Geometric Mean (Definition, Formula) | Calculation with Examples

What is Geometric Mean?

The geometric mean can be defined as a type of mean that uses the product of values that are often assigned to a set of numbers for the purpose of indicating the typical values or central tendency of numbers and this method can be used when there is an exponential change in values.

Geometric Mean Formula

For n numbers present, in order to calculate the geometric mean formula, all the numbers are firstly multiplied together and then the n^th root of the same is taken. The formula for Geometric mean is given below-

Geometric Mean Formula = ^N√(X₁*X₂*X³………….X_N)

Here, X refers to the value given and the N refers to the total number of data present.

Geometric Mean Calculation Example

Calculate the geometric mean example of the following different numbers:

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3,7, 8, 11 and 17

Answer

The geometric mean of 3,7, 8, 11 and 17 can be ascertained as follows-

X = ^N√(X₁*X₂*X³………….X_N)

So, the geometric mean of the data set given is 7.93

Advantages

There are several different advantages of the Geometric Mean are as follows:

Rigidly Defined – It is not very much flexible in nature or in other words, it is rigidly defined. This means in the geometric mean method the values are always going to remain fixed.
Based on Observations – This method is based on the items and observations of various series.
Minimum Level of Impact – Sampling fluctuations have lesser or no impact on geometric mean.
Eases the Measuring Mechanism – Geometric mean is of great use for measuring the changes and it also helps in determining the most appropriate average with respect to percentage and ratio.
Useful for Mathematical Calculation – Geometric mean can also be used for further calculations with respect to algebraic and other mathematical calculations.
More Preference to Small Values – In the geometric mean method, the higher level of weights is vested upon small values while large values are given less significance.
Multiple Purposes – The geometric mean method can be used for averaging ratios, percentages, and also for evaluating the gradual rise and fall in rates.

Disadvantages

The different limitations and drawbacks of the Geometric Mean include the following:

Complex in Nature – This method is a very complicated method. The users of the same must have a thorough mathematical knowledge in ratios, roots, logarithms, etc. This is also one of the very important reasons behind the less popularity of this method. The method is highly difficult for users with ordinary knowledge to understand and its computation is also highly complicated.
Difficulty in Calculating the Method – The method is highly complicated in nature as it requires the users to find out the roots of various products of certain values. Therefore, it is very difficult for users to understand how the same can be calculated.
Not Applicable – The aforesaid method is not applicable for the cases that have zero or negative value of any series. The method can also not be calculated when the negative value of any series is odd.
Lacks Compatibility with Open-end Distribution – Geometric mean can’t be obtained in the case of an open-end distribution. The aforesaid method may also give certain values that are absent from the series.

Important Points

Geometrical mean, harmonic mean and arithmetic mean are the three Pythagorean means. Unlike the arithmetic mean method, geometric mean measures evenness. It helps in normalizing the ranges to disallow the impact of the dominance of the same on the weighting itself. Values that are very large have no influence to make in a skewed pattern of distribution.
Unlike other medians, the geometric mean method handles the ratios in a very consistent manner.
The order in which a user does his or her calculation matters and this helps in generating two results that are different from each other. Both the results have two different interpretations.
With the geometric mean method, a user calculates the average rate of compound interest, inflations and investment returns.
In real life, this method can be used in computer science, aspect ratios, geometry, medicine, proportional growth, water quality standards, and the Human Development Index.
This is used specifically for calculating portfolio returns. The aforesaid method is mostly used in accounting and finance.
It helps in normalizing the ranges to disallow the impact of the dominance of the same on the weighting itself. Values that are very large have no influence to make in a skewed pattern of distribution.
This method is more accurate and effective in a data set that is more volatile in nature. However, it is a complicated method in comparison to the arithmetic mean.
When there are two or more numbers in the series then Geometric mean = (x*y*……) ^1/n
This is considered either growth or compounding returns. Also, it considers the compounding effect. A non-mathematical user may find it difficult to use and understand the geometric mean.
It becomes imaginary when any of the observation earns a negative value.

Conclusion

Geometric mean is used with time-series data such as calculating investment returns since the geometric mean accounts only for the compounding of returns. This is also why the geometric returns are always lesser than or equal to the arithmetic mean return. It is also considered as a power mean and it is mostly used for comparing different items. This has been an exponential relationship with the arithmetic mean of logarithms. It is more or less related to data’s logarithmic transformation.

It helps in normalizing the ranges to disallow the impact of the dominance of the same on the weighting itself. Values that are very large have no influence to make in a skewed pattern of distribution. The aforesaid method is more appropriate in the calculation of the mean and it provides more accurate and effective results in the presence of such variables that are highly dependent and widely skewed.