The most common form of statistical indicators used in economic research is the average value, which is a generalized quantitative characteristic of a characteristic in a statistical population. The average value provides a general characteristic of similar phenomena according to one of the varying characteristics. It reflects the level of this characteristic assigned to a unit of the population. The widespread use of averages is explained by the fact that they have a number of positive properties that make them an indispensable tool for analyzing phenomena and processes in the economy.
The most important property of the average value is that it reflects what is common to all units of the population under study. The attribute values of individual units of a population fluctuate in one direction or another under the influence of many factors, among which there may be both basic and random. For example, the stock price of a corporation as a whole is determined by its financial position. At the same time, on certain days and on certain exchanges, these shares, due to prevailing circumstances, may be sold at a higher or lower rate. The essence of the average lies in the fact that it cancels out the deviations of the characteristic values of individual units of the population caused by the action of random factors, and takes into account the changes caused by the action of the main factors. This allows the average to abstract from the individual characteristics inherent in individual units.
Let's look at some general principles for using averages.
1. When determining the average value in each specific case, one must proceed from the qualitative content of the characteristic being averaged, taking into account the relationship of the characteristics being studied, as well as the data available for calculation.
2. The average value must first of all be calculated from a homogeneous population. Qualitatively homogeneous populations make it possible to obtain a grouping method, which always involves the calculation of a system of generalizing indicators.
3. Overall averages must be supported by group averages. For example, let's say that an analysis of the dynamics of individual crop yields shows that the overall average yield is decreasing. However, it is known that the yield of this crop depends on soil, climatic and other conditions and varies in individual areas. Having grouped districts according to differences and analyzed the dynamics of group averages, one can find that in some districts the average yield either has not changed or is increasing, and the decrease in the overall average for the republic as a whole is due to an increase in the share of areas with lower yields in the total production of this agricultural crop . Obviously, the dynamics of group averages more closely reflects the patterns of changes in yield, while the dynamics of the overall average shows only the overall result.
A reasonable choice of the population unit for which the average is calculated is necessary.
The category of average can be revealed through the concept of its defining property. According to this concept, the average, being a generalizing characteristic of the entire population, should focus on a certain value associated with all units of this population. This value can be represented as a function: (x 1,x 2,…x n).
Since this value in most cases reflects the real economic category, the concept of the defining property of the average is sometimes replaced by the concept of the defining indicator.
If in the above function all values x 1, x 2, x n are replaced by their average value x͞, then the value of this function should remain the same:
ƒ(x 1 ,x 2 ,…,x n)=ƒ(x͞, x͞, …, x͞)
Based on this equality, the average is determined. In practice, it is possible to determine the average in many cases through the initial ratio of the average(ISS) or its logical formula:
So, for example, to calculate the average salary of an enterprise’s employees, it is necessary to divide the total wage fund by the number of employees:
The numerator of the initial ratio of the average is its defining indicator. For average wages, such a determining indicator is the wage fund. Regardless of what primary information we have - whether we know the total wage fund or wages and the number of workers employed in individual positions, or any other initial data - in any case, the average wage can only be obtained through this initial ratio average
For each indicator used in economic analysis, only one true initial ratio can be compiled to calculate the average. If, for example, you need to calculate the average deposit in a bank, then the initial ratio will be as follows:
ISS= 
Let us now consider the types of averages. The choice of the type of average is determined by the economic content of the indicator and the source data. In each specific case, one of the average values is used:
Arithmetic
Harmonic
Geometric
Quadratic
Cubic, etc.
The listed averages belong to the class sedate averages and are combined by the general formula (for different values of c):
where x i is the i-th variant of the characteristic under consideration (i=1͞,k); f i is the specific gravity of the i-th option.
Let us first consider power averages.
Average values are widely used in statistics. Average values characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.
Average - This is one of the common generalization techniques. A correct understanding of the essence of the average determines its special significance in a market economy, when the average, through the individual and random, allows us to identify the general and necessary, to identify the trend of patterns of economic development.
average value - these are generalizing indicators in which the effects of general conditions and patterns of the phenomenon being studied are expressed.
Statistical averages are calculated on the basis of mass data from correctly statistically organized mass observation (continuous and selective). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). For example, if you calculate the average wage in cooperatives and state-owned enterprises, and extend the result to the entire population, then the average is fictitious, since it was calculated for a heterogeneous population, and such an average loses all meaning.
With the help of the average, differences in the value of a characteristic that arise for one reason or another in individual units of observation are smoothed out.
