## Harmonic Mean

Harmonic mean is a type of numerical average that is usually used in situations when the average rate or rate of change needs to be calculated. Harmonic mean is one of the three Pythagorean means. The remaining two are the arithmetic mean and the geometric mean. These three means or averages are very important as they see widespread use in the field of geometry and music.

if we are given a data series or a set of observations then the harmonic mean can be defined as the reciprocal of the average of the reciprocal terms. This implies that the harmonic mean of a particular set of observations is the reciprocal of the arithmetic mean of the reciprocals. In this article, we will take a detailed look at the different aspects of harmonic mean along with the relationship between all these three means (AM, GM, HM).

## What Is a Harmonic Mean?

The harmonic mean is a type of numerical average. It is calculated by dividing the number of observations by the reciprocal of each number in the series. Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.

The harmonic mean of 1, 4, and 4 is:

**The reciprocal of a number n is simply 1 / n.**

## The Basics of a Harmonic Mean

The harmonic mean helps to find multiplicative or divisor relationships between fractions without worrying about common denominators. Harmonic means are often used in averaging things like rates (e.g., the average travel speed given a duration of several trips).

The weighted harmonic mean is used in finance to average multiples like the price-earnings ratio because it gives equal weight to each data point. Using a weighted arithmetic mean to average these ratios would give greater weight to high data points than low data points because price-earnings ratios aren’t price-normalized while the earnings are equalized.

The harmonic mean is the weighted harmonic mean, where the weights are equal to 1. The weighted harmonic mean of x_{1}, x_{2}, x_{3} with the corresponding weights w_{1}, w_{2}, w_{3} is given as:

## Harmonic Mean Versus Arithmetic Mean and Geometric Mean

Other ways to calculate averages include the simple arithmetic mean and the geometric mean. An arithmetic average is the sum of a series of numbers divided by the count of that series of numbers. If you were asked to find the class (arithmetic) average of test scores, you would simply add up all the test scores of the students, and then divide that sum by the number of students. For example, if five students took an exam and their scores were 60%, 70%, 80%, 90%, and 100%, the arithmetic class average would be 80%.

The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as “the *nth* root product of *n* numbers.” The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.

The harmonic mean is best used for fractions such as rates or multiples.

## Example of the Harmonic Mean

As an example, take two firms. One has a market capitalization of $100 billion and earnings of $4 billion (P/E of 25) and one with a market capitalization of $1 billion and earnings of $4 million (P/E of 250). In an index made of the two stocks, with 10% invested in the first and 90% invested in the second, the P/E ratio of the index is:

## What is Harmonic Mean?

Harmonic mean is a type of average that is calculated by dividing the number of values in a data series by the sum of the reciprocals (1/x_i) of each value in the data series. A harmonic mean is one of the three Pythagorean means (the other two are arithmetic mean and geometric mean). The harmonic mean always shows the lowest value among the Pythagorean means.

The harmonic mean is often used to calculate the average of the ratios or rates. It is the most appropriate measure for ratios and rates because it equalizes the weights of each data point. For instance, the arithmetic mean places a high weight on large data points, while the geometric mean gives a lower weight to the smaller data points.

In finance, the harmonic mean is used to determine the average for financial multiples such as the price-to-earnings (P/E) ratio. The financial multiples should not be averaged using the arithmetic mean because it is biased toward larger values. One of the most common problems in finance that uses the harmonic mean is the calculation of the ratio of a portfolio that consists of several securities.

### Formula for Harmonic Mean

The general formula for calculating a harmonic mean is:

**Harmonic mean = n / (∑1/x_i)**

Where:

- n – the number of the values in a dataset
- x_i – the point in a dataset

The weighted harmonic mean can be calculated using the following formula:

**Weighted Harmonic Mean = (∑w_i ) / (∑w_i/x_i)**

Where:

- w_i – the weight of the data point
- x_i – the point in a dataset

### Example of Harmonic Mean

You are a stock analyst in an investment bank. Your manager asked you to determine the P/E ratio of the index of the stocks of Company A and Company B. Company A reports a market capitalization of $1 billion and earnings of $20 million, while Company B reports a market capitalization of $20 billion and earnings of $5 billion. The index consists of 40% of Company A and 60% of Company B.

