Harmonic Mean is a type of statistical average and is calculated by dividing the total number of observations by the series' reciprocal. As a result, the reciprocal of the arithmetic mean of reciprocals is the harmonic mean. A central tendency measure is a single number that describes how data clusters around a central value. To put it simply, the three measures of central tendencies are mean, median, and mode, and Harmonic Mean is a specific important category of mean.
What is Harmonic Mean?
The harmonic mean is a type of Pythagorean 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 Formula
If we 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/x3 + ... + 1/xn]
In this case, the total number of observations is divided by the sum of all observations' reciprocals.
People also read - Mean, Median & Mode.
How to Find a Harmonic Mean?
If the given data values are a, b, c, d,..., then the steps to find the harmonic mean are as follows:
Step 1: Determine the reciprocal of each value (1/a, 1/b, 1/c, 1/d, and so on).
Step 2: Calculate the average of the reciprocals obtained in step 1.
Step 3: Finally, find the reciprocal of the average from step 2.
What Is the Difference Between Harmonic Mean and Arithmetic Mean?
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Harmonic Mean |
Arithmetic Mean |
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The sum of a group of numbers divided by the total number of the group of numbers is the arithmetic mean. |
The reciprocal of the average of the reciprocals of the data values is the harmonic mean. |
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H.M is calculated by dividing the number of observations, or series entries, by the series' reciprocal. |
The arithmetic mean, on the other hand, is simply the sum of a series of numbers divided by the number of numbers in that series |
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Harmonic Mean = n / [(1/a1)+(1/a2)+(1/a3)+…+(1/an)] |
Arithmetic Mean = (a1 + a2 + a3 +…..+an ) / n |
Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean
Pythagorean means are three means: arithmetic mean, geometric mean, and harmonic mean. The following are the formulas for three different types of means:
Arithmetic Mean = (a1 + a2 + a3 +…..+an ) / n
Harmonic Mean = n / [(1/a1)+(1/a2)+(1/a3)+…+(1/an)]
Geometric Mean = a1.a2.a3…ann
If G is the geometric mean, H is the harmonic mean, and A is the arithmetic mean, their relationship is as follows:
G=AH
Or
G2 = A.H
Harmonic Mean of Two Numbers
Let's say that we want to determine the harmonic mean of a and b. Both a and b are positive integers. Hence, by applying the aforementioned formula, we obtain,
n = 2
HM = 2 / [1/a + 1/b]
HM = (2ab) / (a + b)
Properties of Harmonic Mean
The harmonic mean has the following characteristics:
- If all of the perceptions are constants, let's say c, then c is also the value of the harmonic mean of the perceptions.
- When compared to the mathematical mean and the number arithmetic mean, or AM > GM > HM, the harmonic mean is of negligible value.
Advantages and Drawbacks of Harmonic Mean
Advanages
The harmonic mean has the following advantages:
- It is tightly packed.
- It is based on all views of a series, which means that it cannot be computed by ignoring any item in a series.
- It is capable of advancing the algebraic method.
- It produces a more reliable result when the desired outcomes are the same for all methods used.
- It assigns the most weight to the smallest item in a series.
- It can also be calculated when a series contains any negative values.
- It results in a skewed distribution of a normal distribution
- It produces a straighter curve than the A.M and G.M.
Drawbacks
The harmonic series has the following drawbacks:
- All elements of the series must be known in order to calculate this mean. We cannot calculate the harmonic mean when the elements are unknown. Given below are other demerits of harmonic mean.
- The method for calculating the harmonic mean can be time-consuming and complicated.
- This mean cannot be calculated if any term in the given series is 0.
- The harmonic mean is greatly influenced by the extreme values in a series.
Uses of Harmonic Mean
A useful property of harmonic mean is that it can be used to find multiplicative and divisor relationships between fractions without requiring a common denominator. This can be a very useful tool in industries such as finance. Some other real-world applications of harmonic mean 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 kilometers per hour.
- It can also be used to calculate the average rate because it gives equal weight to all data points in a sample.
Important Notes on Harmonic Mean
- When we want to find the reciprocal of the average of the reciprocal terms in a series, we use the harmonic mean.
- The harmonic mean is calculated as n / [1/x1 + 1/x2 + 1/x3 +... + 1/xn].
- The relationship between HM, GM, and AM is HM2 = HM AM.
- The lowest value is HM, the middle value is geometric mean, and the highest value is arithmetic mean.