Average

Averages ( short only means in statistics often average instead of arithmetic mean) occur in mathematics and in particular in the statistics in terms of content different contexts. In general, each average is based on a rule, with one another is calculated from two or more figures between the given numbers.

In the statistics is also often indicated as the mean of the expected value, a so-called location parameter of a frequency or probability distribution, which describes the position of elements in a sample or population in terms of the measurement scale.

• 6.1 Weighted averages
• 6.2 The logarithmic mean
• 6.4 quartile means
• 6.5 range means
• 6.6 The "a means "
• 6.7 Moving Averages
• 6.8 Other Means

• 7.1 Hölder means
• 7.2 Lehmer means
• 7.3 Stolarsky funds
• 7.4 Integral representation according to Chen

Definitions of the most famous and important means

Below are given real numbers, the statistics about the measured values ​​whose average is to be calculated.

Examples of the use of different means

The following is to be exemplified in the seven forms specified right where that definition of the mean value is meaningful.

The mode is useful already in the nominal scale, in which individual features can not be ordered. Are about seven people interviewed three Catholic ( A), two Protestant (B ), a Muslim (C ) and a Hindu (D ), the mode is because this is the most common.

For the median ordinal is provided in which the characteristics can be arranged. For example, three out customers with 'very good' (A ), two "good" (B ) and one each with "medium" and "poor" (C or D) on the question of the quality of the food of a restaurant. After ordering the data as shown in the list on the right can be seen that the mean observation is. The median is so.

The arithmetic mean for example is used to calculate the average speed: Runs a turtle only three meters per hour, then three hours every two meters and speeds for one hour again at three, four and five meters per hour, calculated as the arithmetic mean at a distance of 21 meters in 7 hours:

Also, the harmonic mean can be useful for calculating an average speed, if not equal times, but over the same distances measured: The turtle will run the first meters with 3 meters per hour, an additional 3 m with 2 m / h and accelerates on the last 3 meters further on, respectively 3, 4 and 5 m / h The average speed is obtained at a distance of 7 meters in hours:

With the geometric mean is used to calculate the average growth factor. A bacterial culture is growing, for example, on the first day to five times, the second four times, then twice to three times and the last three days it doubles every day. The number after the seventh day is therefore calculated by alternative can be determined using the geometric mean of the closing balance, because

And thus

A daily growth of the bacterial culture to the 2.83 -fold would thus performed after seven days the same result.

History

In mathematics occur averages, particularly the three classical means (arithmetic, geometric and harmonic mean ), even in ancient times on. Pappus of Alexandria features 10 different mean values ​​of m 2 numbers and () by special values ​​of the distance ratio. Also, the inequality between harmonic, geometric and arithmetic mean is already known and interpreted geometrically in antiquity. In the 19th and 20th century averages play a special role in the analysis, there is essentially related to famous inequalities and important functional properties such as convexity ( Hölder 's inequality, Minkowski 's inequality, Jensen's inequality, etc.). In this case ( see section Holder means below) the conventional means have been generalized in several steps, the first power means and in turn to the quasi- arithmetic means. The classical inequality between harmonic, geometric and arithmetic means thus takes on a more general inequalities between power mean values ​​or quasi - arithmetic mean values.

Common definition of the three classical averages

The idea behind the three classical mean values ​​based, can be generally formulated as follows:

When arithmetic mean you look the number for which

Holds, where the sum extends over the left summands. So the arithmetic mean averages with respect to the arithmetic operation " sum ". Clearly one determined by the arithmetic mean of rods of different length, a with an average or mean length.

The geometric mean of the number one is looking for the

Holds, where the product extends over the left factors. Thus, the geometric mean averages with respect to the arithmetic operation " product".

The harmonic mean solves the equation

Relationships

The inverse of the harmonic mean is equal to the arithmetic mean of the reciprocals of the numbers.

For the mean values ​​are interrelated in the following way:

Or solved for the geometric mean

Inequality of the mean values

The inequality of arithmetic and geometric means compares the value of the arithmetic and geometric mean of two given numbers: It applies to positive variable always

The inequality can be extended to other means, such as (for positive variable)

For two (positive ) variable, there is also a graphic illustration:

The geometric mean follows directly from the Euclidean height set and the harmonic mean of the Euclidean Kathetensatz with the relationship

Other averages and similar functions

Weighted averages

The weighted or weighted mean values ​​arise when the individual values ​​assigns different weights with which these form part of the overall average; For example, if in a test of oral and written performance to different extents included in the overall score.