Firstly, we need to find the P/E ratios of each company. Remember that the P/E ratio is essentially the market capitalization divided by the earnings.

**P/E (Company A) = ($1 billion) / ($20 million) = 50****P/E (Company B) = ($20 billion) / ($5 billion) = 4**

We must use the weighted harmonic mean to calculate the P/E ratio of the index. Using the formula for the weighted harmonic mean, the P/E ratio of the index can be found in the following way:

**P/E (Index) = (0.4+0.6) / (0.4/50 + 0.6/4) = 6.33**

Note that if we calculate the P/E ratio of the index using the weighted arithmetic mean, it would be significantly overstated:

**P/E (Index) = 0.4×50 + 0.6×4 = 22.4**

### Harmonic Mean Definition

Harmonic mean is a type of Pythagorean mean. When we divide the number of terms in a data series by the sum of all the reciprocal terms we get the harmonic mean. The value of the harmonic mean will always be the lowest as compared to the geometric and arithmetic mean.

### Harmonic Mean Example

Suppose we have a sequence given by 1, 3, 5, 7. The difference between each term is 2. This forms an arithmetic progression. To find the harmonic mean, we take the reciprocal of these terms. This is given as 1, 1/3, 1/5, 1/7 (the sequence forms a harmonic progression). Next, we divide the total number of terms (4) by the sum of the terms (1 + 1/3 + 1/5 + 1/7). Thus, the harmonic mean = 2.3864.

## Harmonic Mean Formula

If we have a set of observations given by x1, x2, x3….xn. The reciprocal terms of this data set will be 1/x1, 1/x2, 1/x3….1/xn. Thus, the harmonic mean formula is given by

## How to Find Harmonic Mean?

We can follow the steps given below to find the harmonic mean of the terms in a particular observation set.

**Step 1:**Take the reciprocal of each term in the given data set.**Step 2:**Count the total number of terms whose harmonic mean has to be determined. This will be n.**Step 3:**Add all the reciprocal terms.**Step 4:**Divide the value obtained in step 2 by the value from step 3. The resultant will give us the harmonic mean of the required number of terms.

## Difference Between Geometric Mean and Harmonic Mean

Both harmonic mean and geometric mean are measures of central tendencies. The difference between geometric mean and harmonic mean is given below:

Geometric Mean | Harmonic mean |

When we are given a data set consisting of n number of terms then we can find the geometric mean by multiplying all the terms and taking the n^{th} root. | Given a data set, the harmonic mean can be evaluated by dividing the total number of terms by the sum of the reciprocal terms. |

The value of the geometric mean is always greater than the harmonic mean but lesser than the arithmetic mean. | The value of the harmonic mean is always lesser than the other two means. |

The geometric mean can be thought of as the arithmetic mean with certain log transformations. | The harmonic mean is the arithmetic mean of the data set with certain reciprocal transformations. |

## Harmonic Mean vs Arithmetic Mean

Both types of means fall under the category of Pythagorean means. The table for the difference between harmonic mean vs arithmetic mean is given below.

Harmonic mean | Arithmetic Mean |

To find the harmonic mean we take the reciprocal of the arithmetic mean of the reciprocal terms in that data set. | To calculate the arithmetic mean we take the sum of all the observations in a data set and divide it by the total number of observations. |

The harmonic mean is the lowest value amongst the three means. | The arithmetic mean is the highest value among all three means. |

Harmonic mean cannot be used on a data set consisting of negative or zero rates. | Arithmetic mean can be calculated even if the data set has negative, positive, and zero values. |

## Relation Between Arithmetic Mean, Geometric Mean, and Harmonic Mean

The products of the harmonic mean and the arithmetic mean will always be equal to the square of the geometric mean of the given data set. To understand the relationship between the arithmetic mean, geometric mean, and harmonic mean we will take the help of the formulas. Say we have 2 numbers a and b.

n = 2

According to definition

AM = (a + b) / 2.

