When to use the harmonic mean rather than the arithmetic mean?

The harmonic mean is chosen when it is to average quantities that involve inverses or reciprocals such as speeds, densities, other rates etc.

Does harmonic mean to care about extreme values?

Yes, the harmonic mean is sensitive to the small values and outliers in extreme values can be greatly influenced by small low values

Is it possible for harmonic mean to take in negative values?

No, the harmonic mean is undefined for negative numbers or for zero since the reciprocal of any negative number or zero is undefined

How does the harmonic mean compare to the arithmetic and geometric mean?

The harmonic mean is less than or equal to the geometric mean, which is, in turn, less than or equal to the arithmetic mean.

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Harmonic Mean

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Harmonic Mean

The frequency components of a periodic wave or signal define the concept of harmonics: when a system vibrates at a fundamental frequency, it tends to vibrate also at integer multiples of that frequency, which are known as harmonics; they also contribute to the sound or wave pattern itself.

1.0Insight into Harmonic Mean

The harmonic mean, a type of average value that is used to represent rates or ratios, such as the average speed or work efficiency, has been formulated in such a way that it is defined as the reciprocal of the arithmetic mean of the reciprocals of the data points. Mathematically, the harmonic mean can be expressed for two numbers a and b, by the following Harmonic Mean formula for ungrouped data:

$H M = \frac{2 ab}{a + b}$

For n numbers x₁, x₂, ……, x_n, the harmonic mean is

$H M = \frac{n}{( \frac{1}{x _{1}} + \frac{1}{x _{2}} + ... + \frac{1}{x _{n}} )}$

The Harmonic Mean formula for grouped data is:

$H M = \frac{\sum f _{i}}{\sum \frac{f _{i}}{x _{i}}}$

Here, f_i = frequency of the data points, x_i = class mark or midpoint of the class interval

2.0Arithmetic Mean, Geometric Mean, and Harmonic Mean

In statistics, harmonic mean (HM), geometric mean (GM), & arithmetic mean (AM) are the three main types of means that are widely used to describe a data set. Arithmetic Mean, Geometric Mean, and Harmonic Mean formulas are used in different problems for different purposes depending on the nature of the data.

Arithmetic Mean

The most commonly used average is the arithmetic mean. The formula for calculating AM of two numbers a and b is given as the summation of all the values divided by the number of values.

$A M = \frac{a + b}{2}$

Or,

$A M = \frac{x _{1} + x _{2} + ......... x _{n}}{n}$

Geometric Mean

The geometric mean of the set of values can be computed by taking the n^th root of the product of the numbers. For example, it helps one find rates of growth. The formula for GM of two numbers a and b:

$GM = ab$

Or,

$GM = x_{1} \times x_{2} ...... \times x_{n}$

Harmonic Mean

The harmonic mean is computed as the reciprocal of the arithmetic mean of the reciprocals of the data points.

The formula for HM of two numbers a and b:

$H M = \frac{2 ab}{a + b}$

Or,

$H M = \frac{n}{( \frac{1}{x _{1}} + \frac{1}{x _{2}} + ... + \frac{1}{x _{n}} )}$

Relationship between AM, GM, and HM:

The three means, namely – Arithmetic Mean, Geometric Mean, and Harmonic Mean, are related by the following inequality relationship:

$A M \geq GM \geq H M$

It means that the arithmetic mean of any data set is always greater than or equal to the geometric mean, which in turn is greater than or equal to the Harmonic mean.
The equality case only implies when all the data points in a given data set are equal.
The relationship is important in getting to know how the data point behaves and helps in identifying how they are spread out or bound together around a central value.

3.0Harmonic Mean Examples with Solutions

Problem 1: A person runs 3 km in 20 minutes, then 5 km in 30 minutes, and finally 7 km in 40 minutes. What is his average speed for the entire journey using the harmonic mean?

Solution: For calculating the Harmonic mean of the speed, we need to calculate the speed of each data point:

$Sp ee d = \frac{D i s t an ce}{t im e}$

$Sp ee d 1 = \frac{3}{20} = 0.15 km / min$

$Sp ee d 2 = \frac{5}{30} = 0.167 km / min$

$Sp ee d 3 = \frac{7}{40} = 0.175 km / min$

$H M = \frac{n}{( \frac{1}{x _{1}} + \frac{1}{x _{2}} + ... + \frac{1}{x _{n}} )}$

$H M = \frac{3}{( \frac{1}{0.15} + \frac{1}{0.167} + \frac{1}{0.175} )}$

$H M = \frac{3}{18.381}$

$H M = 0.1636 km / min$

or

$0.1636 \times 60 = 9.82 km / h$

Problem 2: The following table shows the time taken by students to complete an assignment. Find the harmonic mean time taken:

Time	Frequency
1-2	12
2-3	18
3-4	15
4-5	10

Solution:

Class Interval	f_i	x_i	f_i/x_i
1-2	12	1.5	8
2-3	18	2.5	7.2
3-4	15	3.5	4.3
4-5	10	4.5	2.2
Total	55		21.7

Here, $\sum f_{i} = 55, \sum f_{i} / x_{i} = 21.7$ Now, put these values in the formula:

$H M = \frac{\sum f _{i}}{\sum \frac{f _{i}}{x _{i}}}$

$H M = \frac{55}{21.7} = 2.53 h o u rs$

Problem 3: A boat travels 60 kilometres downstream in 3 hours and 60 kilometres upstream in 4 hours. Find the average speed for the entire journey using the harmonic mean.

Solution: Let the speed of the boat upstream = u

Let the speed of the boat downstream = v

Speed of boat upstream = $\frac{60}{4} = 20 km / h$

Speed of boat downstream = $\frac{60}{15} = 15 km / h$

Average speed of the boat:

$v_{a vg} = \frac{2 uv}{u + v}$

$v_{a vg} = \frac{2 \times 20 \times 15}{20 + 15} = \frac{600}{35}$

$v_{a vg} = 17.14 km / h$

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Harmonic Mean

The frequency components of a periodic wave or signal define the concept of harmonics: when a system vibrates at a fundamental frequency, it tends to vibrate also at integer multiples of that frequency, which are known as harmonics; they also contribute to the sound or wave pattern itself.

1.0Insight into Harmonic Mean

The harmonic mean, a type of average value that is used to represent rates or ratios, such as the average speed or work efficiency, has been formulated in such a way that it is defined as the reciprocal of the arithmetic mean of the reciprocals of the data points. Mathematically, the harmonic mean can be expressed for two numbers a and b, by the following Harmonic Mean formula for ungrouped data:

$H M = \frac{2 ab}{a + b}$

For n numbers x₁, x₂, ……, x_n, the harmonic mean is

$H M = \frac{n}{( \frac{1}{x _{1}} + \frac{1}{x _{2}} + ... + \frac{1}{x _{n}} )}$

The Harmonic Mean formula for grouped data is:

$H M = \frac{\sum f _{i}}{\sum \frac{f _{i}}{x _{i}}}$

Here, f_i = frequency of the data points, x_i = class mark or midpoint of the class interval

2.0Arithmetic Mean, Geometric Mean, and Harmonic Mean

In statistics, harmonic mean (HM), geometric mean (GM), & arithmetic mean (AM) are the three main types of means that are widely used to describe a data set. Arithmetic Mean, Geometric Mean, and Harmonic Mean formulas are used in different problems for different purposes depending on the nature of the data.

Arithmetic Mean

The most commonly used average is the arithmetic mean. The formula for calculating AM of two numbers a and b is given as the summation of all the values divided by the number of values.

$A M = \frac{a + b}{2}$

Or,

$A M = \frac{x _{1} + x _{2} + ......... x _{n}}{n}$

Geometric Mean

The geometric mean of the set of values can be computed by taking the n^th root of the product of the numbers. For example, it helps one find rates of growth. The formula for GM of two numbers a and b:

$GM = ab$

Or,

$GM = x_{1} \times x_{2} ...... \times x_{n}$

Harmonic Mean

The harmonic mean is computed as the reciprocal of the arithmetic mean of the reciprocals of the data points.

