Back To Course

Geometry: High School15 chapters | 160 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how different the geometric mean is from the arithmetic mean, or average, that we are all so familiar with. Learn in what circumstance the geometric mean is preferred. Also, learn the formula and how to use it.

The **geometric mean** is the *nth* root when you multiply *n* numbers together. It is not the same as the arithmetic mean, or average, that we know. For the arithmetic mean, we add our numbers together and divide by how many numbers we have. The geometric mean uses multiplication and roots. For example, for the product of two numbers, we would take the square root. For the product of three numbers, we take the third root.

When would we use the geometric mean as opposed to the arithmetic mean? We would use the geometric mean when we want to figure out the average rate of growth if the growth rate is determined by multiplication. For example, the average percentage amount of growth in a bank account per year uses the geometric mean since the growth each year is determined by multiplying the amount in the bank account by the percentage growth.

The arithmetic mean is used when the growth is determined by addition. For example, the average amount it costs to feed people at a party uses the arithmetic mean because the growth in cost is determined by addition. The geometric mean does have one limitation - it cannot be used for negative numbers; all the numbers have to be positive. You can, however, find the geometric mean for as many numbers as you want.

The formula to find the geometric mean is this:

It tells us to first multiply all the numbers we have together and then to take the *nth* root of that product. If we had two numbers, we take the square root; if we have four numbers, we take the fourth root. Most calculators have a button next to the square root button that lets you choose the root. This button lets you calculate the third root, fourth root, fifth root, etc. - whichever root you need. Now some calculators will have you choose the root before pushing the button, while others have you choose the root after pushing the button. Get familiar with your calculator first so you know how yours will calculate your roots.

Let's look at some examples to see how we can use this formula. Say we want to find the geometric mean of 3, 4, and 9. What does the formula tells us to do first? It tells us to multiply our numbers together first. So let's do that: 3 * 4 * 9 = 108. Then the formula tells us to take the third root since we have three numbers. Doing that, we get 4.7622 for our answer.

Now, what if we complicated things a little bit and we wanted to get the geometric mean of the numbers 3, 4, 3, 2, 8, and 10? What do we do first? We do the same as before by multiplying all our numbers together. We don't stress out even if the numbers get really big. Multiplying everything together, we get 3 * 4 * 3 * 2 * 8 * 10 = 5760.

Do you see how we multiplied the 3 twice? That is because our data has the number 3 twice. Remember, whatever numbers the data shows, that is what you multiply together. Even if the numbers repeat, you still multiply them together. Okay, so we've multiplied all our numbers together - now what? Now we figure out which root we need. Let's see - how many numbers do we have? We have six numbers, so that means we will be taking the sixth root. Okay, so the sixth root of 5760 is 4.234, and that is our answer.

To review, the **geometric mean** is the *nth* root when you multiply *n* numbers together. It is used to calculate the average rate of growth when the growth is determined by multiplication as in the case of annual percentage growth of a bank account. The formula for geometric mean is found from its definition. You first multiply your numbers together and then you take the *nth* root where *n* is the number of numbers you multiplied together. So if you are multiplying three numbers together, you will take the third root. If you are multiplying eight numbers together, then you will take the eighth root.

Following this lesson, you should have the ability to:

- Define geometric mean
- Explain when you would want to use the geometric mean instead of the arithmetic mean
- Identify the formula for finding geometric mean

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
2 in chapter 7 of the course:

Back To Course

Geometry: High School15 chapters | 160 lessons

- Ratios and Proportions: Definition and Examples 5:17
- Geometric Mean: Definition and Formula 5:15
- Solving Problems Involving Proportions: Definition and Examples 5:22
- Similar Polygons: Definition and Examples 8:00
- The Transitive Property of Similar Triangles 4:50
- Triangle Proportionality Theorem 4:53
- Constructing Similar Polygons 4:59
- Properties of Right Triangles: Theorems & Proofs 5:58
- The Pythagorean Theorem: Practice and Application 7:33
- The Pythagorean Theorem: Converse and Special Cases 5:02
- Similar Triangles & the AA Criterion 5:07
- What is a Polygon? - Definition, Shapes & Angles 6:08
- Go to High School Geometry: Similar Polygons

- AFOQT Information Guide
- ACT Information Guide
- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- FTCE Middle Grades Math: Connecting Math Concepts
- Social Justice Goals in Social Work
- Developmental Abnormalities
- Overview of Human Growth & Development
- ACT Informational Resources
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- What's the Difference Between Polytheism and Monotheism?
- Ethnic Groups in America
- What Are the 5 Ws in Writing? - Uses & Examples
- Phenol: Preparation & Reactions
- Plant Life Cycle Project Ideas
- Medieval Castle Project Ideas
- Samurai Project Ideas
- Quiz & Worksheet - Solvay Process
- Quiz & Worksheet - Kinds of Color Wheels
- Quiz & Worksheet - Understanding Abbreviations
- Quiz & Worksheet - Act & Rule Utilitarianism Comparison
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- Lesson Plan Templates
- Active Learning | Definition & Strategies for Teachers

- Human & Cultural Geography for Teachers: Professional Development
- Common Core ELA - Informational Text Grades 11-12: Standards
- US History from Reconstruction for Teachers: Professional Development
- NYSTCE Physics (009): Practice and Study Guide
- College World History Textbook
- Matrices & Absolute Value: Homeschool Curriculum
- 11th Grade Grammar Usage: Help and Review
- Quiz & Worksheet - Hallucinogenic Drug Dependence Signs
- Quiz & Worksheet - Characteristics of Animal Tissues
- Quiz & Worksheet - Features of Osteomalacia and Rickets
- Quiz & Worksheet - Characteristics of Leiomyomas
- Quiz & Worksheet - Specific Gravity of Urine

- Patterns & Methods of Tobacco Use
- Metaphors in The Outsiders
- The Highwayman Lesson Plan
- How Much Does a Smart Board Cost?
- Social Studies Games for Kids
- Magna Carta Lesson Plan
- Easter Activities for Kids
- Time Management Lesson Plan
- 504 Plans in Massachusetts
- Texas Teacher Certification Test Limit Waiver
- Poetry Activities for Kids
- 504 Plans in Georgia

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject