Geometric Mean

The Geometric Mean (GM) is a type of average that helps us understand the central tendency of a set of numbers through their multiplicative properties. The Geometric Mean is calculated by multiplying all the numbers in a dataset together and then taking the n-th root of the product, where n is the number of values. This mean is particularly useful when dealing with datasets where values are multiplied together or when the numbers vary widely.

The different types of mean include Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM).

What does Geometric Mean?

The Geometric Mean (GM) represents the average value or mean that indicates the central tendency of a set of numbers by using the product of their values raised to the root. Simply, we multiply all ‘n’ values and extract the nth root, where n stands for a total number of values.

Geometric Mean Formula

The Geometric Mean or G.M of a dataset with ‘n’ observations is the n-th root of the value product. For a dataset with n values, x₁, x₂, x₃, . . . x_n, the Geometric Mean (GM) is calculated using the following formula:

GM = n√x₁ x x₂ x x₃ x ….x_n

For example: let’s find the geometric mean for a given pair: 8 and 1,

The geometric mean is equal to GM = √x1 x x2 = √8 x 1 = √8 = 2 √2

Difference Between Arithmetic Mean And Geometric Mean

Below is a table showing the difference between arithmetic mean and geometric mean.

Relation Between AM, GM, and HM

Before we can analyze how AM, GM and HM are related, we need to know the formulas for all three types of means. Suppose that “a” and “b” are two numbers with number of values equal to 2, then

AM = (a+b)/2

GM = √(ab)

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

Now, the relation between AM, GM and HM.

HM = GM² / AM

HM x AM = GM²

GM = √[ AM × HM]

Therefore, HM x AM = GM²

This shows that the square of the geometric mean is equal to the product of the arithmetic mean and the harmonic mean.

Properties Of Geometric Mean

The Geometric Mean (GM) has several distinct properties that set it apart from other measures of central tendency, such as the Arithmetic Mean (AM). Here’s an overview of these key properties:

• Geometric Mean is Always Less Than or Equal to the Arithmetic Mean
• The ratio between the relevant aspects of the geometric means in two different series is similar to the ratio of their geometric means.
• The product of relevant aspects of two geometric means is equal to the products of their two geometric means.
• If in a given data set each observation is substituted by the geometric mean, then there will be no change in the product of the observations.

Algebraic Properties of the Geometric Mean

• The Geometric Mean is always less than or equal to the Arithmetic Mean, except when all values are identical.
• If a dataset contains a zero value, the Geometric Mean is zero and not defined.
• The Geometric Mean cannot be computed if there are an odd number of negative values.
• The product of all values equals the Geometric Mean raised to the power of n.
• For any set with equal N, the Geometric Mean remains the same regardless of the absolute magnitudes of the numbers.
• The product of pair-wise ratios equals the ratio of their Geometric Means.
• Replacing each element with the Geometric Mean maintains the product of the observations.
• The sum of the logarithms of values above and below the Geometric Mean are equal, relative to the Geometric Mean’s logarithm.

Applications of Geometric Mean

The following are the key advantages of Geometric mean over arithmetic mean and its various applications in different fields.

• In stock indexes, because many of the value line indexes which are used by financial departments make use of G.M.
• To determine the yearly yield on a given investment portfolio.
• To determine the average growth rates called compounded annual growth rate (CAGR) using geometric mean, in finance.
• Biological studies such as cell division and bacterial growth rate, among others also employ geometric mean.

Tips and Tricks on the Geometric Mean

Here are some tips and tricks on G.M:

• For any particular dataset, the geometric mean is always smaller than the arithmetical mean.
• If all values in the data set are replaced with a G.M., then their multiplication would still yield the same results.
• In two series, if we take any available observations of their G.M., it will be found that they have a ratio equal to the ratios of their geometric means.
• Multiplying corresponding items of both GMs produces a product that equals that of their geometric mean.

FAQs on Geometric Mean

What is the Geometric Mean

The Geometric Mean (GM) is the nth root of the product of n values. For a dataset with n values, x1, x2, x3, . . . xn

When should I use the Geometric Mean?

Use the Geometric Mean for data involving multiplication, growth rates, or compounding effects, such as in finance and biological studies.

Can the Geometric Mean be used with negative values or zeros?

No, the Geometric Mean cannot be used with negative values or zeros. If a dataset contains zeros, the GM is zero. With negative values, if there is an odd number, the GM is undefined.