1.NounA type of measure of central tendency calculated as the reciprocal of the mean of the reciprocals, i.e.,
H
=
n
1
x
1
+
1
x
2
+
⋯
+
1
x
n
{\displaystyle H={n \over {1 \over x_{1}}+{1 \over x_{2}}+\cdots +{1 \over x_{n}}}}
.
"The harmonic mean of two numbers
a
{\displaystyle a}
and
c
{\displaystyle c}
is the number
b
{\displaystyle b}
such that
a
−
b
a
=
b
−
c
c
{\displaystyle {a-b \over a}={b-c \over c}}
. Combining the two ratios:
b
−
c
c
=
(
a
−
b
)
+
(
b
−
c
)
a
+
c
=
a
−
c
a
+
c
{\displaystyle {b-c \over c}={(a-b)+(b-c) \over a+c}={a-c \over a+c}}
. Rotating the proportion 90° clockwise:
c
a
+
c
=
b
−
c
a
−
c
{\displaystyle {c \over a+c}={b-c \over a-c}}
; transposing twice to isolate the
b
{\displaystyle b}
:
b
=
c
+
c
(
a
−
c
a
+
c
)
{\displaystyle b=c+c\left({a-c \over a+c}\right)}
. Likewise
b
=
a
−
a
(
a
−
c
a
+
c
)
{\displaystyle b=a-a\left({a-c \over a+c}\right)}
; these are both equivalent to the more typical, symmetrical equations:
b
=
2
a
c
a
+
c
=
2
1
a
+
1
c
{\displaystyle b={2ac \over a+c}={2 \over {1 \over a}+{1 \over c}}}
."