Calculating averages for various series had been a part of most of the mathematics practices well, like the arithmetic mean, the geometric mean of series and sequences could be worthwhile calculations.
Geometric Mean
As the name suggests, the geometric mean formula is useful for calculating the geometric mean of a considerable set of numbers. Recall that geometric mean is the mean that indicates and signifies the central tendency of a considerable set of numbers, utilizing the product of their values.
Geometric Mean has been described as the nth root of the products of n numbers in a sequence. It is, however, important to note that geometric mean cannot be calculated from the arithmetic mean of any sequence. As per statistics, geometric mean has been defined only for a positive set of numbers.
Definition of Geometric Mean
Geometric Mean could be described as a type of average that is generally utilized for recognizing growth rates like interest rates or population growth. While arithmetic mean is based on adding items, the geometric mean is known to multiply items. Also, you already know that geometric mean could be obtained for positive numbers only.
While you may be seeking a formal definition for geometric mean, it would be defined as: “the nth root of the product of n numbers”. In simpler words, for any set of numbers$ {x_{i}}^{N_{i=1}}$, the geometric mean is defined as:
$\bar{x}_{geom}=\sqrt[n]{\prod_{i=1}^{n}{x_{i}}}=\sqrt[n]{x_{1}.x_{1}…..x_{n}}$
This is how the formula for geometric mean is obtained.
Notation of the GM Formula
- $\bar{x}_{geom}$ is known to represent geometric mean.
- “n” represents the number of total observations.
- $\sqrt[n]{\prod_{i=1}^{n}{x_{i}}}$ in that case, is the nth root square of the product of all the given numbers.
Important Points Regarding Geometric Mean
- If a GP includes three terms, the middle term is referred as the geometric mean (G.M.) among the two. Therefore, if a, b, and c are the parts of a geometric progression then b= $\sqrt{ac}$ would be recognized as the geometric mean of a and c.
- In case, a1, a2, …….an are non-positive numbers, then in that case, their GM would be recognized as $G= (a_{1}a_{2}a_{3},…. a_{n})^{\frac{1}{n}}$. Also, if G1, G2, G3, ……..Gn are n geometric means between a and b then in that case, a, G1, G2, G3, ………Gn, b will be an individual geometric mean.