HM = 2ab / (a + b) or (ab)^{2}/_{a+b}

GM = **√**(ab).

Now taking the square we get GM^{2} = (ab). Using this value,

HM = GM^{2} . [2 / (a + b)]

HM = GM^{2} / AM.

Thus, we get

GM^{2} = HM × AM.

Also, HM ≤GM ≤AM.

Arithmetic mean is used when the data values have the same units. The geometric mean is used when the data set values have differing units. When the values are expressed in rates we use harmonic mean.

## Merits and Demerits of Harmonic Mean

Harmonic mean is a mathematical mean that is usually used to find the average of variables when they are expressed as a ratio of different measuring units. Given below are the merits and demerits of harmonic mean:

### Merits of Harmonic Mean

The harmonic mean is completely based on observations and is very useful in averaging certain types of rates. Other merits of the harmonic mean are given below.

- As the value of harmonic mean remains fixed thus, it is rigidly defined.
- Even if there is a sample fluctuation, the harmonic mean does not get significantly affected.
- All items of the series are required to determine the harmonic mean.

### Demerits of Harmonic Mean

To calculate the harmonic mean, all elements of the series must be known. In case of unknown elements, we cannot determine the harmonic mean. Given below are other demerits of harmonic mean.

- The method to calculate the harmonic mean can be lengthy and complicated.
- If any term of the given series is 0 then the harmonic mean cannot be calculated.
- The extreme values in a series greatly affect the harmonic mean.

## Uses of Harmonic Mean

An important property of harmonic mean is that without taking a common denominator it can be used to find multiplicative and divisor relationships between fractions. This can be a very helpful tool in industries such as finance. Given below are some other real-life applications of harmonic mean.

- The patterns in the Fibonacci series can be determined using the harmonic mean.
- Harmonic mean is used in finance when average multiples have to be evaluated.
- It can be used to calculate quantities such as speed. This is because speed is expressed as a ratio of two measuring units such as km/hr.
- It can also be used to find the average of rates as it assigns equal weight to all data points in a sample.

## Weighted Harmonic Mean

Weighted harmonic mean is used when we want to find the average of a set of observations such that equal weight is given to each data point. Let x1, x2, x3….xn be the set of observations and w1, w2, w3….wn be the corresponding weights. Then the formula for weighted harmonic mean is given as follows:

**Important Notes on Harmonic Mean**

- The harmonic mean is used when we want to find the reciprocal of the average of the reciprocal terms in a series.

The relationship between HM, GM, and AM is GM^{2} = HM × AM. Harmonic mean will have the lowest value, geometric mean will have the middle value and arithmetic mean will have the highest value.

## Examples on Harmonic Mean

**Example 1:**Find the harmonic mean of 7 and 9.

**Solution:**Using the formula we have HM = 2ab / (a + b)

a = 7 and b = 9

Thus, HM = (2 × 7 × 9) / (7 + 9) = 7.875.

**Answer:**HM = 7.875

**Example 3:** Calculate the harmonic mean if the arithmetic mean = 9.4, and geometric mean = 8.1649.**Solution:** We know that GM^{2} = HM × AM. Thus, HM = GM^{2} / AM = 8.1649^{2}/ 9.4 = 7.09.**Answer:** Harmonic mean = 7.09.

## FAQs on Harmonic Mean

### What is Harmonic Mean in Statistics?

When we take the reciprocal of the arithmetic mean of the reciprocal terms in a data set we get the harmonic mean. Furthermore, if there are certain weights associated with each observation then we can calculate the weighted harmonic mean.

