Geometric Mean

In mathematics, an information set may be summed up with the assistance of central tendencies. The foremost important central tendencies are median, mean, mode, and range. The information set’s mean provides an overall picture of the info. The mean is the average of the info set’s numbers. The arithmetic mean (AM), the harmonic mean (HM), and also the geometric mean (GM) are the three types of means.

The mean, its formula, differences, and applications with other varieties of means are all covered during this article. 

What is Geometric mean?

A geometric mean is a mean value that sums up the central tendency of a collection of numbers.

In simple words, the mean value could be a form of average within which the numbers are multiplied together, so the root (for two digits), cube root (for three numbers), and so on are computed. It is also known as the multiplicative mean. for instance, if we wanted to calculate the geometric mean of 9 and 2, the result would be √(9×2) =√18

As a result, the geometric mean value is additionally called the nth root of n numbers’ product. 

Geometric Mean Formula

To calculate Geometric mean, i.e., if we consider x1,x2  … xn for observation, then the geometric means of these values are 

GM = √x1x2  … xn     

 OR 

GM = x1x2  … xn1/n

It can also be written as  –

log GM = 1nlog (x1, x2,… xn )

=1n(logx1 +  logx2 + … + logxn )=logxn

Therefore, 

GM = Antilog logxn

Where n =(f1 +  f2 + … + fn )

It is also represented as –

GM = ni=1nxi 

For any grouped Data, GM can be written as  –

GM = Antilog logxin

Difference between Geometric Mean and Arithmetic mean

Relation between Geometric Mean, Arithmetic mean and Harmonic Mean

Assume that “x” and “y” are the two number and the number of values = 2, then 

AM = (a+b)/2

⇒ 1/AM = 2/(a+b) ……. (1)

GM = √(ab)

⇒GM2 = ab ……. (2)

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

⇒HM = 2/[(a+b)/ab

⇒ HM = 2ab/(a+b) ….. (3)

Now, substitute (1) and (2) in (3), 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

Applications of the mean value

The Geometric mean’s most fundamental assumption is that data can truly be interpreted as a scaling factor. Before trying this, we must first understand when to use the GM. The solution is that it should only be used with positive numbers. It is usually applied to many numbers whose values are exponential and data set speculated to be multiplied together. This suggests there’ll be no zero and negative values, which we will be unable to use. The mean has various advantages and is employed during a kind of field. The subsequent are a number of the applications:

• It is utilized in the securities market and various other financial indexes. Therefore, it’s beneficial in financial institutes
• Also, it’s generally used to calculate returns for an investor’s portfolio
• Likewise, it’s a vital concept in biotechnology because it is employed extensively to study bacteria regeneration and more
• GM is employed to calculate various United Nations’ indexes, like the Human Development Index, Water Quality Index, and plenty of more

Tips to solve geometric mean Questions

• The GM for a data set is always less than or equal to the data set’s arithmetic mean and is always greater than or equal to the harmonic mean.
• The product of the values remains unchanged if the GM is swapped for every value within the data set
• The geometric means of the two series are proportional to the ratio of their common ratios.

Geometric Mean Examples 

Example 1

Find the Geometric mean of 2, 3, 4, 5, 6

Solution:

The geometric mean is given as 

= (x1 x x2 x x3 x xn)1/n      

= (2 x 3 x 4 x 5 x 6)1/5 

= (720)⅕

=  5.36

Example 2

Over ten years, the average monthly pay in a city increased from 2000 rupees to 4000 rupees. What is the average annual increase using the geometric mean?

Solution:

The annual increase can be calculated in 2 steps.

Step 1 – Finding Geometric mean

(2000 x 4000)½  = 1,78,885.43

Step 2  = To calculate the average over ten years, dividing by 10

= 1,78,885.43/10 = 17,888.543

Example 3 

Find the geometric mean of the following grouped data for the frequency distribution of weights.

Solution:

From the given data, n = 170

We know that the G.M for the grouped data is

GM=Antilog⁡∑of log⁡x

GM = Antilog logxin

GM = Antilog (402.12 / 170)

GM = Antilog (2.12625)

GM =110.12

Therefore, the GM =110.12

Conclusion

The geometric mean is an essential concept in mathematics and statistics and forms complex arithmetic. Besides, use cases for geometrical means are vast in various fields of study. Students should understand the concept well enough to utilise it across multiple arithmetic questions.