Pythagorean means Average

1 pythagorean means

1.1 geometric mean
1.2 harmonic mean
1.3 inequality concerning am, gm, , hm

pythagorean means

along arithmetic mean above, geometric mean , harmonic mean known collectively pythagorean means.

geometric mean

the geometric mean of n positive numbers obtained multiplying them , taking nth root. in algebraic terms, geometric mean of a1, a2, …, an defined as

g
m
=




∏

i
=
1


n



a

i




n



=




a

1



a

2


⋯

a

n




n





{\displaystyle gm={\sqrt[{n}]{\prod _{i=1}^{n}a_{i}}}={\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}}}}

geometric mean can thought of antilog of arithmetic mean of logs of numbers.

example: geometric mean of 2 , 8



g
m
=


2
⋅
8


=
4


{\displaystyle gm={\sqrt {2\cdot 8}}=4}

harmonic mean

harmonic mean non-empty collection of numbers a1, a2, …, an, different 0, defined reciprocal of arithmetic mean of reciprocals of ai s:

h
m
=


1




1
n





∑

i
=
1


n




1

a

i








=


n



1

a

1




+


1

a

2




+
⋯
+


1

a

n









{\displaystyle hm={\frac {1}{{\dfrac {1}{n}}\displaystyle \sum \limits _{i=1}^{n}{\frac {1}{a_{i}}}}}={\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}}

one example harmonic mean useful when examining speed number of fixed-distance trips. example, if speed going point b 60 km/h, , speed returning b 40 km/h, harmonic mean speed given by

2



1
60


+


1
40





=
48


{\displaystyle {\frac {2}{{\frac {1}{60}}+{\frac {1}{40}}}}=48}



inequality concerning am, gm, , hm

a known inequality concerning arithmetic, geometric, , harmonic means set of positive numbers is

a
m
≥
g
m
≥
h
m


{\displaystyle am\geq gm\geq hm}

it easy remember noting alphabetical order of letters a, g, , h preserved in inequality. see inequality of arithmetic , geometric means.

thus above harmonic mean example: = 50, gm ≈ 49, , hm = 48 km/h.