Definition

Geometric mean: the $n$ th root of the product

Statistics • Mean • Geometric Mean

Formula

G = (x_{1} x_{2} \dots x_{n})^{1/ n}

The geometric mean of

n

positive values

x_{1}, x_{2}, \dots, x_{n}

G = (x_{1} x_{2} \dots x_{n})^{1/ n}

, so the statement '

G

is the geometric mean' is equivalent to the product equation

x_{1} x_{2} \dots x_{n} = G^{n}

; it is defined only for positive values, and for two numbers it reduces to

G = x_{1} x_{2}

How to use it

1Convert every geometric-mean statement to its product form at once, by raising both sides to the power $n$ ; the algebra is then ordinary.
2Two such statements about overlapping sets divide into each other, which isolates the extra value without ever finding the individual terms.
3Adjoining to a set a value equal to its geometric mean leaves the geometric mean unchanged.
4The geometric mean is the right average for ratios and growth factors, where the arithmetic mean would overstate.

Worked example

The geometric mean of

2

x

and

9

6

. Find

x

Step-by-step

1
$2 \cdot x \cdot 9 = 6^{3} = 216$
Why: Cube both sides to clear the cube root.
2
$18 x = 216 \Rightarrow x = 12$

Answer

12

Watch out

Every value must be positive — a zero forces

G = 0

and a negative value makes the even-order root meaningless. The geometric mean is never larger than the arithmetic mean of the same values, so a 'mean' above the arithmetic one has been computed wrongly.

Builds on

Power of a power, and the move to a common base

Also known as

geometric mean, GM, nth root of the product, G.M. of n numbers

Appears in past papers

Formula

Worked example

Appears in past papers

2020 (II)

2019 (I)

2018 (I)

2017 (II)