How do you find the geometric mean of an arithmetic mean?

To find the geometric mean of a set of numbers that includes their arithmetic mean, you first calculate the arithmetic mean, then include it in the set of numbers when calculating the geometric mean.

Here's a breakdown:

1. Calculate the Arithmetic Mean (AM):

The arithmetic mean is the average of a set of numbers. You sum all the numbers and then divide by the count of numbers.

• Formula: AM = (x₁ + x₂ + ... + x_n) / n

2. Include the Arithmetic Mean in the Original Set:

Add the calculated arithmetic mean to the original set of numbers. This creates a new set.

3. Calculate the Geometric Mean (GM) of the New Set:

The geometric mean is the nth root of the product of n numbers.

• Formula: GM = (x₁ x₂ ... * x_n+1)^1/(n+1) (where x_n+1 is the arithmetic mean).

Example:

Let's say you have the numbers 2, 8, 12, 14, and 20.

1. Arithmetic Mean (AM): (2 + 8 + 12 + 14 + 20) / 5 = 56 / 5 = 11.2
2. New Set: 2, 8, 12, 14, 20, 11.2
3. Geometric Mean (GM): (2 8 12 14 20 * 11.2)^1/6 = (451,584)^1/6 ≈ 9.20

Therefore, to find the geometric mean that incorporates the arithmetic mean, calculate the arithmetic mean, include it in the original data set, and then compute the geometric mean of the expanded data set.