The subject of arithmetic mean and geometric mean is included in the mathematics program of grades 6-7. Since the paragraph is quite simple to understand, it quickly passes, and by the end of the school year, students forget it. But the knowledge in basic statistics is needed to pass the exam, as well as for international SAT exams. And for everyday life, developed analytical thinking will never hurt.
How to calculate the arithmetic mean and geometric mean of numbers
Suppose there are a number of numbers: 11, 4, and 3. The arithmetic mean is the sum of all numbers divided by the number of given numbers. That is, in the case of numbers 11, 4, 3, the answer will be 6. How is 6 obtained?
Solution: (11 + 4 + 3) / 3 = 6
The denominator should have a number equal to the number of numbers whose average needs to be found. The sum is divided by 3, since there are three terms.
Now we need to deal with the geometric mean. Suppose there are a number of numbers: 4, 2, and 8.
The geometric mean of numbers is the product of all given numbers, located under the root with a degree equal to the number of given numbers. That is, in the case of numbers 4, 2 and 8, the answer is 4. Here's how it turned out:
Solution: ∛ (4 × 2 × 8) = 4
In both cases, whole answers were obtained, since special numbers were taken as an example. This does not always happen. In most cases, the answer has to be rounded or left under the root. For example, for the numbers 11, 7 and 20, the arithmetic mean is ≈ 12.67, and the geometric mean is ∛1540. And for numbers 6 and 5, the answers, respectively, will be 5.5 and √30.
Could it happen that the arithmetic mean becomes equal to the geometric mean?
Of course it can. But only in two cases. If there is a series of numbers consisting only of either units or zeros. It is also noteworthy that the answer does not depend on their number.
Proof with units: (1 + 1 + 1) / 3 = 3/3 = 1 (arithmetic mean).
∛ (1 × 1 × 1) = ∛1 = 1 (geometric mean).
1 = 1
Proof with zeros: (0 + 0) / 2 = 0 (arithmetic mean).
√ (0 × 0) = 0 (geometric mean).
0 = 0
There is no other option and cannot be.