Geometric Mean Function
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An Geometric Mean Function is a mean function that is based on the nth root of the multiplication of the members.
- Context:
- range: Geometric Mean Values.
- It only applies to positive numbers.
- …
- Counter-Example(s):
- See: Nth Root.
References
2009
- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Geometric_mean
- The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average," except that instead of adding the set of numbers and then dividing the sum by the count of numbers in the set, [math]\displaystyle{ n }[/math], the numbers are multiplied and then the nth root of the resulting product is taken. For instance, the geometric mean of two numbers, say 2 and 8, is just the square root (i.e., the second root) of their product, 16, which is 4. As another example, the geometric mean of 1, 1/2, and 1/4 is the cube root (i.e., the third root) of their product (0.125), which is 1/2.
The geometric mean can be understood in terms of geometry. The geometric mean of two numbers, [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], is simply the side length of the square whose area is equal to that of a rectangle with side lengths [math]\displaystyle{ a }[/math] and b. That is, what is [math]\displaystyle{ n }[/math] such that n2 = a × b? Similarly, the geometric mean of three numbers, [math]\displaystyle{ a }[/math], b, and [math]\displaystyle{ c }[/math], is the side length of a cube whose volume is the same as that of a cuboid with side lengths equal to the three given numbers.
The geometric mean only applies to positive numbers.[1] It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment. The geometric mean is also one of the three classic Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.
- The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average," except that instead of adding the set of numbers and then dividing the sum by the count of numbers in the set, [math]\displaystyle{ n }[/math], the numbers are multiplied and then the nth root of the resulting product is taken. For instance, the geometric mean of two numbers, say 2 and 8, is just the square root (i.e., the second root) of their product, 16, which is 4. As another example, the geometric mean of 1, 1/2, and 1/4 is the cube root (i.e., the third root) of their product (0.125), which is 1/2.
- ↑ The geometric mean only applies to positive numbers in order to avoid taking the root of a negative product, which would result in imaginary numbers, and also to satisfy certain properties about means, which is explained later in the article.