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# Geometric Mean: Definition, Formula, Calculation

The Geometric Mean (GM) is an average value or mean that represents the central tendency of a group of numbers by calculating the product of their values. Measures of central tendencies are used in mathematics and statistics to characterise the summary of complete data set values.

By taking the root of the product of their values, the Geometric Mean (GM) represents the central tendency of a set of numbers. Basically, we add all of the ‘n’ values together and subtract the nth root, where n is the total number of values. For example, the geometric mean of a pair of numbers such as 8 and 1 is √(8×1) = √8 = 2√2.

As a result, the geometric mean is also the nth root of the product of n integers. This is not to be confused with the arithmetic mean. The arithmetic mean is calculated by adding data values and then dividing them by the total number of values. However, in the geometric mean, the given data values are multiplied, and the final product of data values is calculated by taking the root with the radical index. Take the square root if you have two data values, the cube root if you have three data values, and so on. If you have four data values, take the fourth root, and so on.

## Geometric Mean Formula

The nth root of the product of the values is the Geometric Mean (G.M) of a data set with n observations. Consider the case where x1, x2,…, xn are the observations for which the geometric mean is to be calculated. The following is the formula for calculating the geometric mean:

GM = √x1, x2,…, xn

or

GM = (x1, x2,…, xn)1/n

It is also represented as:

G.M. = √∏ᵢ₌₁x

Taking logarithm on both sides,

log GM = log (x1 · x2 · …)1/n

= (1/n) log (x1 · x2 · …)

= (1/n) [log x1 +log x2 + … ]

= (∑ log x) / n

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

This is a different GM equation (that represents the same formula as in the image).

G.M can be calculated for any grouped data using:

GM = Antilog (∑ f log x) / n, where n = f1 + f2 + …

## Geometric Mean Calculator

The Geometric Mean Calculator is an online tool for calculating the geometric mean of numbers.

The centre tendency of a group of numbers is represented by the mean or average. The nth root of the product of n numbers is what it’s called. The formula for calculating the geometric mean is:

Geometric mean = (x1 × x2 × x3…× xn)1/n,where n = total number of terms,

x1,x2,x3, . . . , xn= Different n terms

## Application of Geometric Mean

The G.M’s most fundamental assumption is that data can truly be interpreted as a scaling factor. Before that, we must understand when to employ the G.M. The answer is that it should only be used with positive numbers and is frequently applied to a group of numbers whose values are exponential in nature and are supposed to be multiplied together. This means there will be no zero and negative values, which we will be unable to use. The geometric mean has various advantages and is utilised in a variety of fields. The following are some of the applications:

• It’s a component of stock indexes. G.M. is employed in many of the value line indexes used by financial departments.

• It is used to compute the annual return on the portfolio.

• It is often known as compounded annual growth rates and is used in finance to compute average growth rates.

• It’s also utilised in research on cell division and bacterial development, among other things.

## Conclusion

The mean, median, mode, and range are the most essential metrics of central tendency. The mean of a data set, for example, provides an overview of the data. It’s the average of the data collection’s numbers.

There are various forms of the mean.

1. Mean Arithmetic (AM)

2. Mean Geometric (GM)

3. Mean Harmonic (HM)

The geometric mean is the average value or mean that depicts the central tendency of a group of numbers or data by applying the root of the product of the values.

Get answers to the most common queries related to the JEE Examination Preparation.

## What's the difference between geometric and arithmetic means?

Ans. The arithmetic mean is the ratio of the total number of values to the sum of the provided valu...Read full

## Explain how accurate the arithmetic and geometric means are.

Ans. When the data set is more volatile, the geometric mean is more accurate and effective. When the data se...Read full

## Determine the geometric mean of the numbers 2 and 8.

Ans. Assume a = 2 and b = 8. The number of terms is n ...Read full

## Mention the AM, GM, and HM relationships.

Ans. GM2 = AM × HM is the relationship between AM, GM, and HM. GM = √ [AM × ...Read full

## If AM and HM of the data sets are 4 and 25 respectively, then find the GM.

Ans. Given that, AM = 4 HM = 25. We know that the relation between AM, GM and HM is GM ...Read full

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

Ans. Given, x1 = 10 , x2 = 5, x3 = 15, x4 = 8, x5 ...Read full