In this quick and easy post, we’ll explore the concept of the mean function and its different variations. Traditionally, the mean is defined as:
\[\frac{1}{n} \sum_{i=1}^{n} x_i\]This is known as the arithmetic mean, which provides a measure of centrality. However, there are other measures of centrality, such as the geometric mean, and harmonic mean.
The Geometric Mean
The geometric mean is a measure of centrality that calculates the side length of a cube that has the same volume as an arbitrary box with given dimensions. For example, if we have a box with a height of 2 units, a length of 1 unit, and a depth of 5 units, we can find the side length of a cube that has the equivalent volume. The geometric mean is calculated as:
\[(\prod_{i=1}^{n} x_i)^\frac{1}{n} = \sqrt[\leftroot{10} \uproot{5} n]{x_1 \cdot x_2 \cdot ... \cdot x_n}\]In our example, the volume of the box is 10 units. Therefore, the cube with the same volume would have a side length of \(a = \sqrt[\leftroot{10} \uproot{5} 3]{10}\). The geometric mean helps us find the square, cube, or hypercube that approximates the volume of the given dimensions.
Application in Finance
The geometric mean is commonly used in finance to describe proportional growth. It is particularly useful when dealing with compounding effects of percentages. For instance, when a value grows by 20%, it can be represented as 1.2 in the context of geometric means.
The Harmonic Mean
The harmonic mean is another measure of centrality, which is particularly useful in situations involving rates or ratios. It is defined as the reciprocal of the arithmetic mean of the reciprocals of a set of numbers. Mathematically, the harmonic mean is given by:
\[\text{Harmonic Mean} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}\]Unlike the arithmetic and geometric means, the harmonic mean tends to reduce the impact of extreme values. It is commonly used when dealing with rates, speeds, or other inversely proportional quantities. For example, if we have two numbers, 4 and 6, representing speeds, the harmonic mean would provide a measure of their overall average speed, considering the time taken for each speed.
Application in Finance
The harmonic mean is also employed in finance, specifically in situations where averaging ratios or rates is required. For instance, when calculating average investment returns over multiple periods, the harmonic mean can provide a more accurate measure by considering the compounding effects of rates of return.