Geometric Mean

What is a Geometric Mean?

The geometric mean is a method of averaging a list of n numbers by taking the nth root of the products of the numbers. This is in contrast to an arithmetic mean in which the list of n numbers is summed and then divided by n. Equation 1 is the general form of the geometric mean.

   Equation 1

is the geometric mean of the list of numbers a from 1 to n. represents the product of all the numbers in the list of numbers a. As an example, assume we have a list . As there are three numbers in the list, n = 3. The geometric mean of the list a is the cubed root of the products of a.

Why is this Useful?

A geometric mean is useful in machine learning when comparing items with a different number of properties and numerical ranges. The geometric mean normalizes the number ranges giving each property equal weight in the average. This contrasts with arithmetic mean where a larger number range would more greatly affect the average than a smaller number range. To better understand this try doing a geometric mean calculation compared with an arithmetic mean calculation using two numbers. Make one number be chosen from 0 to 5 and the other number from 0 to 100. Vary the two numbers to see how each affects the average.

Practical Uses of Geometric Mean:

  • Proportional growth – When growth is proportional, exponential or varied, a geometric mean is more appropriate to calculate average growth because of how this method handles difference in number ranges. Arithmetic mean would describe a linear growth generally resulting in higher-than-true average.
  • Aspect ratios – The geometric mean was used to choose a compromise aspect ratio between film and digital video, a major influence in the design of modern movie theaters.