harmonic mean

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., ${\displaystyle H={n \over {1 \over x_{1}}+{1 \over x_{2}}+\cdots +{1 \over x_{n}}}}$.

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

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

• contraharmonic mean

• arithmetic mean
• geometric mean
• quadratic mean

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