### What is the Harmonic Mean of a and b?

If we want to find the harmonic mean of two non-zero numbers a and b then we use the general formula with n = 2. Thus, the formula for the harmonic mean of a and b is 2ab/a + b

### How to Calculate Harmonic Mean?

We first take the sum of the reciprocals of each term in the given data set. Then we divide the total number of terms (n) in the data set by this value to get the harmonic mean.

### What is the Difference between Geometric Mean and Harmonic Mean?

When we have a data set, the geometric mean can be determined by taking the n^{th} root of the product of all the n terms. To find the harmonic mean we divide n by the sum of the reciprocals of the terms. The harmonic mean will always have a lower value than the geometric mean.

### Is Harmonic Mean the Reciprocal of Arithmetic Mean?

Harmonic mean is not the reciprocal of the arithmetic mean. Harmonic mean is the reciprocal of the arithmetic mean of the reciprocal terms in a given set of observations.

### What are the Merits and Demerits of Harmonic Mean?

The harmonic mean has a rigid value and does not get affected by fluctuations in the sample however if the sample contains a zero term we cannot calculate the harmonic mean. Also, the formula to determine the harmonic mean can result in complex computations.

### What are Applications of Harmonic Mean?

Harmonic mean sees widespread use in geometry and music. It is also used to calculate the average of ratios as it equalizes all the data points.

## Harmonic Mean Definition

The Harmonic Mean (HM) is defined as the reciprocal of the average of the reciprocals of the data values.. It is based on all the observations, and it is rigidly defined. Harmonic mean gives less weightage to the large values and large weightage to the small values to balance the values correctly. In general, the harmonic mean is used when there is a necessity to give greater weight to the smaller items. It is applied in the case of times and average rates.

## Harmonic Mean Formula

Since the harmonic mean is the reciprocal of the average of reciprocals, the formula to define the harmonic mean “HM” is given as follows:

If x_{1}, x_{2}, x_{3},…, x_{n} are the individual items up to n terms, then,

Harmonic Mean, HM = n / [(1/x_{1})+(1/x_{2})+(1/x_{3})+…+(1/x_{n})]

## How to Find a Harmonic Mean?

If a, b, c, d, … are the given data values, then the steps to find the harmonic mean are as follows:

Step 1: Calculate the reciprocal of each value (1/a, 1/b, 1/c, 1/d, …)

Step 2: Find the average of reciprocals obtained from step 1.

Step 3: Finally, take the reciprocal of the average obtained in step 2.

## Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean

The three means such as arithmetic mean, geometric mean, harmonic means are known as Pythagorean means. The formulas for three different types of means are:

Arithmetic Mean = (a_{1} + a_{2} + a_{3} +…..+a_{n} ) / n

Harmonic Mean = n / [(1/a_{1})+(1/a_{2})+(1/a_{3})+…+(1/a_{n})]

## Harmonic Mean Uses

The main uses of harmonic means are as follows:

- The harmonic mean is applied in the finance to the average multiples like price-earnings ratio
- It is also used by the market technicians in order to determine the patterns like Fibonacci Sequences

### Merits and Demerits of Harmonic Mean

The following are the merits of the harmonic mean:

- It is rigidly confined.
- It is based on all the views of a series, i.e. it cannot be computed by ignoring any item of a series.
- It is able to advance the algebraic method.
- It provides a more reliable result when the results to be achieved are the same for the various means adopted.
- It provides the highest weight to the smallest item of a series.
- It can also be measured when a series holds any negative value.
- It produces a skewed distribution of a normal one.
- It produces a curve straighter than that of the A.M and G.M.

The demerits of the harmonic series are as follows:

- The harmonic mean is greatly affected by the values of the extreme items
- It cannot be able to calculate if any of the items is zero
- The calculation of the harmonic mean is cumbersome, as it involves the calculation using the reciprocals of the number.

