# Geometric Mean

Geometric mean is one of the methods to estimate mid-value of some data. In geometric mean the midpoint criteria is based on the geometric progression where the difference between the consecutive values increase exponentially. It is calculated as the nth root of the product of the values.

Geometric mean is generally used to estimate midpoint of multiple rates.

## Formula

The formula is

$$\sqrt[\text{n}]{\text{a} _ \text{1}\cdot \text{a} _ \text{2}\cdot \text{a} _ \text{3}\ldots \text{a} _ \text{n}}$$

Where, n is the number of values.

In simple words geometric mean calculates the average using product instead of sum. We combine all values using multiplication and then separate it by the nth root like in arithmetic mean where we combine the values using addition and then separate them by division.

Geometric mean can also be calculated by the logarithmic version of the above formula.

$$\frac{\log{\text{a} _ \text{1}}+\log{\text{a} _ \text{3}}+\log{\text{a} _ \text{3}+\ldots +\ \log{\text{a} _ \text{n}}}}{\text{n}}$$

In fact, the arithmetic mean of the logarithms of the values of the data is equal to the geometric mean.

## Origin

Geometric mean is related to geometric sequence of numbers where the ratio of any two adjacent elements is the same, as in the arithmetic sequence where the difference of any two adjacent elements is same. Hence the geometric mean of two values, as it promises to find the mid-value, will result in a value which maintains the ratio.

Consider two values a and b where the m is the value which will maintain the geometric progression.

$$\text{a} \text{,} \ \text{m} \text{,} \ \text{b}$$

Since the ratio is supposed to same in two adjacent values so

$$\frac{\text{a}}{\text{m}}=\ \frac{\text{m}}{\text{b}}$$

$$\text{m}=\sqrt{\text{ab}}$$

## Example

To start with a simple example, consider two numbers 3 and 12.

$$\text{Geometric Mean of 3 and 12}=\sqrt{\text{3}*\text{12}}=\sqrt{\text{36}}=\text{6}$$

The numbers 3,6,12 now make a geometric sequence. (12/6 = 2 = 6/3)

### When To Use Geometric Mean

An app developer is using app invitation strategy where the user is rewarded for the bringing in new users. The number of users increase exponentially in the span of 6 months as follows.

Month Users New Users Growth
1 1000 - -
2 1150 150 1.15
3 1500 350 1.30
4 2000 500 1.33
5 2700 700 1.35
6 3600 900 1.33

Arithmetic Mean of Growth = 1.294203
Geometric Mean of Growth = 1.291994

Month Arithmetic Mean Growth Simulation Geometric Mean Rate Growth Simulation
1 1000 1000
2 1294 1291
3 1674 1669
4 2167 2156
5 2805 2786
6 3630 3600

In the above table we have simulated the growth in the number of users using arithmetic and geometric mean. We don’t get the original final value in case of the arithmetic mean growth simulation while the simulated growth using geometric mean gives us the original final value. The reason behind the error is that the same growth rate result in unequal increase in the number of users. The number of new users in month is dependent on the number of users in the previous month.

Hence geometric mean is used when the successive value depends on the current value and the rate/proportion as in popluation growth and investment returns etc.