harmonic mean

English

Noun

harmonic mean (plural harmonic means)

1. (mathematics) A type of measure of central tendency calculated as the reciprocal of the mean of the reciprocals, i.e., $H={n \over {1 \over x_{1}}+{1 \over x_{2}}+\cdots +{1 \over x_{n}}}$.

   The harmonic mean of two numbers $a$ and $c$ is the number $b$ such that ${a-b \over a}={b-c \over c}$. Combining the two ratios: ${b-c \over c}={(a-b)+(b-c) \over a+c}={a-c \over a+c}$. Rotating the proportion 90° clockwise: ${c \over a+c}={b-c \over a-c}$; transposing twice to isolate the $b$: $b=c+c\left({a-c \over a+c}\right)$. Likewise $b=a-a\left({a-c \over a+c}\right)$; these are both equivalent to the more typical, symmetrical equations: $b={2ac \over a+c}={2 \over {1 \over a}+{1 \over c}}$.

Hypernyms

• (statistics): measure of central tendency
• Pythagorean mean

Derived terms

• contraharmonic mean

Translations

type of average

• Chinese:

  Mandarin: 調和平均數／调和平均数 (tiáohé píngjūnshù)

• Czech: harmonický průměr m
• Finnish: harmoninen keskiarvo
• French: moyenne harmonique
• German: harmonisches Mittel n
• Hindi: हरात्मक माध्य (harātmak mādhya)
• Hungarian: harmonikus közép
• Indonesian: rerata harmonik
• Japanese: 調和平均 (ja) (chōwaheikin)
• Korean: 조화 평균 (johwa pyeonggyun)
• Polish: średnia harmoniczna f
• Portuguese: média harmónica f (Portugal), média harmônica f (Brazil)
• Russian: сре́днее гармони́ческое n (srédneje garmoníčeskoje)
• Serbo-Croatian: harmonijska sredina f
• Spanish: media armónica f
• Swedish: harmoniskt medelvärde (sv) n
• Tagalog: suwatuhing tamtaman

See also

• arithmetic mean
• geometric mean
• quadratic mean

References

• Means, Meaning, and Music: Pythagoras, Archytas, and Plato by Scott Makeig, §The Tetractus of Means