Arithmetic mean

The arithmetic mean , also called the arithmetic mean (colloquially also known as the average ) is a term used in statistics . It is a location parameter . We calculate this average by the sum of the observed numbers by their number divides .
The arithmetic mean of a sample is also called the empirical mean .

definition

Half of the sum of two quantities and is given by: ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle {\overline {x}}={\frac {1}{2}}(a+b)}$.

Since the variables form an arithmetic sequence , the sum of the characteristics of the characteristics is divided by the number of carriers ${\displaystyle a,{\overline {x}},b}$ ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ ${\displaystyle n}$

${\displaystyle {\overline {x}}:={\frac {1}{n}}(x_{1}+x_{2}+\ldots +x_{n})={\frac {1}{n}}\sum _{i=1}^{n}{x_{i}}}$

referred to as "arithmetic mean" (read: across ). If the arithmetic mean is not weighted (see also section Weighted arithmetic mean ), then it is also referred to as simple arithmetic mean or unweighted arithmetic mean . ${\displaystyle {\overline {x}}}$ ${\displaystyle x}$

For example, the arithmetic mean of the two numbers and is : ${\displaystyle 1}$ ${\displaystyle 2}$

${\displaystyle {\overline {x}}={\frac {1+2}{2}}=1{,}5}$.

The arithmetic mean describes the center of a distribution by a numeric value and thus represents a position parameter . The arithmetic mean is meaningfully defined for any metric characteristics. In general, it is not suitable for qualitative characteristics, but it provides dichotomous characteristics with two categories and a meaningful interpretation. In this case, the arithmetic mean is the same as the relative frequency . The average symbol Durchschnitts is occasionally used to denote the arithmetic mean . In contrast to the empirical median, the arithmetic mean is very susceptible to outliers (see median ). The arithmetic mean can be interpreted as the "midpoint" of the measured values. However, there is no information about how strongly the measured values ​​scatter around the arithmetic mean. This problem can be solved with the introduction of the “mean square deviation” of the arithmetic mean, the empirical variance . ${\displaystyle k_{1}=0}$ ${\displaystyle k_{2}=1}$ ${\displaystyle f_{2}=f(k_{2})}$

Definition for frequency data

For frequency data with the characteristics and the associated relative frequencies , the arithmetic mean results as ${\displaystyle a_{1},a_{2},\ldots ,a_{k}}$ ${\displaystyle h_{1},h_{2},\ldots ,h_{k}}$

${\displaystyle {\overline {x}}:=a_{1}h_{1}+a_{2}h_{2}+\ldots +a_{k}h_{k}=\sum _{j=1}^{k}{a_{j}h_{j}}}$.

Arithmetic mean for stratification

If a stratified sample is available, the arithmetic mean of the strata is known, the arithmetic mean for the total survey can be calculated. Let a survey population with feature carriers be divided into strata with the respective number of feature carriers and arithmetic means . The arithmetic mean in is then defined by ${\displaystyle E}$ ${\displaystyle n}$ ${\displaystyle r}$ ${\displaystyle E_{1},E_{2},\ldots ,E_{r}}$ ${\displaystyle n_{1},n_{2},\ldots ,n_{r}}$ ${\displaystyle {\overline {x}}_{1},{\overline {x}}_{2},\ldots ,{\overline {x}}_{r}}$ ${\displaystyle {\overline {x}}}$ ${\displaystyle E}$

${\displaystyle {\overline {x}}:={\frac {1}{n}}(n_{1}{\overline {x}}_{1}+n_{2}{\overline {x}}_{2}+\ldots +n_{r}{\overline {x}}_{r})={\frac {1}{n}}\sum _{j=1}^{r}{n_{j}{\overline {x}}_{j}}}$.

properties

Substitute value property

It follows directly from the definition of the arithmetic mean that

${\displaystyle \sum _{i=1}^{n}{x_{i}}=n{\overline {x}}}$.

