The Harmonic Mean Formula

Formula

H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}

Used to calculate: The average of a set of nonzero values using the reciprocals of those values.

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What Is the Harmonic Mean Formula?

The harmonic mean formula calculates a type of average by taking the reciprocal of the arithmetic mean of the reciprocals of the values. It is particularly useful for averaging rates and ratios when the numerator or reference quantity remains constant.

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Formula and Variables

H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}

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How to Use the Formula

Take the reciprocal of each value in the data set and add the reciprocals together. Determine the number of values, then divide that number by the sum of the reciprocals to obtain the harmonic mean.

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When Is This Formula Used?

The harmonic mean is commonly used to average rates or ratios when the quantities being compared have a common numerator or fixed reference quantity.

Common uses:

– Averaging speeds over equal distances
– Averaging rates and ratios under appropriate fixed-quantity conditions
– Analyzing financial and statistical ratios

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Important Notes

– For the standard statistical use of the harmonic mean, the values are generally positive.
– No value $x_i$ can equal zero because its reciprocal would be undefined; additionally, the sum of the reciprocals must not equal zero for the general real-valued formula.
– The values should represent compatible quantities, and the harmonic mean has the same unit as those values.
– A common point of confusion is using the harmonic mean for any collection of rates. Whether it is appropriate depends on how the underlying quantities are weighted or held constant.

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Alternate Forms

Alternate form:

H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}

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Related Formulas

• The Arithmetic Mean Formula
• The Weighted Mean Formula
• The Geometric Mean Formula