# Harmonic Mean Formula

## Harmonic Mean Formula

Harmonic Mean Formula is an important concept in Mathematics and Physics.  It is very important for students studying Science in higher education.

Mathematics, or the science of structural relationship and order, is a rational discipline that evolved from the primitive practices of counting, measuring, and describing the shapes of objects. Furthermore, logical reasoning and quantitative calculation are covered. As a result, “Mathematics” now simply refers to the study of Mathematics. Mathematical theories help students understand and solve a wide range of problems in both academic and practical contexts. The best brain exercise is probably solving mathematical puzzles.

Mathematics’ subject matter has become increasingly organised and structured as it has evolved. For centuries, mathematicians from many civilizations around the world have studied the subject. Archimedes (287-212 BC) is recognised as the founder of the academic discipline of Mathematics. He devised formulas for calculating the volume and surface area of solids. Aryabhatt, who was born in 476 CE, is known as the Father of Indian Mathematics.

Mathematics has been a crucial part of Technology and the Physical Sciences since the 17th century. Mathematics, it has recently been predicted, will play a comparable role in the quantitative aspects of the Life Sciences.

In the sixth century BC, the Pythagoreans were the first to study Mathematics as a “demonstrative science.” The term “Mathematics” comes from the Greek word “mathema,” which means “educational material.” Axioms, theorems, proofs, and postulates were developed by another mathematician named Euclid. These concepts are still widely used in modern Mathematics.

Physics is the study of matter’s structure and the interactions of the universe’s fundamental constituents. Physics (from the Greek physikos) encompasses all aspects of nature, both macro and micro. Its research interests include the nature and origin of gravitational, electromagnetic, and nuclear force fields, as well as the behaviour of objects subjected to different forces. Its ultimate goal is to develop a few comprehensive principles that connect and explain all of these disparate phenomena.

The fundamental Physical Science is Physics. Until recently, the terms Physics and Natural Philosophy were used interchangeably to refer to the Science that seeks to discover and formulate nature’s fundamental laws. The term “Physics” has come to refer to the area of Physical Science that is not covered by Astronomy, Chemistry, Geology, or Engineering as the modern sciences have developed and become more specialised. However, Physics is important in all natural sciences, and each of them has branches that focus on physical laws and measurements, with names like Astrophysics, Geophysics, Biophysics, and even Psychophysics. Physics is the fundamental study of matter, motion, and energy. Its laws are typically expressed mathematically and precisely.

Students from all classes can prepare for Mathematics and Physics with the help of the resources provided by Extramarks. Extramarks provides study tools for all classes from Class 1 to Class 12. All the resources offered by Extramarks can be downloaded in PDF format from their website and mobile application. Extramarks’ solutions will help them understand the concept better. Students must use the solutions provided by Extramarks as they are written in a step-by-step manner. With the help of Extramarks’ resources, it becomes easy for students to revise important concepts.

## What is Harmonic Mean?

Harmonic Mean Formula is a type of numerical average that is commonly used when calculating the average rate or rate of change. It is one of three Pythagorean methods. The arithmetic and geometric means are the other two. These three means or averages are very important because they are widely used in geometry and music. If one has a data series or a set of observations, one can define the Harmonic Mean Formula as the reciprocal of the average of the reciprocal terms. It is the reciprocal of the arithmetic mean of the reciprocals, in other words.

The Harmonic Mean Formula is used to calculate central tendency. Assume one wants to find a single value that can be used to describe data behaviour around a central value. A value like this is known as a measure of central tendency. There are three measures of central tendency in statistics. The mean, median, and mode are the three values. The mean is divided into three types: arithmetic mean, geometric mean, and harmonic mean.

### Harmonic Mean Definition

A type of Pythagorean mean is the harmonic mean. To calculate it, divide the number of terms in a data series by the sum of all reciprocal terms. When compared to the geometric and arithmetic means, it will always be the lowest.

### Harmonic Mean Example

Assume one has a sequence of 1, 3, 5, 7. The distinction between each term is two. This results in an arithmetic progression. One takes the reciprocal of these terms to find the harmonic mean. This is written as 1, 1/3, 1/5, and 1/7. (the sequence forms a harmonic progression). The total number of terms (4) is then divided by the sum of the terms (1 + 1/3 + 1/5 + 1/7). Thus, the Harmonic Mean Formula is calculated as 4 / (1 + 1/3 + 1/5 + 1/7) = 2.3864.

## Harmonic Mean Formula

If students have a collection of observations denoted by x1, x2, x3,…xn. This data set’s reciprocal terms will be 1/x1, 1/x2, 1/x3….1/xn. As a result, the Harmonic Mean Formula  is

HM = n / [1/x1 + 1/x2 +… + 1/xn]

In this case, the total number of observations is divided by the sum of all observations’ reciprocals.

### Harmonic Mean of Two Numbers

Assume students want to find the Harmonic Mean Formula of any two numbers in a data set, a and b. a and b are both non-zero numbers. Using the aforementioned formula, they would obtain

n = 2

HM = 2 / [1/a + 1/b]

(2ab) / (a + b) = HM

### How to Find Harmonic Mean?

To find the Harmonic Mean Formula of the terms in a specific observation set, students can use the steps outlined below.

• Step 1: Find the inverse of each term in the given data set.
• Step 2: Determine the total number of terms in the provided data set. This is going to be n.
• Step 3: Combine all of the reciprocal terms.
• Step 4: Divide the result of step 2 by the result of step 3. The resultant will give us the Harmonic Mean Formula of the number of terms required.