For example, the average productivity of a salesperson depends on many reasons: qualifications, length of service, age, form of service, health, etc.
Average output reflects the general property of the entire population.
The average value is a reflection of the values of the characteristic being studied, therefore, it is measured in the same dimension as this characteristic.
Each average value characterizes the population under study according to any one characteristic. In order to obtain a complete and comprehensive understanding of the population being studied according to a number of essential characteristics, in general it is necessary to have a system of average values that can describe the phenomenon from different angles.
There are different averages:
arithmetic mean;
geometric mean;
harmonic mean;
mean square;
average chronological.
Let's look at some types of averages that are most often used in statistics.
Arithmetic mean
The simple arithmetic mean (unweighted) is equal to the sum of the individual values of the attribute divided by the number of these values.
Individual values of a characteristic are called variants and are denoted by x(); the number of population units is denoted by n, the average value of the characteristic is denoted by 
. Therefore, the arithmetic simple mean is equal to:
According to the discrete distribution series data, it is clear that the same characteristic values (variants) are repeated several times. Thus, option x occurs 2 times in total, and option x 16 times, etc.
The number of identical values of a characteristic in the distribution series is called frequency or weight and is denoted by the symbol n.
Let's calculate the average salary of one worker 
in rub.:
The wage fund for each group of workers is equal to the product of options and frequency, and the sum of these products gives the total wage fund of all workers.
In accordance with this, the calculations can be presented in general form:
The resulting formula is called the weighted arithmetic mean.
As a result of processing, statistical material can be presented not only in the form of discrete distribution series, but also in the form of interval variation series with closed or open intervals.
The average for grouped data is calculated using the weighted arithmetic average formula:
In the practice of economic statistics, it is sometimes necessary to calculate the average using group averages or averages of individual parts of the population (partial averages). In such cases, group or private averages are taken as options (x), on the basis of which the overall average is calculated as an ordinary weighted arithmetic average.
Basic properties of the arithmetic mean .
The arithmetic mean has a number of properties:
1. The value of the arithmetic mean will not change from decreasing or increasing the frequency of each value of the characteristic x by n times.
If all frequencies are divided or multiplied by any number, the average value will not change.
2. The common multiplier of individual values of a characteristic can be taken beyond the sign of the average:
3. The average of the sum (difference) of two or more quantities is equal to the sum (difference) of their averages:
4. If x = c, where c is a constant value, then 
.
5. The sum of deviations of the values of attribute X from the arithmetic mean x is equal to zero:
Harmonic mean.
Along with the arithmetic mean, statistics uses the harmonic mean, the inverse of the arithmetic mean of the inverse values of the attribute. Like the arithmetic mean, it can be simple and weighted.
Characteristics of variation series, along with averages, are mode and median.
Fashion - this is the value of a characteristic (variant) that is most often repeated in the population being studied. For discrete distribution series, the mode will be the value of the variant with the highest frequency.
For interval distribution series with equal intervals, the mode is determined by the formula:
Where 
- initial value of the interval containing the mode;
- the value of the modal interval;
- frequency of the modal interval;
- frequency of the interval preceding the modal one;
- frequency of the interval following the modal one.
Median - this is an option located in the middle of the variation series. If the distribution series is discrete and has an odd number of members, then the median will be the option located in the middle of the ordered series (an ordered series is the arrangement of population units in ascending or descending order).
This term has other meanings, see average meaning.Average(in mathematics and statistics) sets of numbers - the sum of all numbers divided by their number. It is one of the most common measures of central tendency.
It was proposed (along with the geometric mean and harmonic mean) by the Pythagoreans.
Special cases of the arithmetic mean are the mean (general population) and the sample mean (sample).
Introduction
Let us denote the set of data X = (x 1 , x 2 , …, x n), then the sample mean is usually indicated by a horizontal bar over the variable (x ¯ (\displaystyle (\bar (x))), pronounced " x with a line").
The Greek letter μ is used to denote the arithmetic mean of the entire population. For a random variable for which the mean value is determined, μ is probabilistic average or the mathematical expectation of a random variable. If the set X is a collection of random numbers with a probabilistic mean μ, then for any sample x i from this set μ = E( x i) is the mathematical expectation of this sample.
In practice, the difference between μ and x ¯ (\displaystyle (\bar (x))) is that μ is a typical variable because you can see a sample rather than the entire population. Therefore, if the sample is represented randomly (in terms of probability theory), then x ¯ (\displaystyle (\bar (x))) (but not μ) can be treated as a random variable having a probability distribution on the sample (the probability distribution of the mean).