The exact definitions can be found here:

• Weighted arithmetic mean
• Weighted geometric mean
• Weighted harmonic mean

The logarithmic mean

The logarithmic average between and is defined as:

For is the logarithmic mean of the geometric and the arithmetic mean.

Can one assume that the data "outliers", that is, contaminated few too high or too low values ​​, so you can see the data either by piece or by " Winsorisieren " (named after Charles P. Winsor ) clean and calculate the truncated (English truncated mean) or winsorisierten mean (English Winsorized mean). In both cases, you first sort the observations in ascending order of size. The nozzle is then cut from the same number of values ​​at the beginning and end of the sequence and calculated by the remaining value to the average value. However, the outliers are replaced at the beginning and end of the sequence by the next larger (or smaller - ) value of the remaining data in the " Winsorisieren ".

% Winsorisierte mean equal - Indes of 10

That the truncated means lies between the arithmetic means ( no truncation ) and the median (maximum truncation ). Usually a 20% truncated agent is used, ie 40 % of the data are excluded from the average calculation. The percentage depends essentially on the number of suspected outliers in the data; for conditions for a truncation of less than 20 % is referred to the literature.

Quartile funds

Quartile the agent is defined as the mean value of the 1st and 3rd quartile:

Herein, the 25 % quantile ( 1st quartile ) and the corresponding 75 % quantile (3rd quartile) of the measured values ​​.

The quartile funds is more robust than the arithmetic mean, but less robust than the median.

Area means

The field agent is defined as the mean of the largest and smallest observation value:

Or

The "a - means "

For a given real vector with the expression

Wherein all permutations of summed referred to as " means " [] of the non-negative real numbers.

For the case that results in precise, the arithmetic mean of the numbers; the event yields precisely the geometric mean.

For the funds that Muirhead 's inequality applies.

Example: Let and

This results in

Moving Averages

Moving averages are used in the dynamic analysis of measured values. They are also a common means of technical analysis in financial mathematics. With moving averages the stochastic noise can be filtered out of time progressing signals. Often these are FIR filters. However, it must be noted that the most moving averages lag behind the real signal. For predictive filters, see eg Kalman filter.

Moving averages usually require an independent variable that designates the size of the trailing sample, and the weight of the previous value for the exponential moving averages.

Major moving averages are:

• Arithmetic moving averages ( Simple Moving Average - SMA),
• Exponential moving averages ( Exponential Moving Average - EMA )
• Double exponential moving averages ( EMA Double, DEMA )
• Trisubstituted -fold exponential moving averages ( EMA Triple - TEMA )
• Linear weighted moving averages ( linearly decreasing weighting )
• Square weighted moving averages and
• Other weights: Sine, Triangular, ...

In the financial literature also called adaptive moving averages can be found that ( etc. other volatility / dispersion ) automatically adapt to a changing environment:

• Kaufmann's Adaptive Moving Average (KAMA ) and
• Variable Index Dynamic Average ( VIDYA ).

For the application of moving averages also see Moving averages ( chart analysis ) and MA model.

Other averages

Other averages, which are described in a separate article, the mode is (actually not an average, but the most frequent value ) and the median, the opposite extreme deviations, called outliers, is robust.

In addition, mean values ​​can be combined; The result is approximately the arithmetic- geometric mean, which is located between the arithmetic and geometric mean.

Generalized averages

There are a number of other functions that can be generated, the well-known and other means.

Hölder means

For positive numbers, we define the - Potency value, even Hölder means ( english - th power mean) as

For = 0, the value is defined by constant complement:

Note that both notation and designation are inconsistent.

For = -1, 0, 1, 2 and 3, this results about the harmonic, the geometric, the arithmetic, the square and the cubic agent. For → - ∞ gives the minimum for → ∞ the maximum of the numbers.

Moreover applies to fixed numbers: The larger is, the larger; it then follows the generalized inequality of the mean values

Lehmer means

The Lehmer means is another generalized mean; to the level it is defined by

It has the special cases

• Is the harmonic mean;
• Is the geometric mean of and;
• Is the arithmetic mean;

Stolarsky funds

The Stolarsky - mean of two numbers is defined by

Integral representation according to Chen

The function

Results for different arguments, the known mean values ​​of and:

• Is the harmonic mean
• Is the geometric mean
• Is the arithmetic mean

By the continuity and monotonicity of the function thus defined the Mittelwertungleichung follows

Average value of a function

The arithmetic average of a continuous function in a closed interval is

The root mean square of a continuous function is

These are used in industry considerable attention, see DC and RMS value.