Geometric Mean

Dhristi JEE 2022-24

Introduction

Geometric Mean is an average that explains the typical value of a series of numbers through using the product of their respective values. It is generally used in the field of Statistics and Mathematics to calculate the mean rather than using the method of traditional Arithmetic mean as it does not provide the accurate measure of returns like Geometric Mean.


What is the Geometric Mean?

The geometric mean stands for a special kind of average of a set of things, products, and the quantification of which is most extensively used to identify the comparative analysis or performance outcomes of an investment, interest rate or portfolio. A geometric mean is mathematically denoted as the "the nth root product of n numbers." It has best applications while working with percentages, which are extracted from values, while the usual arithmetic mean operates with the values themselves. Thus, the geometric mean becomes an influential tool to calculate portfolio performance for many reasons, but one of the most notable is that it takes into consideration the effects of compounding.

 

Calculation of the Geometric Mean

In order to calculate the Geometric Mean, we multiply the given numbers altogether and then take a square root (given that there are two numbers), or cube root (for three numbers) 5th root (for five numbers) etc.

 

Thus, the formula for geometric mean is as in the below image:

 

(Image to be added soon)

 

Calculate Compound Interest Using The Geometric Mean — Instantaneous

Let’s get you to calculate compound interest or any investment portfolio as-easy-as- ABC using the geometric mean. If you or any investor wants to compute the compounding interest of 25 years using the geometric mean of an investment's return, you need to first calculate the interest in year one, then add the interest rate to next year principal’s amount and the chain will follow until 25years.

 

Suppose you want to calculate a principal amount of Rs. 10,000 on 10% interest rate. Then, for year one, you will have to do Rs.10, 000 * 10%, or 1,000. In second year, the updated principal amount will be 11,000, and 10% of 11,000 is 1,100. The new principal amount is now 11,000 +1,100, or Rs.12, 100. For the third year, the new principal amount is 12,100 plus 10% of 12,100 which is $1,210, so it will be Rs. 13310. At the end of 25 years, Rs. 10,000 turns into Rs. 108,347.06, which is an addition of Rs. 98,347.05 more than the original investment.

 

However, geometric means offers you a quick and accurate shortcut which is to multiply the current principal by 1 + the interest rate, and then increase the factor to the number of years compounded. Thus, the calculation is Rs. 10,000 × (1+0.1) 25 = Rs. 108,347.06. Isn’t it really a kid’s play?

 

Solved Examples

Problem 1: You look to purchase a new DSLR camera. One camera has a zoom of 250 and receives a 10 in reviews,the other has a zoom of 290 and gets a 7 in reviews. What is the geometric mean of a camera?

 

Solution:


On using the general arithmetic mean we gwt (250+10)/2 = 175 vs. (290+7)/2 = 148.5

However, the zoom is such a huge number that the user rating gets disoriented.

Therefore we use the geometric means of the two cameras, which are:

√ (250 × 10) = 50

√ (290 × 7) = 41.71...

Therefore, though the zoom is about 10% bigger, the lower user rating of 7 is still imperative.

 

Note: Remember that geometric mean is quite useful when comparing two items with different properties.

 

Problem 2:


What is the Geometric Mean of 1, 2, 4, 8, and 16?

 

Solution:


First we have to multiply the given numbers:  1 × 2 × 4 × 8 × 16 = 1024

Now, (since there are 5 numbers) take the 5th root: 5√1024 = 4

In one line:

Geometric Mean = 5√ (1 × 2 × 4 × 8 × 16) = 4

Thus it comes true that:

 1, 2, 4, 8, 16 = 4 × 4 × 4 × 4 × 4

 

(Image to be added soon)

 

Fun Facts/ Key Takeaways

  • The longer the time span, the more complex compounding becomes and the more accurate the uses of geometric mean.

  • Geometric mean is most workable for series that showcase serial correlation, particularly true for investment portfolios, yields on stocks, bond returns and market risk premiums.

  • Geometric mean is always ≤ the arithmetic mean (equality bearing only when A=B {supposing two quantities}.

  • For unsteady, wayward numbers, the geometric average gives a far more accurate computation of the true return by considering year-over-year compounding that steamrolls the average.

  • Whenever you have a number of factors presenting a product, and you seek to determine the "average" factor, the “way out” is the geometric mean.

