Concept

Geometric Mean

The geometric mean is a mean or average that indicates the central tendency or typical value of a set of numbers. Unlike the arithmetic mean, the geometric mean calculates the

n th

root of the product of

n

values.

Geometric Mean $= n a_{1} \cdot a_{2} \cdot \dots \cdot a_{n}$

Consider an example where the geometric mean of

2,

4,

and

8

is the cube root of the product of

2,

4,

and

8 .

3 2 \cdot 4 \cdot 8 = 364 = 4

Additionally, calculating the geometric mean of two values can be interpreted as calculating the side length of a square with the same area as a rectangle whose dimensions are the given values. Calculating the geometric mean of

2

and

4.5,

for example, is finding the side length of a square with the same area as a rectangle whose dimensions are

2

and

4.5 .

The side length

s

of the square can be calculated by taking the square root of the area.

s^{2} = 2 \cdot 4.5

Solve for

s

SqrtEqn

$LHS = RHS$

s^{2} = 2 \cdot 4.5

SqrtPowToNumber

$a^{2} = a$

s = 2 \cdot 4.5

Evaluate right-hand side

Multiply

s = 9

CalcRoot

Calculate root

s = 3

Therefore, the geometric mean of

2

and

4.5

3,

which is the side length of the square.

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Concept

Geometric Mean

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