Harmonic Mean: Definition, Formula, Weighted Harmonic Mean, Examples

Harmonic Mean is a form of numerical average. It is computed by dividing the total number of observations by the reciprocal of each number in the series. As a result, harmonic mean is the reciprocal of the arithmetic mean of reciprocals. A central tendency measure is a single number that describes how a set of data clusters around a core value. To simplify, mean, median, and mode are the three measures of central tendencies, and Harmonic Mean is a specific important category of mean.

Key words- harmonic mean, mean, arithmetic, numerical average, central tendency

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What is Harmonic Mean?

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Harmonic Mean (HM) is defined as the reciprocal of the average of the data values' reciprocals. It is rigorously defined and based on all observations. To appropriately balance the numbers, harmonic mean provides less weight to high values and more weight to small values. In general, harmonic mean is applied when it is necessary to give more weight to smaller components. 

• It is used to calculate times and average rates.
• It is majorly utilized when there is a need to give more prominent load to the more modest items.
• Consonant mean is regularly used to ascertain the normal of the proportions or paces of the given qualities.
• It is the most suitable measure for proportions and rates since it levels the loads of every information point.

Harmonic Mean Definition

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Harmonic Mean Formula

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Since the harmonic mean is the proportional of the normal of reciprocals, the equation to characterize the consonant signify "HM" is given as follows: 

On the off chance that x1, x2, x3,… , xn are the singular things up to n terms, then, at that point, 

Harmonic Mean, HM = \(\frac{n}{\frac{1}{x1}+\frac{1}{x_2}+\frac{1}{x_3}....+\frac{1}{x_n}}\)

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Properties of Harmonic Mean

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Here are some properties of Harmonic Mean-

• On the off chance that every one of the perceptions taken are constants, say c, the harmonic mean of the perceptions is likewise c.
• The harmonic mean has minimal worth when contrasted with the mathematical mean and the number arithmetic mean i,e AM > GM > HM.

Also Read: Sequence and Series

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Uses of Harmonic Mean

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Here are some of the major uses of Harmonic Mean:

• Under specific conditions, determining average pricing, average speed, and so on.
• In the financial industry, the harmonic mean is important for calculating average multiples such as the price-earnings ratio.
• It is also used in the calculation of Fibonacci sequences.

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Weighted Harmonic Mean

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Weighted harmonic mean is a subset of harmonic mean in which all weights are equal to one. It is analogous to the basic harmonic mean. If the frequencies "f" are assumed to represent the weights "w," the harmonic mean is computed as follows:

If x1, x2,...., xn are n items with corresponding frequencies f1, f2,...., fn, the weighted harmonic mean is:

HMw = \(\frac{n}{\frac{f1}{x1}+\frac{f2}{x2}+\frac{f3}{x3}+\frac{f4}{x4}....+\frac{fn}{xn}}\)

Or,  Wwx = HM

Where w denotes weight and x denotes the variable

Individual values are represented by a1, a2,...an.

Also Read: Section Formula in Coordinate Geometry

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Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean

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Pythagorean means are three means in statistics that include arithmetic, geometric, and harmonic means.

The following are the formulae for three different classifications of means:

\(\frac{a_1+a_2+a_3...+a_n}{n}\)

\(\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}....+\frac{1}{a_3}}\)

Geometric Mean = na1 x a2 x a3 xan

If G is the geometric mean, H is the harmonic mean, and A is the arithmetic mean, their connection is given by-

AH = \({\sqrt{GH}}\)

If a and b are positive numbers, then

Arithmetic Mean (AM) =\(\frac{a+b}{2}\)

Geometric Mean (GM) = \({\sqrt{ab}}\)

Harmonic Mean(HM) = \(\frac{2ab}{a+b} = \frac{(GM)^2}{AM}\)

Also Read: Mean of Grouped Data

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Harmonic Mean: Merits and Demerits

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Following are the benefits and drawbacks of harmonic mean:

Merits

1. It is narrowly defined
2. It is based on all of the observations.
3. It is least impacted by sampling variation.
4. It emphasises the value of minor details.
5. It can be further algebraically treated.

Demerits

1. Calculation is tough.
2. It does not provide equal weight to each item.
3. It is possible that it is not represented in the real data.
4. It does not have a negative value defined.

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Things to Remember

• In the financial industry, the harmonic mean is important for calculating average multiples such as the price-earnings ratio.
• The harmonic mean has minimal worth when contrasted with the mathematical mean and the number arithmetic mean i,e AM > GM > HM
• Harmonic Mean, HM = n/[(1/x1)+(1/x2)+(1/x3)+… +(1/xn)]
• The harmonic mean is by and large utilized when there is a need to give more prominent load to the more modest items.

Also Read:

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Previous Year’s Questions

1. If the value of mode and mean is 60 …? (JEEA 2013)
2. The mean square deviation of a set of nn observation x₁,x₂,...x_n about a point cc is defined as 1n∑i=1n(xi−c)₂…? (BITS Admission Test 2015)
3. The arithmetic mean of the data 0,1,2,……,n with frequencies…?(BITS Admission Test 2015)
4. If coefficient of variation is 60 and standard deviation is 24, then Arithmetic mean is?  (KCET 2017)