Geometric mean

The geometric mean is a mean value; it is in the statistics a suitable mediocrity for sizes, of which the product is interpreted instead of the sum, such as ratios or growth rates.

Definition

The geometric mean of the numbers represented by the - th root of the product of the numbers:

Properties

Unlike the arithmetic mean, the geometric mean is defined only for non-negative numbers and mostly only useful for strictly positive real numbers, because if a factor is equal to zero, the entire product is already zero. For complex numbers, it is not used, since the complex roots are ambiguous.

The inequality of the arithmetic and geometric means indicates that the geometric mean is never greater than the arithmetic mean. Equivalently, the following applies:

That is, the logarithm of the geometric mean of the arithmetic mean of the logarithms, wherein the base of the logarithm may be arbitrary, but must be the same on both sides.

Similar to the weighted arithmetic mean can be defined a weighted geometric mean of the weights:

Said. The arithmetic- geometric mean is a number which lies between the arithmetic and geometric mean.

Moreover, for and

With the arithmetic and the harmonic mean.

Application Examples

• The geometric mean of two values ​​, e.g., of and: .
• The mean of a doubling and subsequent eight-fold increase of a bacterial culture is a fourfold increase (not a multiplication by a factor of 5).
• A balance will bear interest during the first year of two percent in the second year with seven and in the third year of five percent. Which of the three years constant interest rate would have resulted in the conclusion the same capital?

Balances at the end of the third year:

Written or interest factors

With a constant interest rate and related interest factor is calculated at the end of a balance of

With results

And thus the average interest factor is computed as

The average interest rate is thus approx. In general, the average interest factor thus calculated from the geometric mean of the interest factors of the individual years. Because of the inequality of the arithmetic and geometric mean is the average rate less than or at best equal to the arithmetic mean of the rates, which in this example.

Geometric interpretation

The geometric mean of two numbers, and provides the side length of a square, which has the same area as a rectangle with the side lengths. This fact is illustrated by the geometric quadrature of the rectangle.

Corresponds to the geometric mean of the same three numbers in the side length of a cube, the volume is equal to the square of the three side lengths, and accordingly the dimensional figures in the side lengths of hypercubes.