# Geometric Mean Formula

## What is the Geometric Mean?

The geometric mean is the average growth of investment calculated by taking the product of n variables and then finding the nth root. In other words, geometric mean is the average return of an investment over time. It is a metric unit used to calculate the performance of a single investment or an investment portfolio. For example, for a set of 2 numbers such as 24 and 1. The geometric mean for the given set of two numbers is equal to $\sqrt{(24 + 1)} = \sqrt{25} = 5$. The geometric mean is also written as G.M. Along with the geometric mean, there are two more important metric measurements, such as Arithmetic suggest and Harmonic mean, which is used to calculate the average value of a given data. In this article, we will discuss the geometric mean, geometric mean definitions, and formula, the geometric mean formula for grouped data, properties of geometric mean, etc. is.

### Geometric Mean Definition and Formula

The geometric mean definition and formula given below will clear your concepts of geometric mean and help you to calculate the geometric mean for a given data.

The Geometric mean (G.M.) of a series, including n observation, is the nth root of the product of values.

For example, if y₁, y₂, y₃, ...yn are the observation, then the geometric formula is defined as

$G.M = \sqrt[n]{y_{1}, y_{2}, y_{3},... y_{n}}$

or

G.M = (y₁,y₂,y₃,...yn)1/n

### What is the Formula of Geometric Mean?

The formula to find the geometric mean for the observation such as x₁,x₂, x₃...xn is given as :

$\text{Geometric Mean Formula } = \sqrt[n]{x_{1}, x_{2}, x_{3}.....x_{n}}$

Or

$(x_{1},x_{2},...x_{n})^{\frac{1}{n}}$

The geometric mean formula can also be represented in the following way:

Log GM = I/n log (x₁,x₂,...xn)

= 1/n (log x₁ + logx₂ +.......+ log xn)

= Σ log xi / n

Hence, Geometric Mean, GM is equaled to

Antilog Σ log xi / n

Where n = f1+ f₂ +….+ fn

The formula to find geometric mean can also be written as

$G.M. = \sqrt[n]{\prod_{i=1}^{n} x_{i}}$

The geometric mean formula for grouped data can be written as:

$GM = \frac{Antilog \sum \text{f log }x_{i}}{n}$

### Geometric Mean Formula for Grouped Data

If we have a set of n positive values with some repeated values such as x₁,x₂,x3…..xn, and the values are repeating s₁,s₂,s3…….sk times, then the geometric mean formula for grouped data is defined as.

$G.M. of X = \overline{X} = \sqrt[n]{x_{1}^{s_{1}}, x_{2}^{s_{2}}, x_{3}^{s_{3}} . . . . x_{k}s_{k}}$

### Geometric Mean Formula for Ungrouped Data

The geometric mean formula for ungrouped data for value X consisting n observation such as x₁,x₂, x₃...xn is represented by G, M of X and is derived as

$G.M. of X = \overline{X} = \sqrt[n]{x_{1}, x_{2}, x_{3} . . . . x_{n}}$

### Properties of Geometric Mean

The following are the properties of Geometric mean:

1. The geometric mean for a given data is always less than the arithmetic means for a given data set.

2. The ratio of the associated observation of the geometric mean in two series is equivalent to the ratio of their geometric means.

3. The product of the associated observation of the geometric mean in two series is equivalent to the product of their geometric means.

4. If the geometric mean replaces each observation in the given data set, then the product of observations does nor change.

### Solved Examples

1. Find the geometric mean for the following data.

 X Observations 10 3 11 12 12 18 13 12 14 3

Solution:

 x f f log(x) f/x 10 3 6.9078 0.3 11 12 28.7747 1.0909 12 18 44.7283 1.5 13 12 30.7794 0.9231 14 3 7.9172 0.2143 n = 48 Σ f log x = 119.1074 Σ f/x = 4.0283

Geometric mean of X = Antilog (Σ f log x/Σ f)

= Antilog (119.1074)/48

= Antilog (2.4814)

= 11.958

2. Calculate the geometric mean of 10,5,15,8,12.

Solution:

Given, x₁ = 10 ,x₂= 5,x₃ = 15,x₄ = 8,x₅= 12

N= 5

Using the geometric mean formula,

$\sqrt[n]{x_{1}, x_{2}, x_{3}. . . . x_{n}}$

$G.M. of X = \overline{X} = \sqrt[5]{10 \times 5 \times 15 \times 8 \times 12}$

$\sqrt[5]{72000}$

= $(72000)^{\frac{1}{5}}$

= 9.36.

3. Find the geometric mean for the following data.

 Weight of book (x) Log x 45 1.653 60 1.778 48 1.681 100 2.000 65 1.813 Total 8.925

Solution: Here n = 5

Geometric Mean = Σlog xi /n

= Antilog 8.925/5

= Antilog  1.785

### Quiz Time

1. The geometric mean is calculated by

1. (a + b)²

2. (a - b)/2

3. ± √ab

4. ± ab

2. The three geometric means between 2 and 32 are

1. 6,10,14

2. 10,12,14

3. 6,8,10

4. 4,8,16

3. If a, G and b are in geometric progression the ‘G’ is considered as

1. Arithmetic mean

1. Geometric mean

2. Standard deviation

3. None of the above

1. What is the Difference Between Arithmetic Mean and Geometric Mean.

Ans. The difference between the arithmetic mean and Geometric mean is given below in the tabulated form.

 Geometric Mean Arithmetic Mean Meaning It is known as the multiplicative name. It is known as additive mean. Values The geometric mean is always less than the arithmetic mean because of the compounding effect. The arithmetic mean is always greater than the arithmetic mean because it is computed as a simple average. Data set The geometric mean formula applied only on the positive set of numbers. The arithmetic mean formula can be applied on both the positive set of numbers and the negative sets of numbers. Uses The geometric mean is widely used by biologists, economists, and financial analysts. It is most accurate for the dataset that manifests correlation. The arithmetic mean is used to represent average temperature as well as determine the average speed of a car. Effect of outliers The effect of the outliers on Geometric is moderate. For example, in the given data set 11,13,17 and 1000. In this case, 1000 is the outlier and the average is 39.5. The effect of the outliers on arithmetic is severe. For example, in the given data set 11,13,17 and 1000. In this case, 1000 is the outlier and the average is 260.25.

Ans. Some Advantages of the Geometric Mean are:

1. The geometric mean is rigidly defined as their values are always fixed.

2. It is based on all the elements of the series.

3. The geometric mean is not much affected by sampling fluctuations.

4. Geometric means are accurate for algebraic calculations and other mathematical operations.

5. It gives more weightage to small items.