There are several ways to express summaries in mathematics and statistics using measures of major trends, such as using a simple formula. Mean, median, mode and range of central tendencies are crucial metrics of the primary orientations.
If you look at the data set’s mean, for example, you may get a rough idea of what the data is about, and you may use two or more numbers to calculate the mean.
The Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) are three examples of means (HM). This article will cover the definition, formula, qualities, applications, and connection between AM, GM, and HM. Applying such factors for comparison will help in ensuring a comprehensive understanding of the three concepts.
Here, we will understand the relationship between AM, GM and HM with the help of examples and formulae.
Mathematical definition of the Geometric mean
In mathematics, the Geometric Mean (GM) is a term that refers to the average value that represents the central tendency of a collection of numbers by taking the product of the significance of the numbers into account.
There are several different ways to calculate the geometric mean for a set of two numbers, such as the product of 3 and 1. Assuming that n is the total number of data values in the set, we multiply the numbers and then extract the nth root.
In other words, the geometric mean is defined by taking the product of all n numbers, then taking the nth root. The arithmetic mean does not match the geometric mean. Arithmetic mean is calculated by summing up all values and dividing the total by the number of data points.
Using a geometric mean, on the other hand, we multiply the number of data values supplied and then divide by the radical index to get the root.
If we have two data points, we can take the square root; if we have three data points, we can take the cube root; and if we have four data points, we can take the fourth root, and so on.
Difference Between Arithmetic Mean and Geometric Mean
|Arithmetic Mean||Geometric Mean|
|To get the arithmetic mean, divide the total number of data points in a data set by the total number of data points in a dataset and multiply the result by one hundred.||Finding it may be accomplished by multiplying all of the integers in the supplied data set and then taking the nth root of the resulting number.|
|For example, the data sets that have been provided are 5, 10, 15, and 20.The number of data points, in this case, is four.The arithmetic mean, often known as the mean, is (5 10 15 20)/4.50/4 = 12.5 is the mean.||Take, for example, the following data set: 4, 10, 16, 24.In this case, n=4.The G.M = 4th root of (4 10 16 24) = 4th root of 15360 G.M = 11.13 is obtained as a result of this formula.|
To understand the relationship between AM, GM, and HM, we must first understand the formulae for these three sorts of mean values.
Based on the assumption that the variables “x” and “y” are the two integers in question and that the number of potential values is equal to 2, the solution is AM = (a b)/2.
1/AM = 2/(a b)…………………….. (1)
GM is an abbreviation for General Motors (ab)
GM2 = ab…………………… (2)
HM = [(1/a) + (1/b)] = [(1/a) + (1/b)] [(1/a) + (1/b)] = 2/[(1/a) + (1/b)]
In mathematics, the HM formula is 2[(a b)/ab].
HM = 2ab/(a + b)…………………….. (3)
As a result, if we substitute (1) and (2) for (3), we get
HM = GM2/AM
GM2 equals AM + HM
Unless, of course, there is a problem.
GM = [AM + HM]
In this example, AM, GM, and HM are all equivalent to GM2, equal to AM + HM.
The GM has several notable characteristics, including the following:
The arithmetic mean is always greater than the geometric mean for that data set for every given data collection.
There is no difference in the outcome, no matter how many GMs are substituted for each item in the data gathering.
A series’ geometric mean is the same for both series if the geometric mean is the same for both series.
The product of the geometric mean of the two series is equal to the product of the appropriate components of the GM in both series.
The geometric mean may be utilised in several contexts.
For the sake of this discussion, the GM assumes that data may be conceived of as a scaling factor in real life. Before anything else, we need to know when the G-Matrix is appropriate to employ. They should only be used for positive values, according to the general view.
One of the most common uses of this technique is to collect integers with exponentially increasing values. We won’t use zero or negative numbers in our calculations since there will be no such values to work with.
The geometric mean has several benefits, and it is employed in a variety of contexts. Here are a few real-world examples of how you may put them to use:
It is featured in indexes of the stock market. It is vital to highlight that many of the value line indices used by finance departments are based on data from General Motors, which should be noted.
This component is used to compute the portfolio’s yearly return on investment, expressed as a percentage. It is used to determine average growth rates for financial purposes, also known as the compounded annual growth rate (CAGR).
Aside from these applications, it is also used in investigations involving cell division and bacterial growth.
This article provided an overview of the many aspects of the relationship between the AM GM and the General Manager. We hope that you have gotten a better grasp of various issues due to reading this article.