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, n, 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 which equals 4; that is . As another example, the geometric mean of three numbers 1, ½, ¼ is the cube root (i.e., the third root) of their product (0.125), which is 1/2; that is .

The geometric mean can also be understood in terms of geometry. The geometric mean of two numbers, a and 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 and b. That is, what is n such that . Similarly, the geometric mean of three numbers, a, b, and c, is the length of one side of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers; that is, .

The geometric mean only applies to positive numbers. 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.

Calculation

The geometric mean of a data set [a₁, a₂, ..., a_n] is given by

The geometric mean of a data set is less than or equal to the data set's arithmetic mean (the two means are equal if and only if all members of the data set are equal). This allows the definition of the arithmetic-geometric mean, a mixture of the two which always lies in between.

The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (a_n) and (h_n) are defined:

and

then a_n and h_n will converge to the geometric mean of x and y.

This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by Bolzano-Weierstrass theorem) and the fact that geometric mean is preserved:

Replacing arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.

Relationship with arithmetic mean of logarithms

By using logarithmic identities to transform the formula, we can express the multiplications as a sum and the power as a multiplication.

This is sometimes called the log-average. It is simply computing the arithmetic mean of the logarithm transformed values of $a\_i$ (i.e., the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale, i.e., it is the generalised f-mean with f(x) = log x.

Applications

The geometric mean may be more appropriate than the arithmetic mean for describing percentage growth.

Suppose an orange tree yields 100 oranges one year, then 180, 210 and 300 the following years, so the growth is 80%, 16.7% and 42.9% for each of the years. Using the arithmetic mean, we can calculate an average growth as 46.5% (80% + 16.7% + 42.9% divided by 3). However, if we start with 100 oranges and let it grow with 46.5% for three years, the result is 314 oranges, not 300.

Instead we can use the geometric mean. Growing with 80% corresponds to multiplying with 1.80, so we take the geometric mean of 1.80, 1.167 and 1.429, i.e. $\backslash sqrt[3]\{1.80\; \backslash times\; 1.167\; \backslash times\; 1.429\}\; =\; 1.443$, thus the "average" growth per year is 44.3%. If we start with 100 oranges and let the number grow with 44.3% each year, the result is 300 oranges.

Probably the most common example of percentage growth is interest rates, e.g. a savings account where the bank pays a certain percentage growth per year.

Notes and references

See also