What is Harmonic Mean?

So, the harmonic mean is a type of numerical average. It’s usually used in situations where the average rate or rate of change needs to be calculated. The harmonic mean, often denoted as HM, can be defined as the reciprocal of the arithmetic mean of the reciprocals. This might sound a bit complicated, but don’t worry! We’ll break it down further in the following sections.

Key Components of Harmonic Mean

Before we jump into how to find the harmonic mean, let’s understand its key components. The harmonic mean is a measure of central tendency. In statistics, there are three primary measures of central tendency – mean, median, and mode. The mean can be further classified into arithmetic mean, geometric mean, and harmonic mean.

To find it, we divide the number of terms in a data series by the sum of all the reciprocal terms. It will always be the lowest as compared to the geometric and arithmetic mean.

Harmonic Mean: A Practical Example

To make things clearer, let’s take an example. Let’s say we have a sequence given by 1, 3, 5, 7. The difference between each term is 2, forming an arithmetic progression. To find the harmonic mean, we take the reciprocal of these terms:

1, 1/3, 1/5, 1/7

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 = 4 / (1 + 1/3 + 1/5 + 1/7) = 2.3864.

The Harmonic Mean Formula

Now, let’s delve into the formula for the harmonic mean. 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:

[layex]НМ\;=\;\frac n{{\displaystyle\frac1{x_1}}+{\displaystyle\frac1{x_2}}+{\displaystyle\frac1{x_3}}+…+{\displaystyle\frac1{x_n}}}[/latex]

Here, the total number of observations is divided by the sum of reciprocals of all observations.

Finding the Harmonic Mean: A Step-by-Step Guide

Wondering how to find the harmonic mean? Just follow these simple steps:

Step 1: Take the reciprocal of each term in the given data set.
Step 2: Count the total number of terms in the given data set. 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.

Example

A car travels 120 miles at a speed of 40 mph for the first part of the journey and 60 miles at a speed of 60 mph for the second part. What is the average speed of the entire journey?

Solution:

Average speed = (2 * 40 * 60) / (40 + 60) = (4800) / (100) = 48 mph

Now, let’s look at how the harmonic mean compares to the geometric mean and the arithmetic mean.

Harmonic Mean vs. Geometric Mean vs. Arithmetic Mean

Harmonic mean (HM)	Geometric Mean (GM)	Arithmetic Mean (AM)
Given a data set, the harmonic mean can be calculated by dividing the total number of terms by the sum of the reciprocal terms.	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 nth root.	To calculate this, we take the sum of all the observations in a data set and divide it by the total number of observations.
Its value is always lesser than the other two means.	Its value is always greater than the geometric mean but lesser than the arithmetic mean.	It is the highest value among all three means.
HM Formula = 1 / ((1/x₁) + (1/x₂) + … + (1/xₙ))	GM Formula = (x₁ * x₂ * … * xₙ)^(1/n)	AM Formula = (x₁ + x₂ + … + xₙ) / n

Relationship Between AM, GM, and HM

The products of the harmonic mean and the arithmetic mean (link to arithmetic article) will always be equal to the square of the geometric mean (link to geometric article) of the given data set.

Also, HM ≤ GM ≤ AM.

Problems: Find Harmonic Mean

1: A car travels 120 miles at a speed of 40 mph for the first part of the journey and 60 miles at a speed of 60 mph for the second part. What is the average speed of the entire journey?

2: A cyclist travels 20 km at a speed of 10 km/h and then 40 km at a speed of 20 km/h. What is the average speed for the entire journey?

3: A machine produces 120 units per hour, and another produces 60 units per hour. What is the average production rate of the two machines together?

4: In a pond, the population of fish doubles in size in the first year and halves in size in the second year. What is the average population size over this period?

5: An investment grows by 10% in the first year, 20% in the second year, and 5% in the third year. What is the average annual return on investment?

Wrapping Up

The harmonic mean is a helpful computation in statistics, finance, geometry, and music respectively. One aspect of it is that the formula allocates the same importance to all data points thus making it the best option for averaging formulas and ratios.

If you’re interested in exploring more about this and other mathematical concepts, head over to Mathema for a comprehensive learning experience.

Frequently Asked Questions

What is harmonic mean in statistics?

The harmonic mean in statistics is the reciprocal of the arithmetic mean of the reciprocals of a data set.

How to find the harmonic mean?

To find the harmonic mean, take the reciprocal of each term in the given data set, add all the reciprocal terms, and then divide the total number of terms by this value.

What is the harmonic mean calculator?

A harmonic mean calculator is an online tool that helps you quickly and accurately calculate the harmonic mean of a data set.

What is the difference between geometric mean and harmonic mean?

The geometric mean is found by multiplying all the terms in a data set and taking the nth root, while the harmonic mean is found by dividing the total number of terms by the sum of the reciprocals of the terms.

What is Harmonic Mean?