The formula for HM of two numbers a and b:

$H M = \frac{2 ab}{a + b}$

Or,

$H M = \frac{n}{( \frac{1}{x _{1}} + \frac{1}{x _{2}} + ... + \frac{1}{x _{n}} )}$

Relationship between AM, GM, and HM:

The three means, namely – Arithmetic Mean, Geometric Mean, and Harmonic Mean, are related by the following inequality relationship:

$A M \geq GM \geq H M$

It means that the arithmetic mean of any data set is always greater than or equal to the geometric mean, which in turn is greater than or equal to the Harmonic mean.
The equality case only implies when all the data points in a given data set are equal.
The relationship is important in getting to know how the data point behaves and helps in identifying how they are spread out or bound together around a central value.

3.0Harmonic Mean Examples with Solutions

Problem 1: A person runs 3 km in 20 minutes, then 5 km in 30 minutes, and finally 7 km in 40 minutes. What is his average speed for the entire journey using the harmonic mean?

Solution: For calculating the Harmonic mean of the speed, we need to calculate the speed of each data point:

$Sp ee d = \frac{D i s t an ce}{t im e}$

$Sp ee d 1 = \frac{3}{20} = 0.15 km / min$

$Sp ee d 2 = \frac{5}{30} = 0.167 km / min$

$Sp ee d 3 = \frac{7}{40} = 0.175 km / min$

$H M = \frac{n}{( \frac{1}{x _{1}} + \frac{1}{x _{2}} + ... + \frac{1}{x _{n}} )}$

$H M = \frac{3}{( \frac{1}{0.15} + \frac{1}{0.167} + \frac{1}{0.175} )}$

$H M = \frac{3}{18.381}$

$H M = 0.1636 km / min$

or

$0.1636 \times 60 = 9.82 km / h$

Problem 2: The following table shows the time taken by students to complete an assignment. Find the harmonic mean time taken:

Time	Frequency
1-2	12
2-3	18
3-4	15
4-5	10

Solution:

Class Interval	f_i	x_i	f_i/x_i
1-2	12	1.5	8
2-3	18	2.5	7.2
3-4	15	3.5	4.3
4-5	10	4.5	2.2
Total	55		21.7

Here, $\sum f_{i} = 55, \sum f_{i} / x_{i} = 21.7$ Now, put these values in the formula:

$H M = \frac{\sum f _{i}}{\sum \frac{f _{i}}{x _{i}}}$

$H M = \frac{55}{21.7} = 2.53 h o u rs$

Problem 3: A boat travels 60 kilometres downstream in 3 hours and 60 kilometres upstream in 4 hours. Find the average speed for the entire journey using the harmonic mean.

Solution: Let the speed of the boat upstream = u

Let the speed of the boat downstream = v

Speed of boat upstream = $\frac{60}{4} = 20 km / h$

Speed of boat downstream = $\frac{60}{15} = 15 km / h$

Average speed of the boat:

$v_{a vg} = \frac{2 uv}{u + v}$

$v_{a vg} = \frac{2 \times 20 \times 15}{20 + 15} = \frac{600}{35}$

$v_{a vg} = 17.14 km / h$

Frequently Asked Questions

When to use the harmonic mean rather than the arithmetic mean?

Does harmonic mean to care about extreme values?

Is it possible for harmonic mean to take in negative values?

How does the harmonic mean compare to the arithmetic and geometric mean?

Frequently Asked Questions

When to use the harmonic mean rather than the arithmetic mean?

Does harmonic mean to care about extreme values?

Is it possible for harmonic mean to take in negative values?

How does the harmonic mean compare to the arithmetic and geometric mean?

Join ALLEN!

Harmonic Mean

1.0Insight into Harmonic Mean

2.0Arithmetic Mean, Geometric Mean, and Harmonic Mean

Relationship between AM, GM, and HM:

3.0Harmonic Mean Examples with Solutions

Table of Contents

Join ALLEN!

Harmonic Mean

1.0Insight into Harmonic Mean

2.0Arithmetic Mean, Geometric Mean, and Harmonic Mean

Relationship between AM, GM, and HM:

3.0Harmonic Mean Examples with Solutions

Table of Contents