If you multiply the arithmetic mean by the sample size , you get the sum of features . This calculation rule is known as the substitute value property or extrapolation property and is often used in mathematical proofs. It can be interpreted as follows: The sum of all individual values can be thought of as being replaced by equal values ​​of the size of the arithmetic mean. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

Focus property

The deviations of the measured values from the mean ${\displaystyle \nu _{i}}$ ${\displaystyle x_{i}}$ ${\displaystyle {\overline {x}}}$

${\displaystyle \nu _{i}=x_{i}-{\overline {x}}\quad i=1,\ldots ,n}$

are also known as "apparent errors". The center of gravity property (also called zero property ) means that the sum of the apparent errors or the sum of the deviations of all observed measured values ​​from the arithmetic mean is equal to zero

${\displaystyle \sum \nolimits _{i=1}^{n}\nu _{i}=\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)=0}$or in the case of frequency . ${\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)f_{i}=0}$

This can be shown using the substitute value property as follows:

${\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)=\sum _{i=1}^{n}x_{i}-\sum _{i=1}^{n}{\overline {x}}=n{\overline {x}}-n{\overline {x}}=0}$

The center of gravity plays a major role in the concept of degrees of freedom . Due to the center of gravity of the arithmetic mean , the last deviation is already determined by the first . Consequently, only deviations vary freely and one therefore averages, e.g. B. with the empirical variance, by dividing by the number of degrees of freedom . ${\displaystyle \sum \nolimits _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)=0}$ ${\displaystyle \left(x_{n}-{\overline {x}}\right)}$ ${\displaystyle (n-1)}$ ${\displaystyle (n-1)}$ ${\displaystyle (n-1)}$

Optimality property

In statistics one is often interested in minimizing the sum of the squares of deviations from a center . If you want to define the center by a value on the horizontal axis that is the sum of the squared deviations ${\displaystyle Q}$ ${\displaystyle z}$

${\displaystyle Q(z;x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\left(x_{i}-z\right)^{2}}$

minimized between data and center , then the minimizing value is. This result can be shown by simply deriving the objective function according to: ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle z}$ ${\displaystyle z={\overline {x}}}$ ${\displaystyle Q}$ ${\displaystyle z}$

${\displaystyle \partial \,Q(z;x_{1},\ldots ,x_{n})/\partial \,z=-2\sum _{i=1}^{n}(x_{i}-z)\;{\overset {\mathrm {!} }{=}}\;0\Rightarrow z={\overline {x}}}$.

This is a minimum, since the second derivative of after is equal to 2, i.e. greater than 0, which is a sufficient condition for a minimum. ${\displaystyle Q}$ ${\displaystyle z}$

This results in the following optimality property (also called minimization property):

${\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)^{2}<\sum _{i=1}^{n}\left(x_{i}-z\right)^{2}}$for all or in other words ${\displaystyle z\neq {\overline {x}}\;}$ ${\displaystyle \;{\underset {z\in \mathbb {R} }{\rm {arg\,min}}}\,\sum _{i=1}^{n}\left(x_{i}-z\right)^{2}={\overline {x}}}$

Linear transformation property

Depending on the scale level , the arithmetic mean is equivariant to special transformations. It applies to the linear transformation

${\displaystyle y_{i}=a+b\cdot x_{i}\Rightarrow {\overline {y}}=a+b\cdot {\overline {x}}}$,

there

${\displaystyle {\overline {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}={\frac {1}{n}}\sum _{i=1}^{n}(a+b\cdot x_{i})=a+b\cdot {\overline {x}}_{i}}$.

Triangle inequalities

→ Main article : Inequality of the arithmetic and geometric mean

The following triangular inequality applies to the arithmetic mean : The arithmetic mean of positive characteristic values ​​is greater than or equal to the geometric mean of these characteristic values ${\displaystyle n}$ ${\displaystyle x_{i}>0}$

${\displaystyle {\frac {x_{1}+x_{2}+\ldots +x_{n}}{n}}\geq {\sqrt[{n}]{x_{1}\cdot x_{2}\cdot \ldots \cdot x_{n}}}}$.

