Harmonic Mean

Harmonic Mean[]

Formula[]

$HM={\frac {n}{({\tfrac {1}{x_{1}}}+{\tfrac {1}{x_{2}}}+{\tfrac {1}{x_{3}}}...+{\tfrac {1}{x_{n}}})}}$

Where

HM = Harmonic Mean

n = number of values

x = value (1, 2, 3 … n)

Overview[]

The Harmonic Mean (HM) is one of the three Pythagorean means (other two being arithmetic mean and geometric means). It will have the lowest value of the three, as it deprioritizes larger values and emphasizes smaller values. This is well suited to ratios and rates. Examples of rates are speed (m/s, km/hr) and cost ($/item).

HM is calculated by taking the sum of the number of values present in a set or series and dividing it by the sum of the reciprocals of all data points, In other words, the reciprocal of the arithmetic average of the reciprocals. All numbers in the data set or series must be greater than zero.

History[]

The HM (and other Pythagorean means) were founded by Pythagoras (570 BCE to 495 BCE). He used strings of varying lengths to come up with a proportional relationship between sound wave harmonies. This proportional relationship was the foundation the HM is built on.

Uses/Examples[]

HM enjoys many uses across a variety of different sectors, from general purposes to finance to physics and data science.

Simpler applications involve calculating speed. As HM gives more weight to slower speeds, it more readily reflects the excess time spent traveling that speed over a fixed difference. In finance, one of its more popular uses is that of the price to earnings ratio. This helps determine the value of shares for investors. Electrical fields can use this to calculate the total value of resistance in a parallel circuit involving individual resistors.

A specific example of HM being used in math is the Harmonic Series. This is a divergent series that is represented by the following formula;

$\sum _{n=1}^{\infty }=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}...$

It is a HM where there are an infinite amount of data values where the denominators of the reciprocals increase by one for every value. This is useful in the field of mathematics, for example, by proving that there is an infinite number of prime numbers.