Both of these quantities are calculated in the same way:
X ¯ = 1 n ∑ i = 1 n x i = 1 n (x 1 + ⋯ + x n) . (\displaystyle (\bar (x))=(\frac (1)(n))\sum _(i=1)^(n)x_(i)=(\frac (1)(n))(x_ (1)+\cdots +x_(n)).)
If X is a random variable, then the mathematical expectation X can be considered as the arithmetic mean of values in repeated measurements of a quantity X. This is a manifestation of the law of large numbers. Therefore, the sample mean is used to estimate the unknown expected value.
In elementary algebra it has been proven that the mean n+ 1 numbers above average n numbers if and only if the new number is greater than the old average, less if and only if the new number is less than the average, and does not change if and only if the new number is equal to the average. The more n, the smaller the difference between the new and old averages.
Note that there are several other "averages" available, including the power mean, the Kolmogorov mean, the harmonic mean, the arithmetic-geometric mean, and various weighted averages (e.g., weighted arithmetic mean, weighted geometric mean, weighted harmonic mean).
Examples
- For three numbers, you need to add them and divide by 3:
- For four numbers, you need to add them and divide by 4:
Or simpler: 5+5=10, 10:2. Because we were adding 2 numbers, which means how many numbers we add, we divide by that many.
Continuous random variable
For a continuously distributed quantity f (x) (\displaystyle f(x)), the arithmetic mean on the interval [ a ; b ] (\displaystyle ) is determined through a definite integral:
F (x) ¯ [ a ; b ] = 1 b − a ∫ a b f (x) d x (\displaystyle (\overline (f(x)))_()=(\frac (1)(b-a))\int _(a)^(b) f(x)dx)
Some problems of using the average
Lack of robustness
Main article: Robustness in statisticsAlthough arithmetic means are often used as averages or central tendencies, this concept is not a robust statistic, meaning that the arithmetic mean is heavily influenced by "large deviations." It is noteworthy that for distributions with a large coefficient of skewness, the arithmetic mean may not correspond to the concept of “mean”, and the values of the mean from robust statistics (for example, the median) may better describe the central tendency.
A classic example is calculating average income. The arithmetic mean can be misinterpreted as a median, which may lead to the conclusion that there are more people with higher incomes than there actually are. “Average” income is interpreted to mean that most people have incomes around this number. This “average” (in the sense of the arithmetic mean) income is higher than the incomes of most people, since a high income with a large deviation from the average makes the arithmetic mean highly skewed (in contrast, the average income at the median “resists” such skew). However, this "average" income says nothing about the number of people near the median income (and says nothing about the number of people near the modal income). However, if you take the concepts of “average” and “most people” lightly, you can draw the incorrect conclusion that most people have incomes higher than they actually are. For example, a report of the "average" net income in Medina, Washington, calculated as the arithmetic average of all annual net incomes of residents, would produce a surprisingly large number due to Bill Gates. Consider the sample (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five out of six values are below this mean.
Compound interest
Main article: Return on InvestmentIf the numbers multiply, but not fold, you need to use the geometric mean, not the arithmetic mean. Most often this incident occurs when calculating the return on investment in finance.
For example, if a stock fell 10% in the first year and rose 30% in the second, then it is incorrect to calculate the “average” increase over those two years as the arithmetic mean (−10% + 30%) / 2 = 10%; the correct average in this case is given by the compound annual growth rate, which gives an annual growth rate of only about 8.16653826392% ≈ 8.2%.
The reason for this is that percentages have a new starting point each time: 30% is 30% from a number less than the price at the beginning of the first year: if a stock started out at $30 and fell 10%, it is worth $27 at the start of the second year. If the stock rose 30%, it would be worth $35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock has only risen by $5.1 over 2 years, the average growth of 8.2% gives a final result of $35.1:
[$30 (1 - 0.1) (1 + 0.3) = $30 (1 + 0.082) (1 + 0.082) = $35.1]. If we use the arithmetic average of 10% in the same way, we will not get the actual value: [$30 (1 + 0.1) (1 + 0.1) = $36.3].
Compound interest at the end of 2 years: 90% * 130% = 117%, that is, the total increase is 17%, and the average annual compound interest is 117% ≈ 108.2% (\displaystyle (\sqrt (117\%))\approx 108.2\%) , that is, an average annual increase of 8.2%.
