Harmonic Mean

Harmonic mean is one of the measures of central tendency. We can find the harmonic mean by dividing the total number of terms, or observations in a series, by the sum of their reciprocals. Among the three means, the harmonic mean will always be the lowest. In finance, multiples such as price-to-earnings ratio are usually averaged using harmonic mean. The following are the key takeaways of the harmonic mean:

• We can define it as the reciprocal of the arithmetic mean of the reciprocals. 
   
• In finance and physics, the harmonic mean is used to average values like price multiples and other variables. 
   
• To spot patterns, including the Fibonacci sequences, market technicians often use harmonic. 
   
• When compared to other Pythagorean means, the harmonic mean gives the lowest value. 

The harmonic mean is used for analyzing data involving rates, ratios, or quantities such as speed, time, and financial multiples. It plays a vital role in the fields of physics, statistics, and finance. The harmonic mean is very helpful when a data set’s smaller values have a greater impact or significance. Suppose we have a set of observations such as x₁, x₂, x₃… x_n. Then, the reciprocal terms of this data set will be 1/x₁, 1/x₂, 1/ x₃…1/x_n. So, the formula for the harmonic mean is:

HM = n / (1/x₁ + 1/x₂ + 1/ x₃ +…1/x_n)

Here, n is the number of terms in the given data set. 

x₁, x₂, x₃… xn are the values in the given data set.  

In the formula, the total number of terms is divided by the sum of the reciprocal of each number. 

For a better understanding, take a look at the given example.

Imagine we have a sequence given by 2, 6, 10, 14. The difference between each term is 4, creating an arithmetic progression. To calculate the harmonic mean, first, we can take the reciprocals of these terms.

1/2, 1/6, 1/10, 1/14

This creates a harmonic progression. Next, we can divide the total number of terms, i.e., 4 by the sum of the reciprocal terms:

HM = 4 / (1/2, 1/6, 1/10, 1/14)

1/2, 1/6, 1/10, 1/14 = 0.5 + 0.1667 + 0.1 + 0.0714 = 0.8381

HM = n / ( 1/x₁ + 1/x₂ + 1/ x₃ +…1/x_n)

HM = 4 / 0.8381 ≈ 4.773

So, the harmonic mean of the sequence is approximately 4.77. 

The harmonic mean is a type of average that helps to calculate the average of rates like speed, price, and time. In order to calculate the harmonic mean, we have to follow certain steps. They are:

Step 1: Take each term’s reciprocal in the data set. 

Step 2: Determine how many terms are there in the given data set. Then denote the value as n. 

Step 3: Sum all the reciprocal values.

Step 4: Divide the total number of terms(n) by the sum of the reciprocal values. This will result in the harmonic mean of the data set. 

By following these steps, we can determine the harmonic mean of any given data set. 

The geometric mean and the harmonic mean are the measures of central tendencies. Both types of averages are differentiated according to their different usages and calculation processes. The major difference between these two measures of central tendency is given below:
 

• By dividing the total number of values or terms by the reciprocals of those values, we can determine the harmonic mean of a given data set. 
• Whereas, we can determine the geometric mean by multiplying all the n terms in a given data set and getting the nth root. 
   
• Compared to the arithmetic mean and the geometric mean, the value of the harmonic mean is always lower. The value of the geometric mean is less than the arithmetic mean but higher than the harmonic mean.
   
• After applying specific reciprocal transformations, we get the harmonic mean, which is the arithmetic mean of the given data set . While the geometric mean can be viewed as the arithmetic mean with specific logarithmic transformations. 
   
• The formula for the harmonic mean is:
  
  HM = n / ( 1/x₁ + 1/x₂ + 1/ x₃ +…1/x_n)
  
  Whereas, the formula for the geometric mean is:
  
  GM = (x₁ × x₂ ×… x_n)^1/n

The harmonic mean, the arithmetic mean, and the geometric mean are the three Pythagorean means. These three averages or means are very crucial in mathematics, and the relationship is referred to as AM-GM-HM inequality. 

The square of the geometric mean of a given data set is always equal to the product of the harmonic mean and the arithmetic mean.  

The arithmetic mean is always greater than or equal to the geometric mean. Also, the geometric mean is always greater than or equal to the harmonic mean.

When every value in the given data set is the same, we get AM = GM = HM. If every number is equal, all the three means will be equal. 

The formulas for a set of numbers x₁, x₂, x₃… x_n are:

AM = x₁ + x₂ + x₃+ …+ x_n / n

GM = (x1 × x2 ×… xn)1/n

HM =  n / ( 1/x₁ + 1/x₂ + 1/ x₃ +…1/x_n)

We use arithmetic mean when the values in the data share the same units.However, if the values in a given data set have different units, then we use geometric mean.. Also, we use the harmonic mean, when the values are represented in rates. 

Weighted harmonic mean is a variant of the harmonic mean. It assigns a weight to each term in the data set based on its significance. In the weighted version, each term’s impact is multiplied by a weight, whereas, in the normal harmonic mean, all terms are considered equally. The formula for weighted harmonic mean is:

WHM =  ∑ n_i =1 ​w_i  / ∑n_i = 1 w_i / x_i

Here, w_i = Weights corresponding to each term x_i

x₁, x₂,... x_n =  Values in the data set

n = total number of terms 

The weighted harmonic mean is useful when some values in a given data set are more significant than others. 

In our daily lives, we often encounter rates, ratios, and proportions, and we use the harmonic mean. When different quantities or rates need to be given equal weight, we can use the harmonic mean. It gives a greater weight to smaller values in the given data set. Here are some of the real-life applications of the harmonic mean:
 

• If we are travelling, we can use the harmonic mean to determine the average speed when covering different distances at different speeds. 
   
• In the field of electrical engineering, engineers utilize the harmonic mean to find the effective resistance of resistors that are connected in parallel. 
   
• Business professionals employ the harmonic mean when dealing with different companies and stock markets, particularly to calculate the average price-to-earnings ratio. 
   
• To calculate the oscillations and speed of a wave, scientists in the field of physics rely on this fundamental concept. 
   
• Also, in the fields of machine learning, economics, and chemistry, the harmonic mean is used to calculate the average rates and ratios.