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: 

$HM=\frac{2ab}{a+b}$

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

$HM=\frac{n}{(\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n})}$

The Harmonic Mean formula for grouped data is: 

$HM=\frac{\sum f_i}{\sum \frac{f_i}{x_i}}$

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. 

Relationship between AM, GM, and HM: 

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

$AM\ge GM\ge HM$

• 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: 

$Speed=\frac{Distance}{time}$

$Speed1=\frac{3}{20}=0.15km/min$

$Speed2=\frac{5}{30}=0.167km/min$

$Speed3=\frac{7}{40}=0.175km/min$

$HM=\frac{n}{(\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n})}$

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

$HM=\frac{3}{18.381}$

$HM=0.1636km/min$

or

$0.1636\times 60=9.82km/h$

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

Solution: 

Here, $\sum f_i=55,\sum f_i/x_i=21.7$ Now, put these values in the formula:

$HM=\frac{\sum f_i}{\sum \frac{f_i}{x_i}}$

$HM=\frac{55}{21.7}=2.53hours$

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 = $\frac{60}{4}=20km/h$

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

Average speed of the boat: 

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

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

$v_{avg}=17.14km/h$