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Welcome to our Math lesson on The Geometric Mean, this is the third lesson of our suite of math lessons covering the topic of Properties of Proportion. Geometric Mean, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Before explaining what the geometric mean is, it is worth having a quick look at the concept of arithmetic mean, x ̅, which is a concept that represents the simple average of a set of values. Arithmetic mean of a set of values a1, a2, ⋯, an is calculated through the formula
For example, if four measurements taken give the values 20, 23, 19 and 22 respectively, their arithmetic mean is
On the other hand, if we have a proportion that meets the condition
then, the number x is called the geometric mean of a and b.
From the above proportion, we can write
For example, if
where a = 9 and b = 4, then, the geometric mean x of 9 and 4 is
We can extend the meaning of geometric mean for more than two numbers. Thus, if we have a set of n numbers, from a1 to an, we obtain for their geometric mean x:
Geometric mean is a very important concept used in finance and banking. More specifically, the geometric mean is extremely useful when numbers in the series are not independent of each other or if numbers tend to make large fluctuations. In a set of measurements it helps determine the deflection from the true value of the quantity measured when the arithmetic mean (we will explain it below too) is not suitable to use. Let's consider an example which involves the calculation of geometric mean.
A rectangular container has the dimensions 25 cm × 8 cm × 5 cm. The container is melted down and the material is used to produce a cube with the same volume. What is the side length of the cube?
From geometry, it is known that the volume of rectangular container (cuboid) is
where a, b and c are the dimensions of the cube (a = 25 cm, b = 8 cm and c = 5 cm). Therefore, the volume of container is
The cube produced when the container is melted down must have the same volume (1000 cm3). Given that the volume of cube is x3, where x is the side length of the cube, we use the concept of geometric mean to calculate it. Hence,
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