Calculation Average

1 calculation

1.1 arithmetic mean
1.2 pythagorean means

1.2.1 geometric mean
1.2.2 harmonic mean
1.2.3 inequality concerning am, gm, , hm


1.3 statistical location

1.3.1 mode
1.3.2 median

calculation
arithmetic mean

the common type of average arithmetic mean. if n numbers given, each number denoted ai (where i = 1,2, …, n), arithmetic mean sum of divided n or

a
m
=


1
n



∑

i
=
1


n



a

i


=


1
n



(

a

1


+

a

2


+
⋯
+

a

n


)



{\displaystyle am={\frac {1}{n}}\sum _{i=1}^{n}a_{i}={\frac {1}{n}}\left(a_{1}+a_{2}+\cdots +a_{n}\right)}

the arithmetic mean, called mean, of 2 numbers, such 2 , 8, obtained finding value such 2 + 8 = + a. 1 may find = (2 + 8)/2 = 5. switching order of 2 , 8 read 8 , 2 not change resulting value obtained a. mean 5 not less minimum 2 nor greater maximum 8. if increase number of terms in list 2, 8, , 11, arithmetic mean found solving value of in equation 2 + 8 + 11 = a + a + a. 1 finds = (2 + 8 + 11)/3 = 7.

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
=
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1

a

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=


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1

a

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+


1

a

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⋯
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a

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{\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.

statistical location

the mode, median, , mid-range used in addition mean estimates of central tendency in descriptive statistics. these can seen minimizing variation measure; see central tendency § solutions variational problems.

mode

comparison of arithmetic mean, median , mode of 2 log-normal distributions different skewness.

the occurring number in list called mode. example, mode of list (1, 2, 2, 3, 3, 3, 4) 3. may happen there 2 or more numbers occur equally , more other number. in case there no agreed definition of mode. authors modes , there no mode.

median

the median middle number of group when ranked in order. (if there number of numbers, mean of middle 2 taken.)

thus find median, order list according elements magnitude , repeatedly remove pair consisting of highest , lowest values until either 1 or 2 values left. if 1 value left, median; if 2 values, median arithmetic mean of these two. method takes list 1, 7, 3, 13 , orders read 1, 3, 7, 13. 1 , 13 removed obtain list 3, 7. since there 2 elements in remaining list, median arithmetic mean, (3 + 7)/2 = 5.