1 pythagorean means
1.1 arithmetic mean (am)
1.2 geometric mean (gm)
1.3 harmonic mean (hm)
1.4 relationship between am, gm, , hm
pythagorean means
arithmetic mean (am)
the arithmetic mean (or mean ) of sample
x
1
,
x
2
,
…
,
x
n
{\displaystyle x_{1},x_{2},\ldots ,x_{n}}
, denoted
x
¯
{\displaystyle {\bar {x}}}
, sum of sampled values divided number of items in sample:
x
¯
=
1
n
(
∑
i
=
1
n
x
i
)
=
x
1
+
x
2
+
⋯
+
x
n
n
{\displaystyle {\bar {x}}={\frac {1}{n}}\left(\sum _{i=1}^{n}{x_{i}}\right)={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}}
geometric mean (gm)
the geometric mean average useful sets of positive numbers interpreted according product , not sum (as case arithmetic mean) e.g. rates of growth.
x
¯
=
(
∏
i
=
1
n
x
i
)
1
n
=
(
x
1
x
2
⋯
x
n
)
1
/
n
{\displaystyle {\bar {x}}=\left(\prod _{i=1}^{n}{x_{i}}\right)^{\tfrac {1}{n}}=\left(x_{1}x_{2}\cdots x_{n}\right)^{1/n}}
for example, geometric mean of 5 values: 4, 36, 45, 50, 75 is:
(
4
×
36
×
45
×
50
×
75
)
1
/
5
=
24
300
000
5
=
30.
{\displaystyle (4\times 36\times 45\times 50\times 75)^{1/5}={\sqrt[{5}]{24\;300\;000}}=30.}
harmonic mean (hm)
the harmonic mean average useful sets of numbers defined in relation unit, example speed (distance per unit of time).
x
¯
=
n
⋅
(
∑
i
=
1
n
1
x
i
)
−
1
{\displaystyle {\bar {x}}=n\cdot \left(\sum _{i=1}^{n}{\frac {1}{x_{i}}}\right)^{-1}}
for example, harmonic mean of 5 values: 4, 36, 45, 50, 75 is
5
1
4
+
1
36
+
1
45
+
1
50
+
1
75
=
5
1
3
=
15.
{\displaystyle {\frac {5}{{\tfrac {1}{4}}+{\tfrac {1}{36}}+{\tfrac {1}{45}}+{\tfrac {1}{50}}+{\tfrac {1}{75}}}}={\frac {5}{\;{\tfrac {1}{3}}\;}}=15.}
relationship between am, gm, , hm
am, gm, , hm satisfy these inequalities:
a
m
≥
g
m
≥
h
m
{\displaystyle \mathrm {am} \geq \mathrm {gm} \geq \mathrm {hm} \,}
equality holds if , if elements of given sample equal.
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