Directions
Main article: Destination statisticsWhen calculating the arithmetic mean of some variable that changes cyclically (such as phase or angle), special care must be taken. For example, the average of 1° and 359° would be 1 ∘ + 359 ∘ 2 = (\displaystyle (\frac (1^(\circ )+359^(\circ ))(2))=) 180°. This number is incorrect for two reasons.
- First, angular measures are defined only for the range from 0° to 360° (or from 0 to 2π when measured in radians). So the same pair of numbers could be written as (1° and −1°) or as (1° and 719°). The average values of each pair will be different: 1 ∘ + (− 1 ∘) 2 = 0 ∘ (\displaystyle (\frac (1^(\circ )+(-1^(\circ )))(2))=0 ^(\circ )) , 1 ∘ + 719 ∘ 2 = 360 ∘ (\displaystyle (\frac (1^(\circ )+719^(\circ ))(2))=360^(\circ )) .
- Second, in this case, a value of 0° (equivalent to 360°) will be a geometrically better average value, since the numbers deviate less from 0° than from any other value (the value 0° has the smallest variance). Compare:
- the number 1° deviates from 0° by only 1°;
- the number 1° deviates from the calculated average of 180° by 179°.
The average value for a cyclic variable calculated using the above formula will be artificially shifted relative to the real average towards the middle of the numerical range. Because of this, the average is calculated in a different way, namely, the number with the smallest variance (the center point) is selected as the average value. Also, instead of subtraction, the modular distance (that is, the circumferential distance) is used. For example, the modular distance between 1° and 359° is 2°, not 358° (on a circle between 359° and 360°==0° - one degree, between 0° and 1° - also 1°, in total - 2 °).
Average value
Average value- numerical characteristics of a set of numbers or functions (in mathematics); - a certain number between the smallest and largest of their values.
Basic information
The starting point for the development of the theory of averages was the study of proportions by the school of Pythagoras. At the same time, no strict distinction was made between the concepts of average size and proportion. A significant impetus to the development of the theory of proportions from an arithmetic point of view was given by Greek mathematicians - Nicomachus of Geras (late 1st - early 2nd century AD) and Pappus of Alexandria (3rd century AD). The first stage in the development of the concept of average is the stage when the average began to be considered the central member of a continuous proportion. But the concept of average as the central value of a progression does not make it possible to derive the concept of average in relation to a sequence of n terms, regardless of the order in which they follow each other. For this purpose it is necessary to resort to a formal generalization of averages. The next stage is the transition from continuous proportions to progressions - arithmetic, geometric and harmonic ( English).
In the history of statistics, for the first time, the widespread use of averages is associated with the name of the English scientist W. Petty. W. Petty was one of the first to try to give the average value a statistical meaning, linking it with economic categories. But Petty did not describe the concept of average size or distinguish it. A. Quetelet is considered to be the founder of the theory of averages. He was one of the first to consistently develop the theory of average values, trying to provide a mathematical basis for it. A. Quetelet distinguished two types of averages - actual averages and arithmetic averages. Actually, the average represents a thing, a number, that actually exists. Actually, averages or statistical averages should be derived from phenomena of the same quality, identical in their internal meaning. Arithmetic averages are numbers that give the closest possible idea of many numbers, different, although homogeneous.
Each type of average can appear either in the form of a simple or in the form of a weighted average. The correct choice of the middle form follows from the material nature of the object of study. Simple average formulas are used if the individual values of the characteristic being averaged are not repeated. When in practical research individual values of the characteristic being studied occur several times in units of the population under study, then the frequency of repetitions of individual values of the characteristic is present in the calculation formulas of power averages. In this case, they are called weighted average formulas.
Hierarchy of averages in mathematics
- The average value of a function is a concept defined in many ways.
- More specifically, but based on arbitrary functions, Kolmogorov means are determined for a set of numbers.
- power average is a special case of Kolmogorov averages with ϕ (x) = x α (\displaystyle \phi (x)=x^(\alpha )) . Averages of different degrees are connected by inequality about averages. The most common special cases:
- arithmetic mean (α = 1 (\displaystyle \alpha =1));
- mean square (α = 2 (\displaystyle \alpha =2));
- harmonic mean (α = − 1 (\displaystyle \alpha =-1));
- by continuity as α → 0 (\displaystyle \alpha \to 0) the geometric mean is defined further, which is also the Kolmogorov mean for ϕ (x) = log x (\displaystyle \phi (x)=\log x)
- power average is a special case of Kolmogorov averages with ϕ (x) = x α (\displaystyle \phi (x)=x^(\alpha )) . Averages of different degrees are connected by inequality about averages. The most common special cases:
- More specifically, but based on arbitrary functions, Kolmogorov means are determined for a set of numbers.
