# What Is Geometric Mean?

The geometric mean is defined as the n^{th} root of the product of n numbers. Mathematically this can be expressed as:

Because the n^{th} root of a negative value can be imaginary, the values’ product has to be positive. Thus, it is usually required that all values are positive.

For example, consider a set of three values defined as 3, 5, and 6. The geometric mean would be:

The geometric mean represents a central tendency or typical values of a set of values, and it is one of the three Pythagorean means. The other two are the arithmetic mean and the harmonic mean. When comparing these three means, the geometric mean is always in between the other two, the harmonic mean the lowest, and the arithmetic mean the greatest.

### How Is The Geometric Mean Useful?

The geometric mean is used in many applications in finance, engineering, and science. For example, in finance, it has been used to calculate specific financial indices, such as the former FT 30 index, and it is also widely used to calculate portfolio performance.

In economics, it is considered a better description than the arithmetic means for proportional growth, exponential growth, and varying growth.

Its use extends to socio-economic measures. For example, in 2010, the United Nations Development Index changed to a geometric mean because it reproduces the statistic’s non-substitutable nature better.

In businesses, it’s used for the calculation of the compound annual growth rate (CAGR).

### Geometric Mean and LogicPlum

LogicPlum’s platform relies heavily on mathematics and statistics. Therefore, the geometric mean calculation is an integral aspect of it.

Moreover, the platform uses it during certain neural network definitions. For example, the Dropout algorithm is a state-of-the-art method for neural network training aimed at avoiding overfitting. It works by randomly dropping units during training to prevent their co-adaptation and approximating logistic functions’ expectations via normalized geometric means.

###### Additional Resources

- Baldi, P and Sadowski, P. (2014).
*The dropout learning algorithm.*Available athttps://www.sciencedirect.com/science/article/pii/S0004370214000216 - Larochelle, H. (2013).
*Neural networks [7.5]: Deep learning – dropout*. Available athttps://www.youtube.com/watch?v=UcKPdAM8cnI - WHY NOT MATH? (2020) Geometric Means Available athttps://www.youtube.com/watch?v=CUd8I_0Cq_o