series is replaced by its geometric mean, the products of the series continue to have the same values.

• The total amount of the deviations of the logarithms of the original values above and below the G.M.’s logarithm is equivalent.