Harmonic Mean Formula

A harmonic mean formula is an important mathematical concept that is widely used to calculate what is the average of a data set. In the financial market, it is often used in cases involving ratios, rates, etc, which leads to the identification of patterns.

The harmonic mean is the reciprocal of the arithmetic mean of reciprocals, i.e., the average calculated by dividing the number of observations in the given dataset by the sum of its reciprocals (1/Xi) of every observation in the given dataset.

The harmonic mean formula is a method that is used to find the reciprocals of all the arithmetic means of a particular set of numerical or financial data. Situations which deals with rates, prices or ratios, are users of such a method for finding the average.

In the financial market the main users of this calculation of harmonic mean formula for grouped data are the analysts and investors. They can identify patterns in prices through this method and take investment decisions.

The formula for the same is given below.

• One can see it’s the reciprocal of the normal mean.
• The harmonic mean for the normal mean is ∑ x / n, so if the formula reverses, it becomes n / ∑x. So then all the values of the denominator that one must use should be reciprocal, i.e., for the numerator, it remains “n,” but for the denominator, the values or the observations for them, we need to use reciprocal values.
• The derived value would always be less than the average or the arithmetic mean.

The harmonic mean formula for grouped data is considered to be very useful due to the fact that it takes into account all the data or entries in a particular series and cannot be calculated if any item is eliminated. It also takes into account very small and insignificant values, which may also be negative.  

However, since reciprocals is taken for each value, the entire calculation process may turn out to be complex and time taking. Moreover, a series containing a zero is very difficult to calculate because there is no reciprocal for it. So we see that this method is applicable in many cases but with some restrictions.

Let us understand the concept of harmonic mean formula in statistics with the help of some suitable examples, as given below.

Example #1

Consider a data set of the following numbers: 10, 2, 4, 7. Then, you are required to calculate the harmonic mean using the above-discussed formula.

Solution:

Use the following data for the calculation.

The Harmonic mean =  n / ∑

= 4/ (1/10 + 1/2 + 1/4 + 1/7)

= 4 / 0.99

Example #2

Mr.Vijay is a stock analyst at JP Morgan. His manager has asked him to determine the P/E ratio of the index, which tracks the stock prices of Company W, Company X, and Company Y.

Company W reports earnings of $40 million and a market capitalization of $2 billion. Company X reports $3 billion and a market capitalization of $9 billion; Company Y reports earnings of $10 billion and a market capitalization of $40 billion. Next, calculate the harmonic mean for the P/E ratio of the index.

Solution:

Use the following data for the calculation.

First, we shall calculate the P/E ratio.

P/E ratio is essentially (the market capitalization / the earnings).

• P/E of (Company W) = ($2 billion) / ($40 million) = 50
• P/E of (Company X) = ($9 billion) / ($3 billion) = 3
• P/E of (Company Y) = ($40 billion) / ($10 billion) = 4

Calculation of 1/X value

• Company W = 1/50 = 0.02
• Company X= 1/3 = 0.33
• Company Y= 1/4 = 0.25

The calculation can be done as follows,

The Harmonic mean = n / ∑

• =3/(1/50 + 1/3 + 1/4)
• =3/0.60

Example #3

Rey, a resident of northern California, is a professional sports biker and is on his tour to a beach from his home on Sunday evening around 5:00 PM EST. He drives his sports bike at 50 mph for 1^st half of the journey and 70 mph for 2^nd half from his home to the beach. What will be his average speed?

Solution:

Use the following data for the calculation.

In this example, Rey went on a journey at a certain speed, and here the average would be based on distance.

The calculation is as follows,

Here, we can calculate the Harmonic mean for the average speed of Rey’s sports bike.

The Harmonic mean = n / ∑

• =2/ (1/50 + 1/70)
• =2/ 0.03

The average speed of Rey’s sports bike is 58.33.

From the above examples we can understand how the relevant data can be extracted from financial statements of a company to calculate ratios of any other important information. The method of harmonic mean formula in statistics can also be used to calculate the average speed of any vehicle, etc because it takes into consideration the time that is taken in each part of the journey. In other words, the impact of extreme values can be minimised trough this method.

• Harmonic means, like other average formulas, have several usages. They are mainly used in finance for certain average data, such as price multiples. For example, financial multiples like the P/E ratio must not be averaged using the normal mean or the arithmetic mean because those mean are biased towards the larger values. One can also use harmonic means to identify a certain type of pattern like Fibonacci sequences that market technicians use in technical analysis.
• The harmonic mean also deals with averages of units such as rates, ratios or speed, etc. Also, it is essential to note that it is affected by the extreme values in the given data set or a given set of observations.
• The harmonic mean formula for individual series is defined rigidly and is based upon all values or observations in a given dataset or sample, and it can be suitable for further mathematical treatment. Like the geometric mean, the Harmonic mean is also not affected much by the observations or sampling fluctuations. It would be giving greater importance to the small values or the small observations, and this will be useful only when those small values or those small observations need to be given greater weight.
• Apart from finance and economics, the method of harmonic mean formula for individual series can also be used in the field of medicine or environmental science for monitoring or modelling, like the speed of flow of a river, or the average magnitude of an earthquake to provide a measure of level of seismic activity or getting health related statistics, like prevalence of a particular disease in the field of medical research.
• In the field of quality control for the purpose of manufacturing, this metric is often used to calculate what is the average time taken to produce a particular product, which will help in evaluating the efficiency level of the process. It is also successfully used in data analysis and machine learning or social science.

Thus, we see from the above points that this metric has a number of uses in the modern world, which helps in getting information that are successfully implemented in various aspects of our social and economic fields.