The geometric mean of a collection of positive real numbers is the th root of the product of the numbers. Note that if is even, we take the positive th root. It is analogous to the arithmetic mean (with addition replaced by multiplication) in the following sense: the arithmetic mean of two numbers and is the number such that , while the geometric mean of the numbers and is the number such that .
The geometric mean of the numbers 6, 4, 1 and 2 is .
The geometric mean features prominently in the Arithmetic Mean-Geometric Mean Inequality.
The geometric mean arises in geometry in the following situation: if is a chord of circle with midpoint and divides the diameter passing through it into pieces of length and then the length of line segment is the geometric mean of and .
The geometric mean also arises in the following common word problem: if a driver travels half the distance of a trip at a speed of miles per hour and the other half at a speed of miles per hour, the average speed over the whole trip is the geometric mean of and . (If the driver spent half the time of the trip at each speed, we would instead get the arithmetic mean.)