Harmonic Mean Formula: Formula, Properties, Uses & Examples

Harmonic mean formula in statistics helps in finding the multiplicative or divisor connection between fractions. It does not affect the common denominators. They are often used in averaging the rates.

• The harmonic mean formula can be expressed in terms of the reciprocal of the arithmetic mean.
• It is a type of Schur-concave function.
• The formula is the most common example for average.
• It is just like the weighted harmonic mean but with the weights equivalent to 1.
• The harmonic mean formula cannot be made large just by changing the value to higher ones.
• It is used in the field of finance to calculate the price-to-earnings ratio.
• The formula is a type of Pythagorean mean.

The harmonic mean formula is given as:

Harmonic Mean Formula = n/ (1/x1, 1/x2, 1/x3….1/xn).

Read More: Difference Between Variance and Standard Deviation

Key Terms: Harmonic Mean, Harmonic Mean Formula, Average, Pythagorean Mean, Arithmetic Mean, Weights, Fractions, Denominators, Ratio

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What is Harmonic Mean?

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Harmonic mean is a measure of central tendency and is a type of Pythagorean mean. It can be described as the reciprocal of the arithmetic mean of reciprocals of the observations. 

• It gives less importance to the larger values and more importance to the smaller values. 
• Harmonic Value is generally used when calculating the average of the ratios or rates of given values. 
• It equalizes the weights of each data point; hence, it is the most appropriate measure for ratios and rates.
• The value is less in comparison to arithmetic and geometric mean.

Harmonic Mean

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Harmonic Mean Formula

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Harmonic mean is commonly used for calculating the average set of numbers. For calculating this mean, the total number of values will be averaged and divided by the sum of the reciprocals of all the values of a given set.

• For instance, we are given a set of observations with values as x1,x2,x3,…xn.
• Then the reciprocal terms of the given dataset will be 1/x1, 1/x2, 1/x3….1/xn.
• Therefore, from the given set of values, the harmonic mean formula will be given by:

Harmonic mean, HM = n/ (1/x1, 1/x2, 1/x3….1/xn).

Harmonic Mean Formula

Read More: Mean Value Theorem

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Weighted Harmonic Mean

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Weighted harmonic mean is a mean which is used when trying to calculate the average of a set of observations such that equal weight is provided to each data point. Let’s consider a set with values of observation given as x1,x2,x3,…xn and weights as w1,w2,w3,…wn respectively. Then the formula for weighted harmonic mean can be given as:

Weighted HM = ∑i=1nw_i / (∑i=1nw_i/x_i)

For normalized weights, the sum of all weights will be equal to 1. That is, w1,w2,w3,…wn = 1.

Read More: Statistical Inference

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Properties of Harmonic Mean Formula

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Harmonic means possess few properties which make them different from other types of means. Some important properties of harmonic mean are as follows:

• For all the observations, say c, then the harmonic means calculated of the observations will also be c.
• The harmonic mean can also be evaluated for the series having any negative values.
• If any of the values of a given series is 0 then its harmonic mean cannot be determined as the reciprocal of 0 doesn’t exist.
• If in a given series all the values are neither equal nor any value is zero.
• The harmonic mean calculated will be lesser than the geometric mean and arithmetic mean.
• When compared to geometric mean and arithmetic mean, the harmonic mean possesses the least value i.e. AM > GM > HM.

Harmonic Mean: Properties

Read More: Mode of Grouped Data

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Uses of Harmonic Mean Formula

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One of the important properties of the harmonic mean is that instead of taking a common denominator, it can be utilized in finding the multiplicative and divisor relationships between fractions. Listed below are a few of the common real-life applications of harmonic mean formula:

• Harmonic Mean Formula can be used to determine the patterns of a Fibonacci series.
• It can be used to calculate average speed prices, average speed, etc.
• While evaluating average multiples in finance, harmonic mean can be used.
• The harmonic mean formula can be used for calculating speed as it can be expressed as a ratio of two measuring units, such as km/hr.
• This can also calculate the average rates of a given sample as it assigns equal weight to all data points of the sample.

Read More: Mean of Grouped Data

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Steps for Calculating Harmonic Mean Formula

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Usually, the harmonic mean is the reciprocal of the arithmetic mean. It gives less importance to larger values and more importance to smaller values to maintain the balance of the values properly. Below mentioned are the simple steps to calculate the harmonic mean of a given set of values:

• Begin by taking the reciprocal of every term in the data set provided.
• Consider the count of the total number of terms whose harmonic mean is to be calculated as ‘n’.
• Now, add the reciprocal terms
• At last, divide the value obtained in the 2nd step by the value obtained from the 3rd step.
• The result obtained at the end is the harmonic mean of the required number of terms.

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Things to Remember

• Harmonic Mean Formula is considered a weighted mean with the weights equivalent to 1.
• The reciprocal of the arithmetic mean is termed the harmonic mean.
• It can also be represented as 1/n.
• The application of harmonic mean is used in the field of finance.
• It calculates the average data, such as price multiples.
• Market technicians can also use it for identifying patterns such as Fibonacci sequences.

Read More: Sampling Error Formula

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