In the time of Pythagoras, there were only three means (Bakker 2003; Brown 1975; Huffman 2005), the arithmetic, the geometric, and third that was called subcontrary, but the “name of which was changed to harmonic by Archytas of Tarentum and Hippasus and their followers, because it manifestly embraced the ratios of what is harmonic and melodic” (Huffman 2005, p. 164). The harmonic mean is a measure of location used mainly in particular circumstances – when the data consists of a set of rates, such as prices ($/kilo), speeds (mph), or productivity (output/manhour). lt is defined as the reciprocal of the arithmetic mean of the reciprocals of the values.
The harmonic mean of n numbers x 1, x 2, …, x n is calculated in the following way:
As a simple example, the harmonic mean of three numbers, 2, 5, and 10 is equal to
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References and Further Reading
Bakker A (2003) The early history of average values and implications for education. J Stat Educ 11:1
Brown M (1975) Pappus, Plato and the harmonic mean. Phronesis 20(2):173–184
Chou Y (1989) Statistical analysis with business and economic applications. Elsevier, New York
Ferger WF (1931) The nature and use of the harmonic mean. J Am Stat Assoc 26(173):36–40
Francis A (2004) Business mathematics and statistics, 6th edn. Cengage Learning Business Press
Haans A (2008) What does it mean to be average? The miles per gallon versus gallons per mile paradox revisited. Pract Assess Res Eval 13(3)
Hand DJ (1994) Deconstructing statistical questions (with discussion). J R Stat Soc A 157:317–356
Huffman CA (2005) Archytas of Tarentum: Pythagorean, philosopher, and mathematician king. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Komić, J. (2011). Harmonic Mean. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_645
Download citation
DOI: https://doi.org/10.1007/978-3-642-04898-2_645
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04897-5
Online ISBN: 978-3-642-04898-2
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering