1.NounA type of average calculated as the arithmetic mean of the squares of the values divided by the arithmetic mean of the values, i.e.
C
=
(
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
)
n
(
x
1
+
x
2
+
.
.
.
+
x
n
)
n
o
r
C
(
x
1
,
x
2
,
.
.
.
,
x
n
)
=
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
x
1
+
x
2
+
.
.
.
+
x
n
{\displaystyle C={{(x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}) \over n} \over {(x_{1}+x_{2}+...+x_{n}) \over n}}\ or\ C(x_{1},x_{2},...,x_{n})={{x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}} \over {x_{1}+x_{2}+...+x_{n}}}}
"The contraharmonic mean of two numbers
a
{\displaystyle a}
and
c
{\displaystyle c}
is the number
b
{\displaystyle b}
such that
a
−
b
c
=
b
−
c
a
{\displaystyle {a-b \over c}={b-c \over a}}
. Combining the two ratios:
b
−
c
a
=
(
a
−
b
)
+
(
b
−
c
)
c
+
a
=
a
−
c
a
+
c
{\displaystyle {b-c \over a}={(a-b)+(b-c) \over c+a}={a-c \over a+c}}
. Rotating the proportion 90° clockwise:
a
a
+
c
=
b
−
c
a
−
c
{\displaystyle {a \over a+c}={b-c \over a-c}}
; transposing twice to isolate the
b
{\displaystyle b}
:
b
=
c
+
a
(
a
−
c
a
+
c
)
{\displaystyle b=c+a\left({a-c \over a+c}\right)}
. Likewise
b
=
a
−
c
(
a
−
c
a
+
c
)
{\displaystyle b=a-c\left({a-c \over a+c}\right)}
; these are both equivalent to the more symmetrical
b
=
a
2
+
c
2
a
+
c
{\displaystyle b={a^{2}+c^{2} \over a+c}}
."