- Weighted average is a generalization of the average to the case of an arbitrary linear combination:
- Weighted arithmetic mean.
- Weighted geometric mean.
- Weighted harmonic mean.
- average chronological - generalizes the values of a characteristic for the same unit or population as a whole, changing over time.
- logarithmic mean, determined by the formula a ¯ = a 1 − a 2 ln (a 1 / a 2) (\textstyle (\bar (a))=(\frac (a_(1)-a_(2))(\ ln(a_(1)/a_(2))))), used in heat engineering
- the logarithmic average, determined in electrical insulation in accordance with GOST 27905.4-88, is defined as l o g a ¯ = log a 1 + l o g a 2 + . . . + . . . l o g a n a 1 + a 2 + . . . + a n (\textstyle log(\bar (a))=(\frac (\log a_(1)+loga_(2)+...+...loga_(n))(a_(1)+a_( 2)+...+a_(n)))) (logarithm to any base)
In probability theory and statistics
Main article: Distribution center indicators- nonparametric means - mode, median.
- the average value of a random variable is the same as the mathematical expectation of a random variable. Essentially, it is the average value of its distribution function.
Symbol
This term has other meanings, see Symbol (meanings).Symbol(Ancient Greek σύμβολον - “ (conventional) sign, signal"") is a sign, an image of an object or animal, to indicate the quality of an object; a conventional sign of any concepts, ideas, phenomena 2.
Sometimes a sign and a symbol are different because, unlike a sign, a symbol is attributed a deeper social-normative (spiritual) dimension.
Story
The concept of symbol is closely related to such categories as artistic image, allegory and comparison. For example, in late antiquity, the cross became a symbol of Christianity[ unreputable source?]. In modern times, the swastika has become a symbol of National Socialism.
F. I. Girenok drew attention to the fact that in modern culture the difference “between a sign and a symbol” has been erased, while the specificity of a symbol is an indication of the superreal.
A.F. Losev defined a symbol as “the substantial identity of an idea and a thing.” Every symbol contains an image, but cannot be reduced to it, since it implies the presence of a certain meaning, inseparably fused with the image, but not identical to it. Image and meaning form two elements of a symbol, unthinkable without each other. Therefore, symbols exist as symbols (and not as things) only within interpretations.
In the 20th century, the neo-Kantian Cassirer generalized the concept of symbol and classified as “symbolic forms” a wide class of cultural phenomena, such as language, myth, religion, art and science, through which man organizes the chaos around him. Earlier, Kant argued that art, being an intuitive way of representation, is symbolic in nature.
Interested in what exactly does a pentagram inscribed in a circle of sun rays mean?
Uncle Nikita
After reading the answers of others, it is immediately clear that people immediately see the symbol of the Devil in the pentagram))) People do not want to know, their fear of Satan replaces their knowledge.
The pentagram, and also in the circle, is an ancient protective sign. And the correct pentagram stands at both ends. As I see in the picture, there is no inverted pentagram in the picture. Just stylized a simple pentagram in a circle, something like rays, tentacles, flames (?)
In theory, this is not only a protective sign, but also a symbol of the victory of the spiritual over the material. These are the four alchemical elements, plus ether.
And the inverted pentagram symbolizes the opposite - the victory of the material over the spiritual. And in general, Satanism should not be confused with Devil worship. These are two different things and people like to paint everything with the same brush, because they do not have knowledge, but have fears, conjectures, guesses and fantasies.
Lonesome crow
The most famous magician of the 20th century, Aleister Crowley, interpreted the inverted pentagram as a spirit represented in the form of sun rays that animates matter-Earth. Other esotericists argue that the inverted pentagram pours energy from heaven to earth and is therefore a symbol of materialistic tendencies, while the ordinary pentagram directs energy upward, being a symbol of the spiritual quest of humanity.
Oh, the Masons have so many different symbols...
Most likely, this is something Kabbalistic.
And why are you interested in satanic symbols? ! Get it out of your head - and that's the end of it, as they say.
) and sample mean(s).
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Let us denote the set of data X = (x 1 , x 2 , …, x n), then the sample mean is usually indicated by a horizontal bar over the variable (pronounced " x with a line").
