## Geometric Mean Calculator

#### Result Summary:

 Type Result Given set of numbers 0 Total numbers provided 0 Geometric Mean Value 0

### What is Geometric mean?

In mathematics and statistics, the geometric mean is a measure of central tendency, just like the arithmetic mean. In arithmetic mean, the sum of the values of the data set was considered to find the central value. In geometric mean, we multiply the n values of the data set and then take the nth root of the product obtained. The geometric mean is essentially the nth root of the product of n numbers. If the data set consists of only 2 values, we take the square root of the product. If there are 3 values, we take the cube root, and so on.

The geometric mean is calculated by first multiplying all the values of the data set, and then taking the nth root. For example, if the values are 4 and 5, then we first find the product = 4 × 5 = 20. Then the geometric mean = √20 = 4.472. If the values are 4, 5 and 6, then the geometric mean = ∛(4×5×6) = ∛120 = 4.932.

The geometric mean is quite useful if we were to compare some values which are widely different from each other, which was a limitation of the arithmetic mean. It also works for equally spaced data values.

### Properties of Geometric mean

Given below are some important properties of the geometric mean.

Data set – The values for which the geometric mean is to be calculated.

Product – The number obtained after multiplying all the values in the data set.

Geometric mean – This is the value obtained by taking the nth root of the product of n numbers.

### Geometric mean Formula

The formula for geometric mean is: $$geometric \,mean \,= (product \,of \,the \,n \,values)^{1\over n}$$

This can also be written as $$GM = {n \sqrt{x_1x_2x_3...x_n}} = (\prod_{i=1}^n)^{1\over n}$$

In this formula, each x_i indicates one value of the data set, and n denotes the total number of values.

There is also a formula using logarithms. $$log GM = {1 \over n}log(x_1x_2x_3...x_n)$$

$$log⁡ \,GM \,= {1 \over n}(log⁡ x_1 + log x_2 + log x_3 + ⋯ + log⁡ x_n)$$

$$log \,GM \,= {1 \over n} \sum_{i=1}^n log x_i$$

In other words, $$GM \,= antilog \,({1\over n} \sum_{i=1}^n log x+i)$$

If the data set consists of n discrete values x_1,x_2,x_3,…,x_n with corresponding frequencies f1,f2,f3,…,fn, then the formula for geometric mean is as follows.

$$GM \,= \,antilog \,({1\over N} \sum_{i=1}^n f_i log x+i)$$

where $$N \,= \,f_1 + f_2 + f_3 + ⋯ + f_n$$

The geometric mean is always a positive real number.

### Characteristics of Geometric mean

• For any data set, the geometric mean is always lesser than the arithmetic mean.
• If every value of the data set is replaced by the geometric mean, the product remains unaffected.
• For two different series, the geometric mean of the ratio of any corresponding values is equal to the ratio of their geometric means.
• For two different series, the geometric mean of the product of any corresponding values is equal to the product of their geometric means.
• The geometric mean is not adversely affected if the values of the data set are widely different. It is not affected by any extreme values, which makes it advantageous over the arithmetic mean.
• The geometric mean is quite useful if the values of the data set are subject to large fluctuations.
• The geometric mean may not always coincide with any of the values of the data set.
• It cannot be calculated if any of the values is negative or zero.
• For data sets involving a large number of values, the calculation of the geometric mean can be quite cumbersome.

• ### Areas of application

Given below are some applications of the geometric mean.

• It is commonly used in business and finance when working with percentages to calculate average growth rates. This is known as the compounded annual growth rate.
• It is also used in financial and stock market indexes.
• It is used in biology to study cell division and microbial growth.
• It is extremely useful if the numbers being multiplied are exponentially large.
• The geometric mean is used to calculate nth roots in advanced mathematical calculations.
• It can be used in statistical studies to model the growth of human population.
• Given below are some real-life uses of the geometric mean formula.

Question: The annual percentage growth rate of profits in a corporate from the years 2010 to 2015 is given as: 50, 54, 73, 81, 93. Calculate the geometric mean of the annual percentage growth in profits.

Answer: To solve this problem, we first need to tabulate the growth rate of profits along with their logarithmic values as shown below.

 xi 50 54 73 81 93 Total log xi 1.699 1.7324 1.8633 1.9085 1.9685 9.1717

We can now apply the following formula directly.

$$GM \,= antilog \,({1\over n} \sum_{i=1}^n log x+i)$$

$$GM \,= antilog \,({9.1717 \over 5}) \\ GM = anitlog (1.83434) = 68.29$$

Hence, the geometric mean of the annual percentage growth in profits is 68.29.

Question: Find the geometric mean of the following values: 5, 15, 25, 30, 50.

Answer: In this problem, the x_i are 5, 15, 25, 30, 50.

The geometric mean can be computed using the formula below.

$$GM = {n \sqrt{x_1x_2x_3...x_n}} \\ GM = {5 \sqrt{5* 15 * 25 * 30 * 50}} = GM = {5 \sqrt{2812500}} = 19.49$$

Hence, the geometric mean of this data set is 19.49.