FAQs on Geometric Mean

1. How long will it require for the students of Mathematics to understand the concept of Geometric Mean?

This article of Vedantu contains Definition of Geometric Mean, Formula to calculate Geometric Mean, Calculation of compound interest using Geometric Mean, Fun facts on Geometric Mean, Importance of Geometric Mean and various Applications of Geometric mean provided by the team of skilled and dedicated teachers at Vedantu. As a result, students of Maths will require an approximate time of 40 minutes to 60 minutes or an hour to understand the concept of Geometric Mean properly

2. Which Classes include the topic of Geometric Mean as a part of the syllabus of Mathematics for the students?

Students of Class 8, 9, 10, 11 and 12 will find the topic of  Geometric Mean as a part of the syllabus of Maths in CBSE, ICSE and State board as well. As a result, students are advised to read and practice solving questions of the topic Geometric Mean to score high marks in the maths exam of all Classes.

3. What are some learning courses and study material available on Vedantu for the students to learn about Geometric Mean and related topics?

On Vedantu, students can find NCERT solutions, Reference Solutions of Important reference books like RD Sharma, RS Aggarwal and HC Verma, Important Questions on Geometric mean, Revision notes and key notes on the topic Geometric Mean and various Previous Year's question papers and sample question papers to solve questions on the topic of Geometric Mean. Students can also find several Micro courses on various important topics of Mathematics including Geometric Mean with Topic wise explanations of the same. Students can also understand the topic of Geometric Mean from the recorded videos provided at the YouTube channel of Vedantu Math like Sequence and Series and Relation between AM and GM. Vedantu also provides private Home tuitions at selected metropolitan cities of India and abroad to teach maths to the students. Vedantu provides a set of Foundation courses at an affordable price for the students of Classes 6 to 8 to excel and master each topic including Geometric Mean and score high marks in the Maths exam. This includes LIVE and interactive classes, regular doubt sessions, objective tests like V-brainer, V-maths and turbo maths with surprise gifts for the students to motivate and push their limits. Students can learn directly from the best teachers at Vedantu by starting their 7-day free trial on the Vedantu app to get access to all the LIVE Classes by expert teachers of Maths, unlimited access to notes and unlocked test series for straight 7-days, for free. Hurry up and download the Vedantu app to start your free trial of 7-days without providing any bank details, just by signing up at Vedantu app now!

4. What are some quick and useful articles provided by Vedantu to students of Maths for the preparation of Geometric Mean?

Vedantu provides several articles to learn the concept of Geometric Mean like ‘Geometric Mean Formula’. Students can also find the Question Answers Article based on Geometric mean like ‘How do you find Geometric mean’, Conceptual articles like ‘How do you find Geometric Mean for the pair of Class 11’ for the students of CBSE and step by step solution of simple questions like ‘Find the Geometric Mean of 4 and 25’ along with verified answers and related questions. Students can also upload their answers just by signing up at the website of Vedantu, for free. So What are you waiting for? Go and register yourself on the official website of Vedantu and start your biggest success journey now!

5. How is Geometric Mean useful?

A primary advantage of utilising the Geometric Mean is that the actual amounts invested need not be known; since the computation centres around entirely on the return figures and exhibits an "apples-to-apples" comparison when taking into account the two investment options over more than one time horizon. Geometric means are always a bit smaller than the arithmetic mean, which is a simple average.


Using the geometric mean enables economists and analysts to accurately measure the return on an investment that gets paid interest on interest. And this perhaps is one of the major reasons for portfolio managers suggesting clients to reinvest earnings and dividends.

6. What are its various applications? When can you use Geometric Mean Instead Of Arithmetic Mean?

We need Geometric Means, anytime we attempt to calculate average growth rates or population growth, where the growth is identified by multiplication, not addition This critically links Geometric Means to Economics, Statistics, Investment Portfolios, Assessing Present Value and Future Value Cash Flow, Personal Finances, Financial Transactions between banks and countries. That being said, Arithmetic Mean is most appropriate any time several quantities sum up together to generate a total. The Arithmetic Mean resolves the problem/question, "If all the quantities are of the same value, what that value would have to be for the purpose to attain the same total?"


Similarly, the Geometric Mean is relevant any time various quantities multiply together to generate a product. The Geometric Mean resolves the problem/question, "If all the quantities had the same value, what would that value have to be for the purpose to attain the same product.

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