Geometric mean

Geometric mean is similar to the arithmetic mean. The difference is that arithmetic mean adds items, while geometric mean multiplies them. Furthermore, we can only observe geometric mean for positive numbers.

Definition

Definition: Let $\{(x_{1}, f_{1}), \cdots, (x_{k}, f_{k})\}$ be a distribution of numeric variable given on finite population with $N$ elements and whose values are positive. Geometric mean of that numeric variable is number

$$G = \sqrt [N]{x_{1}^{f_{1}}\cdots x_{k}^{f_{k}}}.$$

Note that

$$G = x_{1}^{\frac{f_{1}}{N}} \cdots x_{k}^{\frac{f_{k}}{N}} = x_{1}^{p_{1}}\cdots x_{k}^{p_{k}},$$

where $p_{1}, \cdots, p_{k}$ are relative frequencies of properties $x_{1}, \cdots, x_{k}$, respectively.

If a variable is given with statistical sequence $y_{1}, \cdots, y_{N}$ (with positive values), then its geometric mean is obviously

$$G(y_{1}, \cdots, y_{N}) = \sqrt [N] {y_{1} \cdots y_{N}}.$$

Also,

$$logG = \frac{1}{N}log(y_{1}\cdots y_{N})$$

$$logG =\frac{logy_{1}+ \cdots + logy_{N}}{N}.$$

Examples

Geometric mean can be easily understood with simple numbers, as in the following example:

Example 1: What is the geometric mean of $2, 3$ and $4$?

Solution:

Since we know that $4 = 2^{2}$, we have

$$G = \sqrt [3]{2^{3}\cdot 3} = \sqrt [3] {24} \approx 2.88.$$

Real world applications of the geometric mean include using it as measure of average speed of some changes.

Example 2: If some place had $2000$ residents in $2000.$, $9000$ residents in $2005.$ and $18000$ residents in $2010.$, then the number of residents firstly increased for $4.5$ times, and after that $2$ times.

The average population change is not $\frac{4.5 + 2}{2} = 3.25$, but $\sqrt[2]{4.5 \cdot 2} = \sqrt [2]{9} = 3$. Indeed, $2000 \cdot 3 \cdot 3 = 18 000$.

Example 3: If bacteria increases its population by $10 \%$ in the first hour, $20 \%$ in the next hour and $25\%$ in the third hour, we can calculate the mean growth rate. Let’s begin with $100$ bacteria.

After the first hour we have $100 + 0.1 \cdot 100 = 110$ bacteria. This means that, beacause of $110 = 100 \cdot 1.1$, the growth rate is $1.1$.

After the second hour we have $110 + 0.2 \cdot 110 = 132$ bacteria. This means that, because of $132 = 110 \cdot 1.2$, the growth rate is $1.2$.

After the third hour we have $132 + 0.25 \cdot 132 = 165$ bacteria. This means that, because of $165 = 132 \cdot 1.25$, the growth rate is $1.25$.

Now we need to find the geometric mean:

$$G = \sqrt[3] {1.1 \cdot 1.2 \cdot 1.25}$$

$$G \approx 1.1817.$$

Our result is interpreted as the mean rate of growth of the bacteria over the period of $3$ hours.

Whenever we have an example or situation with percentage growth during some period of time, we must remember that it requires the use of geometric mean.

In economy, the geometric mean is the average return of an investment over a period of time, used in order to evaluate an investment portfolio.

Geometric mean formula for investments is: $$\left(\prod_{i = 1}^{n} (1 + R_{i})\right)^{\frac{1}{n}} – 1,$$

where $R_{i}$ is growth rate for year $i$.

Example 4: If an investor has annual return of $10\%, 5\%, -50\%, 20\%$ and $20\%$, what is his average return during this period?

Solution:

By using geometric mean formula for investments, we get

$$\sqrt [5] {(1 + 0.1)(1 + 0.05 )(1 – 0.5)(1 + 0.2)(1 + 0.2) } – 1$$

$$\sqrt [5] {(1 .1)(1.05 )(0.5)(1.2)(1.2) } – 1$$

$$\sqrt [5] {0.8316} – 1$$

$$\approx -0.03621$$

Therefore, the annual return is $-3.621 \%$.

Geometric interpretation of the geometric mean

The geometric mean of two positive numbers $x$ and $y$ is a positive number $g$ whose square equals the product $xy$:

$$g^{2} = xy.$$

Therefore, geometric mean is an answer to the following problem: given a rectangle with sides $x$ and $y$, find the side of the square whose area is equal to the area of mentioned rectangle. This particular problem gave the geometric mean its name.

The first step in geometric construction of the geometric mean is to draw a semicircle with diameter of length $x + y$.

Next step is to draw a perpendicular to the diameter from the point where the segments of length $x$ and $y$ meet.

Finally, the geometric mean of $x$ and $y$ is the length of the perpendicular from the semicircle to the diameter.

Arithmetic mean – geometric mean inequality (AM – GM)

The Arithmetic mean – geometric mean inequality is an elementary inequality which states that for any set of nonnegative real numbers, the arithmetic mean of the set is greater than or equal to the geometric mean of the set. More formally, for a set of nonnegative real numbers $a_{1}, a_{2}, \cdots, a_{n}$, the following holds:

$$\frac{a_{1} + a_{2} + \cdots + a_{n}}{n} \geq \sqrt [n] {a_{1}a_{2}\cdots a_{n}}.$$

For example, the arithmetic mean of the set $\{10, 15, 20\}$ is $\frac{10 + 15 + 20}{3} = 15$, while the geometric mean is $\sqrt[3]{10 \cdot 15 \cdot 20} \approx 14.42$. Obviously, $15 \geq 14.42$, and AM – GM guarantees this is always the case.