The most predominantly used is Arithmetic Mean(AM), it might not be an ideal choice when we have extreme observations in the sample. Extreme observations influence the mean and they pull the value of mean at a higher rate. Therefore, they might not be the best measure when there are extreme observations in datasets. Hence, we have an alternative like Median and Geometric Mean which can be preferred over mean. Median is a form of ordered statistic., however, I would strike with GM.
GM is the nth root product of the sample observations,
![{\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026cae6801f672b9858d55935ec7397183dc3a36)
This has been closely related to logarithmic transformations for simple calculations which makes the work easier. Let x and y are two real numbers (positive) such that
AM = (x+y)/2 ;
GM=2*sqrt(x+y)
GM=2*sqrt(x+y)
So, we could observe that GM is not greater than AM, as a result, GM can be preferred over AM. In the calculation of GM, we combine our data to be the product. They are commonly used in portfolio management especially for investments, in calculating the growth rate, social and biological science. Another important character of GM is we can compare observations which possess different behaviors.
For example, one student might have scored 90, student 2 might have got 50; the score might differ. for each student. In this scenario, GM might be more suitable for understanding the average scores. GM does not dampen the effect of high and low scores as AM and they reduce the bias in the average score.However, there is also another side under the scenarios when it cannot be used. If the datasets contain zero or negative values GM might be the wrong choice since it would provide the GM to be zero. During this scenario, other transformations can be made to the datasets which can be helpful in understanding the average. Hence, they are more suitable when the datasets has extreme values as in the case of hydrology, stock market, agriculture and so on.
For example, one student might have scored 90, student 2 might have got 50; the score might differ. for each student. In this scenario, GM might be more suitable for understanding the average scores. GM does not dampen the effect of high and low scores as AM and they reduce the bias in the average score.However, there is also another side under the scenarios when it cannot be used. If the datasets contain zero or negative values GM might be the wrong choice since it would provide the GM to be zero. During this scenario, other transformations can be made to the datasets which can be helpful in understanding the average. Hence, they are more suitable when the datasets has extreme values as in the case of hydrology, stock market, agriculture and so on.
Beautifully explained.
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