1 types of mean
1.1 pythagorean means
1.1.1 arithmetic mean (am)
1.1.2 geometric mean (gm)
1.1.3 harmonic mean (hm)
1.1.4 relationship between am, gm, , hm
1.2 statistical location
1.3 mean of probability distribution
1.4 generalized means
1.4.1 power mean
1.4.2 ƒ-mean
1.5 weighted arithmetic mean
1.6 truncated mean
1.7 interquartile mean
1.8 mean of function
1.9 mean of angles , cyclical quantities
1.10 fréchet mean
1.11 other means
types of mean
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.
statistical location
comparison of arithmetic mean, median , mode of 2 skewed (log-normal) distributions.
geometric visualisation of mode, median , mean of arbitrary probability density function.
in descriptive statistics, mean may confused median, mode or mid-range, of these may called average (more formally, measure of central tendency). mean of set of observations arithmetic average of values; however, skewed distributions, mean not same middle value (median), or value (mode). example, mean income typically skewed upwards small number of people large incomes, majority have income lower mean. contrast, median income level @ half population below , half above. mode income income, , favors larger number of people lower incomes. while median , mode more intuitive measures such skewed data, many skewed distributions in fact best described mean, including exponential , poisson distributions.
mean of probability distribution
the mean of probability distribution long-run arithmetic average value of random variable having distribution. in context, known expected value. discrete probability distribution, mean given
∑
x
p
(
x
)
{\displaystyle \textstyle \sum xp(x)}
, sum taken on possible values of random variable ,
p
(
x
)
{\displaystyle p(x)}
probability mass function. continuous distribution,the mean
∫
−
∞
∞
x
f
(
x
)
d
x
{\displaystyle \textstyle \int _{-\infty }^{\infty }xf(x)\,dx}
,
f
(
x
)
{\displaystyle f(x)}
probability density function. in cases, including in distribution neither discrete nor continuous, mean lebesgue integral of random variable respect probability measure. mean need not exist or finite; probability distributions mean infinite (+∞ or −∞), while others have no mean.
generalized means
power mean
the generalized mean, known power mean or hölder mean, abstraction of quadratic, arithmetic, geometric , harmonic means. defined set of n positive numbers xi by
x
¯
(
m
)
=
(
1
n
⋅
∑
i
=
1
n
x
i
m
)
1
m
{\displaystyle {\bar {x}}(m)=\left({\frac {1}{n}}\cdot \sum _{i=1}^{n}{x_{i}^{m}}\right)^{\tfrac {1}{m}}}
by choosing different values parameter m, following types of means obtained:
ƒ-mean
this can generalized further generalized f-mean
x
¯
=
f
−
1
(
1
n
⋅
∑
i
=
1
n
f
(
x
i
)
)
{\displaystyle {\bar {x}}=f^{-1}\left({{\frac {1}{n}}\cdot \sum _{i=1}^{n}{f(x_{i})}}\right)}
and again suitable choice of invertible ƒ give
weighted arithmetic mean
the weighted arithmetic mean (or weighted average) used if 1 wants combine average values samples of same population different sample sizes:
x
¯
=
∑
i
=
1
n
w
i
⋅
x
i
∑
i
=
1
n
w
i
.
{\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{n}{w_{i}\cdot x_{i}}}{\sum _{i=1}^{n}{w_{i}}}}.}
the weights
w
i
{\displaystyle w_{i}}
represent sizes of different samples. in other applications represent measure reliability of influence upon mean respective values.
truncated mean
sometimes set of numbers might contain outliers, i.e., data values lower or higher others. often, outliers erroneous data caused artifacts. in case, 1 can use truncated mean. involves discarding given parts of data @ top or bottom end, typically equal amount @ each end, , taking arithmetic mean of remaining data. number of values removed indicated percentage of total number of values.
interquartile mean
the interquartile mean specific example of truncated mean. arithmetic mean after removing lowest , highest quarter of values.
x
¯
=
2
n
∑
i
=
(
n
/
4
)
+
1
3
n
/
4
x
i
{\displaystyle {\bar {x}}={2 \over n}\sum _{i=(n/4)+1}^{3n/4}{x_{i}}}
assuming values have been ordered, specific example of weighted mean specific set of weights.
mean of function
in circumstances mathematicians may calculate mean of infinite (even uncountable) set of values. can happen when calculating mean value
y
ave
{\displaystyle y_{\text{ave}}}
of function
f
(
x
)
{\displaystyle f(x)}
. intuitively can thought of calculating area under section of curve , dividing length of section. can done crudely counting squares on graph paper or more precisely integration. integration formula written as:
y
ave
(
a
,
b
)
=
∫
a
b
f
(
x
)
d
x
b
−
a
{\displaystyle y_{\text{ave}}(a,b)={\frac {\int \limits _{a}^{b}\!f(x)\,dx\,}{b-a}}}
care must taken make sure integral converges. mean may finite if function tends infinity @ points.
mean of angles , cyclical quantities
angles, times of day, , other cyclical quantities require modular arithmetic add , otherwise combine numbers. in these situations, there not unique mean. example, times hour before , after midnight equidistant both midnight , noon. possible no mean exists. consider color wheel -- there no mean set of colors. in these situations, must decide mean useful. can adjusting values before averaging, or using specialized approach mean of circular quantities.
fréchet mean
the fréchet mean gives manner determining center of mass distribution on surface or, more generally, riemannian manifold. unlike many other means, fréchet mean defined on space elements cannot added or multiplied scalars. known karcher mean (named after hermann karcher).
other means
Comments
Post a Comment