### Harmonic Mean Examples

**Example 1:**

Find the harmonic mean for data 2, 5, 7, and 9.

**Solution:**

Given data: 2, 5, 7, 9

Step 1: Finding the reciprocal of the values:

½ = 0.5

⅕ = 0.2

1/7 = 0.14

1/9 = 0.11

Step 2: Calculate the average of the reciprocal values obtained from step 1.

Here, the total number of data values is 4.

Average = (0.5 + 0.2 + 0.143 + 0.11)/4

Average = 0.953/4

Step 3: Finally, take the reciprocal of the average value obtained from step 2.

Harmonic Mean = 1/ Average

Harmonic Mean = 4/0.953

Harmonic Mean = 4.19

Hence, the harmonic mean for the data 2, 5, 7, 9 is 4.19.

**Example 2:**

Calculate the harmonic mean for the following data:

x | 1 | 3 | 5 | 7 | 9 | 11 |

f | 2 | 4 | 6 | 8 | 10 | 12 |

**Solution:**

The calculation for the harmonic mean is shown in the below table:

x | f | 1/x | f/x |

1 | 2 | 1 | 2 |

3 | 4 | 0.333 | 1.332 |

5 | 6 | 0.2 | 1.2 |

7 | 8 | 0.143 | 1.144 |

9 | 10 | 0.1111 | 1.111 |

11 | 12 | 0.091 | 1.092 |

N =42 | Σ f/x = 7.879 |

The formula for weighted harmonic mean is

HM_{w} = N / [ (f_{1}/x_{1}) + (f_{2}/x_{2}) + (f_{3}/x_{3})+ ….(f_{n}/x_{n}) ]

HM_{w} = 42 / 7.879

HM_{w} = 5.331

Therefore, the harmonic mean, HM_{w} is 5.331.

## Frequently Asked Questions on Harmonic Mean

### Define harmonic mean.

The harmonic mean is defined as the reciprocal of the average of the reciprocals of the given data values.

### Mention the steps to calculate the harmonic mean.

The steps to calculate the harmonic mean are as follows:

Step 1: Find the reciprocal of the given values

Step 2: Calculate the average for the reciprocals obtained in step 1.

Step 3: Finally, calculate the reciprocal of the average obtained from step 2.

### What is the relationship between AM, GM, and HM?

If AM, GM, and HM are the arithmetic mean, geometric mean and harmonic mean, respectively, then the relationship between AM, GM and HM is GM^{2} = AM × HM

### What is the harmonic mean of a and b?

The harmonic mean of a and b is 2ab/(a+b).

As “a” and “b” are the two data values, then the harmonic mean is written as

H.M = 2 /[(1/a)+(1/b)]

H.M = 2/[(a+b)/ab]

H.M = 2ab/(a+b)

### Calculate the harmonic mean of 2 and 4.

Harmonic Mean = [2(2)(4)]/(2+4)

Harmonic Mean= 16/6

Harmonic Mean = 2.67

Hence, the harmonic mean of 2 and 4 is 2.67.

## Harmonic Mean

The harmonic mean is:

the reciprocal of the average of the reciprocals

Yes, that is a lot of reciprocals!

**Examples**

Example #1

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

**Solution:**

Use the following data for the calculation.

**Company W reports earnings of $40 million and the market capitalization of $2 billion, Company X reports earnings of $3 billion and the market capitalization of $9 billion and while Company Y reports earnings of $10 billion and the market capitalization of $40 billion. 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,

#### 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.

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

The Harmonic mean = n / ∑[1/ X_{i}]

- =2/ (1/50 + 1/70)
- =2/ 0.03

### Use and Relevance

Harmonic means, like other average formulas, they also do have several usages. They are mainly used in the field of finance to certain average data such as price multiples. The 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. Harmonic means further can also be used to identify a certain type of pattern like Fibonacci sequences that are majorly used in technical analysis by the market technicians.

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 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 with the fluctuations in observations or sampling. 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.

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