### Harmonic Mean Vs Geometric Mean

The difference between Harmonic Mean Formula and Geometric Mean is given below:

• Harmonic Mean: The Harmonic Mean Formula of a data set can be calculated by dividing the total number of terms by the sum of the reciprocal terms. Its worth is always less than the other two options.  It is the data set’s arithmetic mean after certain reciprocal transformations.
• Geometric Mean: When given a data set with n terms, one can calculate the geometric mean by multiplying all of the terms and taking the nth root. It always has a greater value than the Harmonic Mean Formula but a lower value than the arithmetic mean. It is analogous to the arithmetic mean with certain log transformations.

### Harmonic Mean vs Arithmetic Mean

The Difference between Harmonic Mean Formula and Arithmetic Mean is given below:

• Harmonic Mean Formula: Students calculate this by taking the reciprocal of the arithmetic mean of the reciprocal terms in that data set. It has the lowest value of the three options. It cannot be used on data sets with negative or zero rates.
• Arithmetic Mean:  Students must divide the total number of observations by the sum of all the observations in a data set to get this. It is the most valuable of the three options. It is possible to calculate it even if the data set contains negative, positive, and zero values.

### Relation Between AM, GM, and HM

The Harmonic Mean Formula (HM) and arithmetic mean (AM) products will always equal the square of the geometric mean (GM) of the given data set. One will use formulas to understand the relationship between the AM, GM, and GM. Assume students have two numbers, a and b.

n = 2

In accordance with the definition

AM = (a + b) / 2.

2ab / (a + b) or (ab) [2 / (a + b)] = HM

GM = √(ab) (ab).

Taking the square again, we get GM2 = (ab). Using this number,

[2 / (a + b)] HM = GM2

HM is equal to GM2 / AM.

As a result,

GM2 equals HM AM.

In addition, HM is smaller or equal to GM

GM is smaller or equal to AM.

Take note of the following:

When the data values have the same units, the arithmetic mean is used.

When the data set values have different units, the geometric mean is used.

When students express values in rates, they use the Harmonic Mean Formula.

## Merits and Demerits of Harmonic Mean

It is a mathematical mean that is commonly used to calculate the average of variables when expressed as a ratio of different measuring units. The following are the benefits and drawbacks of the Harmonic Mean Formula:

### Merits of Harmonic Mean

It is entirely based on observations and is extremely useful for averaging specific types of rates. Other advantages of the Harmonic Mean Formula are listed below.

• It is rigidly defined because its value remains fixed.
• Even if there is a sample fluctuation, it has no significant impact.
• The Harmonic Mean Formula must be calculated for each item in the series.

### Demerits of Harmonic Mean

All elements of the series must be known in order to calculate this mean. Students cannot calculate the Harmonic Mean Formula when the elements are unknown. Other disadvantages of the Harmonic Mean Formula are listed below.

• The method for calculating the Harmonic Mean Formula can be time-consuming and complicated.
• This mean cannot be calculated if any term in the given series is 0.
• The Harmonic Mean Formula is greatly influenced by the extreme values in a series.

### Uses of Harmonic Mean

A useful property of the Harmonic Mean Formula is that it can be used to find multiplicative and divisor relationships between fractions without requiring a common denominator. In industries such as finance, this can be a very useful tool. Other real-world applications of the Harmonic Mean Formula are listed below.

• It can be used to determine the Fibonacci series patterns.
• It is used in finance to calculate average multiples.
• It can be used to compute values such as speed. This is due to the fact that speed is expressed as a ratio of two measuring units, such as kilometres per hour.
• It can also be used to calculate the average rate because it gives equal weight to all data points in a sample.

### Weighted Harmonic Mean

When students want to find the reciprocal of the average of the reciprocal terms in a series, they can use the Harmonic Mean Formula.

The Harmonic Mean Formula is calculated as n / [1/x1 + 1/x2 + 1/x3 +… + 1/xn].

HM2 = HM AM is the relationship between HM, GM, and AM.

The lowest value is HM, the middle value is the geometric mean, and the highest value is the arithmetic mean.

### Examples on Harmonic Mean

To understand the concept of Harmonic Mean Formula better, students must see solved examples of this topic. This will help them understand how questions in the exam are asked and how they can properly answer them. Students can also take help from the various resources provided by Extramarks to prepare for their exams. They can enhance their knowledge with the help of Extramarks’ resources.

### Practice Questions on Harmonic Mean

To get fully prepared for any topic students must solve a lot of practice questions. Similarly, to prepare the topic on Harmonic Mean Formula completely, students must solve a lot of questions from this topic. They can revise and practise this topic with the help of the resources provided by Extramarks. Extramarks’ resources can be downloaded from their website and mobile application.

### 1. What is Harmonic Mean?

Harmonic Mean Formula is a type of numerical average that is commonly used when calculating the average rate or rate of change. It is one of three Pythagorean methods. The arithmetic and geometric means are the other two.

### 2. What are the various uses of the Harmonic Mean Formula?

A useful property of the Harmonic Mean Formula is that it can be used to find multiplicative and divisor relationships between fractions without requiring a common denominator. In industries such as finance, this can be a very useful tool. Other real-world applications of the Harmonic Mean Formula are listed below.
Harmonic Mean Formula can be used to determine the Fibonacci series patterns.  Harmonic Mean Formula is used in finance to calculate average multiples etc.

### 3. From where can students download the various study tools provided by Extramarks?

Students can download all the resources provided by Extramarks from their website and mobile application in PDF format.

### 4. What are the various resources offered by Extramarks?

The various resources offered by Extramarks include NCERT solutions, revision notes, past years’ papers and their solutions etc. All the resources are written by expert subject experts to help students in their studies. Students must download the resources from the website and mobile application of Extramarks.