What is the difference between geometric mean and arithmetic mean?
Arithmetic mean of two numbers A and B is:
(A + B)/2
Geometric mean of two numbers A and B is:
sqrt(A*B)
The geometric mean can be rewritten as follows:
sqrt(A*B) = Exp{Log[sqrt(A*B)]} =
Exp{1/2 [Log(A) + Log(B)]} =
Exp[Arithmetic mean of logarithms].
The arithmetic mean and geometric mean are two different ways to calculate the average of a set of numbers.
The arithmetic mean is what most people are familiar with. It is simply the sum of all the numbers in the set divided by the total number of values. For example, to find the arithmetic mean of the numbers 2, 4, and 6, you would add them together (2 + 4 + 6 = 12) and divide by 3 (since there are three numbers) to get an arithmetic mean of 4.
The geometric mean, on the other hand, is calculated by taking the product of all the numbers in the set and then finding the nth root of that product, where n is the total number of values in the set. For example, to find the geometric mean of the numbers 2, 4, and 6, you would multiply them together (2 * 4 * 6 = 48) and then take the square root since there are three numbers (the square root is the 2nd root) to get a geometric mean of approximately 3.63.
The key difference between the two is that the arithmetic mean treats all numbers equally and focuses on their sum, while the geometric mean emphasizes the relationship between the numbers by considering their product and finding the appropriate root.
To summarize:
Arithmetic Mean: Sum of all values divided by the total number of values.
Geometric Mean: Product of all values raised to the appropriate power (root) based on the total number of values.