The equality is only given if all characteristics are the same. Further applies to the absolute value of the arithmetic mean of several characteristic values that he is less than or equal to the root mean square is

${\displaystyle \left|{\frac {x_{1}+x_{2}+\ldots +x_{n}}{n}}\right|\leq {\sqrt {\frac {x_{1}^{2}+x_{2}^{2}+\ldots +x_{n}^{2}}{n}}}}$.

Examples

Simple examples

• The arithmetic mean of 50 and 100 is ${\displaystyle \quad {\overline {x}}={\frac {50+100}{2}}=75}$
• The arithmetic mean of 8, 5 and −1 is ${\displaystyle \quad {\overline {x}}={\frac {8+5+\left(-1\right)}{3}}=4}$

Application example

A car drives 100 km / h for one hour and 200 km / h the following hour. At what constant speed does another car have to drive to cover the same distance in two hours?

The total distance that the first car has covered is ${\displaystyle s_{1}}$

${\displaystyle s_{1}=100\ \mathrm {km/h} \cdot 1\ \mathrm {h} +200\ \mathrm {km/h} \cdot 1\ \mathrm {h} }$

and that of the second car

${\displaystyle s_{2}=v_{2}\cdot 2\ \mathrm {h} ,}$

where is the speed of the second car. From results ${\displaystyle v_{2}}$ ${\displaystyle s_{1}=s_{2}}$

${\displaystyle v_{2}\cdot 2\ \mathrm {h} =100\ \mathrm {km/h} \cdot 1\ \mathrm {h} +200\ \mathrm {km/h} \cdot 1\ \mathrm {h} }$

and thus

${\displaystyle v_{2}={\frac {100\ \mathrm {km/h} \cdot 1\ \mathrm {h} +200\ \mathrm {km/h} \cdot 1\mathrm {h} }{2\ \mathrm {h} }}={\frac {100\ \mathrm {km} +200\ \mathrm {km} }{2\ \mathrm {h} }}=150\ \mathrm {km/h} .}$

Weighted arithmetic mean

You can also define a weighted arithmetic mean (also known as a weighted arithmetic mean). It extends the scope of the simple arithmetic mean to values ​​with different weightings . One example is the calculation of a school grade, in which oral and written achievements have different degrees of influence. When applying Richmann's rule of mixing to determine the mixing temperature of two bodies made of the same material, a weighted arithmetic mean is also calculated.

Descriptive statistics

The weighted average is used, for example, if one averages , from samples of the same population with different sample sizes will combine: ${\displaystyle x_{i}}$ ${\displaystyle i=1,\dots ,n}$ ${\displaystyle n}$ ${\displaystyle w_{i}}$

${\displaystyle {\overline {x}}={\frac {\sum _{i=1}^{n}{w_{i}\cdot x_{i}}}{\sum _{i=1}^{n}w_{i}}}}$.

probability calculation

Sample means

The concrete characteristics can be understood as realizations of random variables . Each value thus represents a realization of the respective random variable after the sample has been drawn . The arithmetic mean of these random variables ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ ${\displaystyle X_{1},X_{2},\ldots ,X_{n}}$ ${\displaystyle x_{i}}$ ${\displaystyle X_{i}}$

${\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}$

is also known as the sample mean and is also a random variable.

Independently distributed random variables

If the independently distributed random variables (i.e. is a random variable with the random variables and is a random variable with the random variables ) with a common expected value but different variances , the weighted mean value also has the expected value and its variance is ${\displaystyle X_{i}}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{11},\dots ,X_{1n}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{21},\dots ,X_{2m}}$ ${\displaystyle \mu }$ ${\displaystyle \sigma _{i}^{2}}$ ${\displaystyle \mu }$

${\displaystyle \sigma _{\overline {x}}^{2}={\frac {\sum _{i=1}^{n}w_{i}^{2}\sigma _{i}^{2}}{\left(\sum _{i=1}^{n}w_{i}\right)^{2}}}}$.

If you choose the weight , the variance is simplified to ${\displaystyle w_{i}=1/\sigma _{i}^{2}}$

${\displaystyle \sigma _{\overline {x}}^{2}={\frac {\sum _{i=1}^{n}{\frac {1}{\sigma _{i}^{4}}}\sigma _{i}^{2}}{\left(\sum _{i=1}^{n}{\frac {1}{\sigma _{i}^{2}}}\right)^{2}}}={\frac {\sum _{i=1}^{n}{\frac {1}{\sigma _{i}^{2}}}}{\left(\sum _{i=1}^{n}{\frac {1}{\sigma _{i}^{2}}}\right)^{2}}}={\frac {1}{\sum _{i=1}^{n}{\frac {1}{\sigma _{i}^{2}}}}}}$.

It follows from the Cauchy-Schwarz inequality

${\displaystyle \left(\sum _{i=1}^{n}w_{i}^{2}\sigma _{i}^{2}\right)\cdot \left(\sum _{i=1}^{n}{\frac {1}{\sigma _{i}^{2}}}\right)\geq \left(\sum _{i=1}^{n}w_{i}\right)^{2}}$.

The choice of the weights or a choice proportional to them thus minimizes the variance of the weighted mean. With this formula, the weights can be appropriately selected depending on the variance of the respective value, which accordingly influences the mean value to a greater or lesser extent. ${\displaystyle w_{i}=1/\sigma _{i}^{2}}$ ${\displaystyle \sigma _{\overline {x}}^{2}}$ ${\displaystyle w_{i}}$

Independently and identically distributed random variables

If random variables are independent and identically distributed with the expected value and variance , then the sample mean also has the expected value , but the smaller variance . If a random variable has finite expectation and variance, it follows from the Chebyshev inequality that the arithmetic mean of a sample converges stochastically to the expectation of the random variable . According to many criteria, the arithmetic mean is therefore a suitable estimate of the expected value of the distribution from which the sample originates. ${\displaystyle X_{1},\dotsc ,X_{n}}$ ${\displaystyle \mu }$ ${\displaystyle \sigma ^{2}}$ ${\displaystyle {\overline {X}}:={\frac {1}{n}}\sum \nolimits _{i=1}^{n}X_{i}}$ ${\displaystyle \mu }$ ${\displaystyle \sigma ^{2}/n}$

If the special sample means are of the same size as the same population, then the variance has , so the choice is optimal. ${\displaystyle X_{i}}$ ${\displaystyle n_{i}}$ ${\displaystyle X_{i}}$ ${\displaystyle \sigma ^{2}/n_{i}}$ ${\displaystyle w_{i}=n_{i}}$

Weighted arithmetic mean as expected value

In the case of a discrete random variable with a countably finite carrier , the expected value of the random variable results as ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X)}$

${\displaystyle \operatorname {E} (X)=p_{1}x_{1}+p_{2}x_{2}+\ldots +p_{n}x_{n}}$.

Here is the probability that it will take the value . This expected value can be interpreted as a weighted mean of the values with the probabilities . In the case of uniform distribution, the following applies and therefore becomes the arithmetic mean of the values ${\displaystyle p_{i}=P(X=x_{i})}$ ${\displaystyle X}$ ${\displaystyle x_{i}}$ ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ ${\displaystyle p_{i}\;(i=1,\ldots ,n)}$ ${\displaystyle p_{1}=p_{2}=\ldots =p_{n}=1/n}$ ${\displaystyle \operatorname {E} (X)}$ ${\displaystyle x_{i}}$

${\displaystyle \operatorname {E} (X)={\frac {1}{n}}(x_{1}+x_{2}+\ldots +x_{n})={\frac {1}{n}}\sum _{i=1}^{n}{x_{i}}={\overline {x}}}$.

Examples of weighted averages

A farmer produces 100 kg of butter as a sideline. He can sell 10 kg for 10 € / kg, another 10 kg for 6 € / kg and the rest he has to sell for 3 € / kg. At what (weighted) average price did he sell his butter? Solution: (10 kg x 10 € / kg + 10 kg x 6 € / kg + 80 kg x 3 € / kg) / (10 kg + 10 kg + 80 kg) = 400 € / 100 kg = 4 € / kg. The average price, weighted with the quantity sold, corresponds to the fixed price at which the total quantity would have to be sold in order to achieve the same revenue as when selling partial quantities at changing prices.

The arithmetic mean of the numbers 1, 2 and 3 is 2, the arithmetic mean of the numbers 4 and 5 is 4.5. The arithmetic mean of all 5 numbers results as the mean value of the partial mean values ​​weighted with the sample size: ${\displaystyle {\overline {x}}_{1}}$ ${\displaystyle n_{1}=3}$ ${\displaystyle {\overline {x}}_{2}}$ ${\displaystyle n_{2}=2}$

${\displaystyle {\overline {x}}={\frac {n_{1}{\overline {x}}_{1}+n_{2}{\overline {x}}_{2}}{n_{1}+n_{2}}}={\frac {3{\frac {1+2+3}{3}}+2{\frac {4+5}{2}}}{3+2}}={\frac {6+9}{3+2}}=3={\frac {1+2+3+4+5}{5}}.}$

If the observations are available as a classified frequency, the arithmetic mean can approximately be determined as a weighted mean, with the class middle as the value and the class size as the weight. For example, if in a school class there is one child in the 20 to 25 kg weight class, 7 children in the 25 to 30 kg weight class, 8 children in the 30 to 35 kg weight class and 4 children in the 35 to 40 kg weight class, the average weight can be calculated as

${\displaystyle {\frac {1\cdot 22{,}5+7\cdot 27{,}5+8\cdot 32{,}5+4\cdot 37{,}5}{1+7+8+4}}={\frac {625}{20}}=31{,}25}$

estimate. In order to determine the quality of this estimate, one then has to determine the minimum / maximum possible mean value by taking the smallest / largest values ​​per interval as a basis. This means that the actual mean value is between 28.75 kg and 33.75 kg. The error of the estimate 31.25 is therefore a maximum of ± 2.5 kg or ± 8%.

The mean of a function

As the mean of the Riemann integrable function , the number ${\displaystyle f\colon [a,b]\to \mathbb {R} }$

${\displaystyle {\overline {f}}:={\frac {1}{b-a}}\int _{a}^{b}f(x)\mathrm {d} x}$

Are defined.

The designation mean value is justified insofar as the arithmetic mean is used for an equidistant division of the interval with the step size ${\displaystyle \{x_{0},x_{1},x_{2},\dotsc ,x_{n}\}}$ ${\displaystyle h={\tfrac {b-a}{n}}}$

${\displaystyle m_{n}(f):={\frac {1}{n}}(f(x_{1})+f(x_{2})+\ldots +f(x_{n}))={\frac {1}{b-a}}\sum _{k=1}^{n}f(x_{k})h}$

against converges. ${\displaystyle {\overline {f}}\;}$

If continuous , then the mean value theorem of integral calculus says that there is a with , so the function assumes its mean value at at least one point. ${\displaystyle f\;}$ ${\displaystyle \xi \in [a,b]}$ ${\displaystyle f(\xi )={\overline {f}}}$

The mean of the function with weight (where for all ) is ${\displaystyle f(x)}$ ${\displaystyle w(x)\;}$ ${\displaystyle w(x)>0\;}$ ${\displaystyle x\in [a,b]}$

${\displaystyle {\overline {f}}={\frac {\int _{a}^{b}f(t)w(t)\mathrm {d} t}{\int _{a}^{b}w(t)\mathrm {d} t}}}$.

