In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers x1, x2, ..., xn, the geometric mean is defined as
(
∏
i
=
1
n
x
i
)
1
n
=
x
1
x
2
⋯
x
n
n
{\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}
For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is,
2
⋅
8
=
4
{\displaystyle {\sqrt {2\cdot 8}}=4}
. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2, that is,
4
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1
⋅
1
/
32
3
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/
2
{\displaystyle {\sqrt[{3}]{4\cdot 1\cdot 1/32}}=1/2}
. The geometric mean applies only to positive numbers.The geometric mean is often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as a set of growth figures: values of the human population or interest rates of a financial investment over time.
The geometric mean can be understood in terms of geometry. The geometric mean of two numbers,
a
{\displaystyle a}
and
b
{\displaystyle b}
, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths
a
{\displaystyle a}
and
b
{\displaystyle b}
. Similarly, the geometric mean of three numbers,
a
{\displaystyle a}
,
b
{\displaystyle b}
, and
c
{\displaystyle c}
, is the length of one edge of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers.
The geometric mean is one of the three classical Pythagorean means, together with the arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between (see Inequality of arithmetic and geometric means.)
Given are a fixed point ##P## and a fixed circle ##c## with the radius ##r##. Point ##P## can be anywhere inside or outside the circle. I now draw two arbitrary lines ##l_1## and ##l_2## through the point ##P## in such a way, that both lines intersect with the circle ##c## in two distinct...
Homework Statement
We have a normal 6 sided dice marked from 1 to 6. There is an equal chance to get each number at every roll. Let's put 1&2 as A type, 3&4 as B type and 5&6 as C type.
We roll the dice over and over until we get a number of every type.
Let X be the number of rolls.
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$$I(z)=I_{0}e^{-\gamma z},$$
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However, if two different wavelengths are present...
Homework Statement
Let ##\{a_n\}## be a sequence of positive numbers such that ##\lim_{n\to\infty} a_n = L##. Prove that $$\lim_{n\to\infty}(a_1\cdots a_n)^{1/n} = L$$
Homework EquationsThe Attempt at a Solution
Let ##\epsilon > 0##. There exists ##N\in\mathbb{N}## such that if ##n\ge N## then...
Homework Statement
The problem is stated as follows:
"The result in Problem 1-7 has an important generalization: If ##a_1,...,a_n≥0##, then the "arithmetic mean" ##A_n=\frac {a_1+...+a_n} {n}##
and "geometric mean"
##G_n=\sqrt[n] {a_1...a_n}##
Satisfy
##G_n≤A_n##
Suppose that ##a_1\lt A_n##...
Homework Statement
Hello.
There is a financial metric called time weighted rate of return, which is computed using the following formula:
1) if we compute daily returns, or other returns within a year:
r tw = (1+r1) x (1+r2) x...x (1+r nth year),
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Homework Statement
Hello!
I am trying to compute the bi-geometrical mean on data that contains negatives.
But before that I wanted the test the formula that accounts only for positive values using the sum of their logarithms. By doing so I don't get the result I compute by using the "usual" geo...
In a geometric mean equation, say 2 x 8 = 16, or a x b = c, what are the words we would use to describe the numbers or terms? Specifically, if you know 'a' and 'c', what do you call 'b'?
For example, in a normal multiplication, a x b = c, 'a' is the multiplicand, 'b' is the multiplier, and 'c'...
Homework Statement
GMR_{hollow cylinder}=Re^{-Kμ} where K=\frac{AR^4-R^2r^2+Br^4+r^4ln(R/r)}{(R^2-r^2)^2}, where R is the outer radius and r is the inner radius, and mu is the relative permeability. We are to determine the numerical values of A and B.
I am stumped on how to begin attempting...
Why is the geometric mean used to define the center frequency of a bandpass filter instead of the arithmetic mean?
I read in this book that
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Hi,
I have the following equation:
\gamma=\frac{1}{\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}}
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I used two properties to bound the above equation...
Homework Statement
Let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Show that the inequality
(1+R_{G})^{n} \leq V
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Homework Equations
The Attempt at a Solution
I've...
Homework Statement
let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to 1+r_{k} at the end of year k (so that r_{k} is the "return on investment" in year k). Then the value of an investment of one dollar at...
I have the following mapping (generalized geometric mean):
y(i)=exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N
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my question is - is this a contraction mapping?
in other words, does the following equation have a unique...
Hey,
(sin A + sin B + sin C)/3 >= \sqrt[3]{}(sin A*sin B*sin C)
I know this is true by Arithmetic mean always greater than geometric mean...
but is there any other way of proving this?
I don't know why I can't figure this one out tonight. I just can't think straight and I am hoping someone can help ASAP.
Here is the question:
In 1998 revenue from gambling was $651 million. In 2001 the revenue increased to $2.4 billion. What is the geometric mean annual increase for the period?
I have a question that I would like your assistance to see if I have the correct info:
In 1990 there were 9.19 million cable TV subscribers. By 2000 the number of subscribers increased to 54.87 million. What is the geometric mean annual increase for the period ?
Answer...