Geometric Mean

The Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. In mathematics and statistics, measures of central tendencies describe the summary of whole data set values. The most important measures of central tendencies are mean, median, mode, and range. Among these, the mean of the data set provides the overall idea of the data. The mean defines the average of numbers in the data set. The different types of mean are Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM).

In this lesson, let us discuss the definition, formula, properties, and applications of geometric mean and also the relation between AM, GM, and HM with solved examples in the end.

The Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by taking the root of the product of their values. Basically, we multiply the 'n' values altogether and take out the n^th root of the numbers, where n is the total number of values. For example: for a given set of two numbers such as 8 and 1, the geometric mean is equal to √(8×1) = √8 = 2√2.

Thus, the geometric mean is also defined as the n^th root of the product of n numbers. Note that this is different from the arithmetic mean. In the arithmetic mean, data values are added and then divided by the total number of values. But in geometric mean, the given data values are multiplied, and then you take the root with the radical index for the final product of data values. For example, if you have two data values, take the square root, or if you have three data values, then take the cube root, or else if you have four data values, then take the 4^th root, and so on.

The Geometric Mean (G.M) of a data set containing n observations is the n^th root of the product of the values. Consider, if x₁, x₂, ..., xₙ are the observations, for which we aim to calculate the geometric mean. The formula to calculate the geometric mean is given below:

GM = √x₁ · x₂ · ... · xₙ

or

GM = (x₁ · x₂ · ... · xₙ)^1/n

It is also represented as:

G.M. = √∏ᵢ₌₁ⁿ xᵢ

Taking logarithm on both sides,

log GM = log (x₁ · x₂ · ... · xₙ)^1/n

= (1/n) log (x₁ · x₂ · ... · xₙ)

= (1/n) [log x₁ +log x₂ + ... + log xₙ]

= (∑ log xᵢ) / n

Therefore, geometric mean, GM = Antilog (∑ log xᵢ) / n

This is an alternative formula of GM (that represents the same formula as in the image).

Note: For any grouped data, G.M can be calculated using:

GM = Antilog (∑ f log xᵢ) / n, where n = f₁ + f₂ + ... + fₙ.

Here is a table representing the difference between arithmetic and geometric mean.

Before we learn the relation between the AM, GM and HM, we need to know the formulas of all these 3 types of mean. Assume that “a” and “b” are the two number and the number of values = 2, then

AM = (a+b)/2

⇒ 1/AM = 2/(a+b) ……. (I)

GM = √(ab)

⇒GM² = ab ……. (II)

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

⇒HM = 2/[(a+b)/ab

⇒ HM = 2ab/(a+b) ….. (III)

Now, substitute (I) and (II) in (III), we get

HM = GM² /AM

⇒GM² = AM × HM

Or else,

GM = √[ AM × HM]

Hence, the relation between AM, GM, and HM is GM² = AM × HM. Therefore the square of the geometric mean is equal to the product of the arithmetic mean and the harmonic mean.

Let us also see why the G.M for the given data set is always less than the arithmetic mean for the data set. Let A and G be A.M. and G.M.

So,

A = (a+b)/2 and G=√ab

Now let’s subtract the two equations

A−G = (a+b)/2 − √ab = (a+b−2√ab)/2 = (√a−√b)²/2 ≥ 0

A−G ≥ 0

This indicates that A ≥ G

Application of Geometric Mean

Geometric mean has many advantages over arithmetic mean and it is used in many fields. Some of the applications are as follows:

• It is used in stock indexes because many of the value line indexes which are used by financial departments make use of G.M.
• To calculate the annual return on the investment portfolio.
• The geometric mean is used in finance to find the average growth rates which are also known as the compounded annual growth rate (CAGR).
• Geometric Mean is also used in biological studies like cell division and bacterial growth rate etc.

Tips & Tricks on Geometric Mean:

Some of the tips and tricks on G.M are as follows:

1. The G.M for the given data set is always less than the arithmetic mean for the data set.
2. If each value in the data set is substituted by the G.M, then the product of the values remains unchanged.
3. The ratio of the corresponding observations of the G.M in two series is equal to the ratio of their geometric means.
4. The products of the corresponding items of the G.M in the two series are equal to the product of their geometric mean.

☛Related Topics:

Given below is the list of topics that are closely connected to the Geometric Mean. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

• Geometric Mean Calculator
• Measures of Central Tendency
• Harmonic Mean Formula

What is the Definition of Geometric Mean?

The geometric mean of n number of data values is the n^th root of the product of all the data values. This is a kind of average used like other means (like arithmetic mean).

What Is the Difference Between the Arithmetic Mean and Geometric Mean?

The arithmetic mean is defined as the ratio of the sum of given values to the total number of values. Whereas in geometric mean, we multiply the “n” number of values and then take the n^th root of the product.

What is Geometric Mean in Statistics?

Geometric Mean is the value or mean of a set of data points which is calculated by raising the product of the points to the reciprocal of the number of the data points.

What Is the Geometric Mean of 2 and 8?

The geometric mean of 2 and 8 can be calculated as

Let a = 2 and b = 8
Here, the number of terms, n = 2
If n =2, then the formula for geometric mean = √(ab)
Therefore, GM = √(2×8)
GM =√16 = 4
Therefore, the geometric mean of 2 and 8 is 4.

What Is Geometric Mean Formula?

Geometric mean (GM) is the nth root of the product of the elements in a sequence. GM formula for given set {x₁, x₂, x₃, ..., x_n} is given by (x₁ × x₂ × x₃ × ... × x_n)^1/n

Why Is Geometric Mean Better Than Arithmetic?

The geometric mean differs from the arithmetic average, or arithmetic mean, in how it is calculated because it takes into account the compounding that occurs from period to period. Because of this, investors usually consider the geometric mean a more accurate measure of returns than the arithmetic mean.