Averages

The arithmetic mean of a set of numbers is the sum of all numbers in the set divided by the number of values in the set. Algebraicly, it can be writte as \(\frac{S_{n}}{n}\). This is a valuable metric for finding the dead center of a set of numbers. It is a key metric in statistics, although for datasets with outliers it can be noticeably skewed.

Chances are, arithmetic mean is what you think of when you first hear of averages. AMC 10 knows this, so when they say average, it will always refer to arithmetic mean, unless otherwise mentioned. The reason that it is called arithmetic mean is because the relationship is completely linear. The alternative, geometric mean, uses powers and inverse reciprocal relationships. It is also conveniently used in geometric shapes whereas the arithmetic mean is exclusively for direct arithmetic relationshpis and statistics

Notice that in an arithmetic mean, you don't need to know each individual value, you just need the sum and number of values. If you have to add values, you just add to the sum and number, not having to recalculate the whole thing. This is a key part of the AMC 10 and their traps and logic.

The geometric mean of a number, \(x\), satisfies \(\frac{x}{a}=\frac{b}{x}\)and visa versa. If you solve this for \(x\), you would get \(x=\sqrt{ab}\). My interpretation is that this is called a geometric mean because it has to do with reciprocals and exponents, characteristic of geometric series, although there are also geometric properties. By the AM-GM inequality, we know that the geometirc mean will always be less than or equal to the arithmetic mean. The arithmetic mean can only be found for two numbers, not a whole set, and rather than being the central focal point, has purposes for trigonometry and triangles

The geometric mean of two numbers has several geometric features as well, particularly in triangles. Consider a right triangle with an altitude drawn from the right angle to the hypotenuse. It splits the hypotenuse into two parts. The geometric means of the lengths of the two sections is equal to the length of the hypotenuse. Furthermore, if you find the geometric mean of the whole hypotenuse and one section of the hypotenuse, it will be equal to the leg adjacnet to thta section. Both of these can be proved by using the fact that the altitude splits the triangle into similar triangles.