Intro to Arithmetic- and Geometric Means

1. A R I T H M E T I C - A N D
G E O M E T R I C M E A N S

2. 5. Geometric Mean - Definition
2. Arithmetic Mean – Definition
3. Use of the Arithmetic Mean (Sample Exercise)
TA B L E O F C O N T E N T
1. Video introduction
4. Application of the Arithmetic Mean

3. 8. Relationship of the AM and the GM
6. Use of the Geometric Mean (Sample Exercise)
TA B L E O F C O N T E N T
7. Application of the Geometric Mean
9. Application of the AM – GM inequality

5. A R I T H M E T I C M E A N
The arithmetic mean (AM) of two non-negative
numbers is
𝑎+𝑏
2
.
The arithmetic mean of the non-negative
numbers
𝑎1; 𝑎2; 𝑎3; … ; 𝑎𝑛 is
𝑎1+𝑎2+𝑎3+⋯+𝑎𝑛
𝑛
.
Definition

6. A R I T H M E T I C M E A N
The table shows the grades received by a
student
in Geography last year. Find the (arithmetic)
mean
of his grades.
Sample
Exercise

7. A R I T H M E T I C M E A N
Application
We use the arithmetic mean in our everyday lives
in many contexts:
• We ask friends about average house rents in
their neighborhoods;
• we calculate the average monthly expenses
before moving to a new flat;
• we work out the average height and weight for
children of a certain age so as to see whether a
child is undernourished or fat, unusually tall or
short;
• it is used to work out the mean temperature
… and so on and so forth.

8. G E O M E T R I C M E A N
The geometric mean (GM) of two non-negative
numbers is 𝑎𝑏.
The geometric mean of the non-negative
numbers
𝑎1; 𝑎2; 𝑎3; … ; 𝑎𝑛 is 𝑛
𝑎1 ∙ 𝑎2 ∙ 𝑎3 … ∙ 𝑎𝑛 .
Definition

9. G E O M E T R I C M E A N
In the table below you can see the data about the
rate of growth of production of a company.
Calculate the „average rate of growth” (geometric
mean).
Sample
Exercise
𝐺𝑀 =
4
1,24 ∙ 1,39 ∙ 1,31 ∙ 1,15 = 1,27
The average rate of growth was 1,27 (127%).
SOLUTION:

10. G E O M E T R I C M E A N
Application
Applications of the geometric mean are
most common in business and
finance, where it is frequently used
when dealing with percentages to
calculate growth rates and returns on
investments or on a portfolio of
securities. It is also used in certain
financial and stock market indices,
such as the Financial Times' Value Line
Geometric index.

11. Work out the arithmetic- and
geometric means of the same
groups of data on the right one
by one, then compare them.
What have you found?
T H E R E L A T I O N S H I P O F T H E A R I T H M E T I C - A N D
G E O M E T R I C M E A N S
Click on the interactive
elements to see the results.

12. Given that 𝑎, 𝑏 ∈ ℝ0
+
,
𝑎 + 𝑏
2
≥ 𝑎𝑏.
T H E R E L A T I O N S H I P O F T H E A R I T H M E T I C - A N D
G E O M E T R I C M E A N S
Theorem
:
Algebraic Proof
Geometric Proof

13. Given that 𝑎, 𝑏 ∈ ℝ0
+
,
𝑎 + 𝑏
2
≥ 𝑎𝑏.
T H E R E L A T I O N S H I P O F T H E A R I T H M E T I C - A N D
G E O M E T R I C M E A N S
Theorem
:
Algebraic Proof
Geometric Proof

14. Given that 𝑎, 𝑏 ∈ ℝ0
+
,
𝑎 + 𝑏
2
≥ 𝑎𝑏.
T H E R E L A T I O N S H I P O F T H E A R I T H M E T I C - A N D
G E O M E T R I C M E A N S
Theorem
:
Geometric Proof
Algebraic Proof

15. Given that 𝑎, 𝑏 ∈ ℝ0
+
,
𝑎+𝑏
2
≥ 𝑎𝑏.
T H E R E L A T I O N S H I P O F T H E A R I T H M E T I C - A N D
G E O M E T R I C M E A N S
The same theorem applies in the case of more values as
well:
Theorem
:
Given that 𝑎1; 𝑎2; 𝑎3; … ; 𝑎𝑛 ∈ ℝ0
+
,
𝑎1+𝑎2+𝑎3+⋯+𝑎𝑛
𝑛
≥ 𝑛
𝑎1 ∙ 𝑎2 ∙ 𝑎3 … ∙ 𝑎𝑛.
Theorem
:

16. Given that 𝑎, 𝑏 ∈ ℝ0
+
,
𝑎+𝑏
2
≥ 𝑎𝑏.
T H E A P P L I C A T I O N O F T H E A M - G M I N E Q U A L I T Y
( M I N I M U M - M A X I M U M P R O B L E M S )