For Lebesgue integrals in the measure space with a finite measure , the mean value of a Lebesgue integrable function can be expressed as ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle \mu (\Omega )<\infty }$

${\displaystyle {\overline {f}}:={\frac {1}{\mu (\Omega )}}\int _{\Omega }f(x)\,\mathrm {d} \mu (x)}$

define. If it is a probability space , that is true , then the mean takes the form ${\displaystyle \mu (\Omega )=1\;}$

${\displaystyle {\overline {f}}:=\int _{\Omega }f(x)\,\mathrm {d} \mu (x)}$

on; this corresponds exactly to the expected value of . ${\displaystyle f\;}$

The mean value of a function is of considerable importance in physics and technology, especially for periodic functions of time, see equivalence .

Quasi-arithmetic mean ( f- mean)

Let be a strictly monotonic continuous (and therefore invertible) function on a real interval and let ${\displaystyle f}$ ${\displaystyle I}$

${\displaystyle w_{i},0\leq w_{i}\leq 1,\sum _{i}w_{i}=1}$

Weight factors. Then for the quasi-arithmetic mean weighted with the weights is defined as ${\displaystyle x_{i}\in I}$ ${\displaystyle w_{i}}$

${\displaystyle {\overline {x}}_{f}=f^{-1}\left(\sum _{i=1}^{n}w_{i}f(x_{i})\right)}$.

Obviously,

${\displaystyle \min(x_{i})\leq {\overline {x}}_{f}\leq \max(x_{i}).}$

For obtained the arithmetic, for the geometric mean and the - Potency . ${\displaystyle f(x)=x}$ ${\displaystyle f(x)=\log(x)}$ ${\displaystyle f(x)=x^{k}}$ ${\displaystyle k}$

This mean value can be generalized to the weighted quasi-arithmetic mean of a function , with the image set of comprehensive intervals being strictly monotonic and continuous in one: ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle x}$

${\displaystyle {\overline {x}}_{f}=f^{-1}\left({\frac {\int f(x(t))w(t)\mathrm {d} t}{\int w(t)\mathrm {d} t}}\right)}$

See also

• Harmonious mean
• Square mean
• Holder means

Web links

Wikibooks: Proof of the arithmetic mean of two numbers

Individual evidence

1. ^ Karl Bosch : Elementary introduction to applied statistics . 8th edition. Vieweg, Wiesbaden 2005, p. 13 .
2. ^ Ludwig Fahrmeir , Rita artist, Iris Pigeot , and Gerhard Tutz : Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 49.
3. ^ ^{A b} Ludwig Fahrmeir, Rita artist, Iris Pigeot, and Gerhard Tutz: Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 50.
4. ↑ Horst Degen, Peter Lorscheid: Statistics textbook: with economic and population statistics. P. 42.
5. ^ Ludwig Fahrmeir, Rita artist, Iris Pigeot, and Gerhard Tutz: Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 65.
6. ^ ^{A b} Ludwig Fahrmeir, Rita artist, Iris Pigeot, and Gerhard Tutz: Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 54.
7. Analogous to ( argument of the maximum ), ↑ denotes the argument of the minimum ${\displaystyle \arg \min(\cdot )}$ ${\displaystyle \arg \max(\cdot )}$
8. ^ IN Bronstein, KA Semendjajew u. a .: Paperback of mathematics . 2nd Edition. 1995, p. 19 ff.
9. ^ IN Bronstein, KA Semendjajew u. a .: Paperback of mathematics. 2nd Edition. 1995, p. 629.
10. ^ H. Heuser: Textbook of Analysis . Part 1. 8th edition. Teubner, Stuttgart 1990, ISBN 3-519-12231-6 .