The Greek letter μ is used to denote the arithmetic mean of the entire population. For a random variable for which the mean value is determined, μ is probabilistic average or mathematical expectation of a random variable. If the set X is a collection of random numbers with a probabilistic mean μ, then for any sample x i from this set μ = E( x i) is the mathematical expectation of this sample.
In practice, the difference between μ and x ¯ (\displaystyle (\bar (x))) is that μ is a typical variable because you can see a sample rather than the entire population. Therefore, if the sample is random (in terms of probability theory), then x ¯ (\displaystyle (\bar (x)))(but not μ) can be treated as a random variable having a probability distribution over the sample (probability distribution of the mean).
Both of these quantities are calculated in the same way:
x ¯ = 1 n ∑ i = 1 n x i = 1 n (x 1 + ⋯ + x n) . (\displaystyle (\bar (x))=(\frac (1)(n))\sum _(i=1)^(n)x_(i)=(\frac (1)(n))(x_ (1)+\cdots +x_(n)).)Examples
- For three numbers, you need to add them and divide by 3:
- For four numbers, you need to add them and divide by 4:
Or simpler: 5+5=10, 10:2. Because we were adding 2 numbers, which means how many numbers we add, we divide by that many.
Continuous random variable
f (x) ¯ [ a ; b ] = 1 b − a ∫ a b f (x) d x (\displaystyle (\overline (f(x)))_()=(\frac (1)(b-a))\int _(a)^(b) f(x)dx)Some problems of using the average
Lack of robustness
Although arithmetic means are often used as averages or central tendencies, this concept is not a robust statistic, meaning that the arithmetic mean is heavily influenced by "large deviations." It is noteworthy that for distributions with a large coefficient of skewness, the arithmetic mean may not correspond to the concept of “mean”, and the values of the mean from robust statistics (for example, the median) may better describe the central tendency.
A classic example is calculating average income. The arithmetic mean can be misinterpreted as a median, which may lead to the conclusion that there are more people with higher incomes than there actually are. “Average” income is interpreted to mean that most people have incomes around this number. This “average” (in the sense of the arithmetic mean) income is higher than the incomes of most people, since a high income with a large deviation from the average makes the arithmetic mean highly skewed (in contrast, the average income at the median “resists” such skew). However, this "average" income says nothing about the number of people near the median income (and says nothing about the number of people near the modal income). However, if you take the concepts of “average” and “most people” lightly, you can draw the incorrect conclusion that most people have incomes higher than they actually are. For example, a report of the "average" net income in Medina, Washington, calculated as the arithmetic average of all annual net incomes of residents, would yield a surprisingly large number due to Bill Gates. Consider the sample (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five out of six values are below this mean.
Compound interest
If the numbers multiply, but not fold, you need to use the geometric mean, not the arithmetic mean. Most often this incident occurs when calculating the return on investment in finance.
For example, if a stock fell 10% in the first year and rose 30% in the second, then it is incorrect to calculate the “average” increase over those two years as the arithmetic mean (−10% + 30%) / 2 = 10%; the correct average in this case is given by the compound annual growth rate, which gives an annual growth rate of only about 8.16653826392% ≈ 8.2%.
The reason for this is that percentages have a new starting point each time: 30% is 30% from a number less than the price at the beginning of the first year: if a stock started out at $30 and fell 10%, it is worth $27 at the start of the second year. If the stock rose 30%, it would be worth $35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock has only risen by $5.1 over 2 years, the average growth of 8.2% gives a final result of $35.1:
[$30 (1 - 0.1) (1 + 0.3) = $30 (1 + 0.082) (1 + 0.082) = $35.1]. If we use the arithmetic average of 10% in the same way, we will not get the actual value: [$30 (1 + 0.1) (1 + 0.1) = $36.3].
Compound interest at the end of 2 years: 90% * 130% = 117%, that is, the total increase is 17%, and the average annual compound interest 117% ≈ 108.2% (\displaystyle (\sqrt (117\%))\approx 108.2\%), that is, an average annual increase of 8.2%. This number is incorrect for two reasons.
The average value for a cyclic variable calculated using the above formula will be artificially shifted relative to the real average towards the middle of the numerical range. Because of this, the average is calculated in a different way, namely, the number with the smallest variance (the center point) is selected as the average value. Also, instead of subtraction, the modular distance (that is, the circumferential distance) is used. For example, the modular distance between 1° and 359° is 2°, not 358° (on a circle between 359° and 360°==0° - one degree, between 0° and 1° - also 1°, in total - 2 °).
The essence and meaning of average values.
Absolute and relative values.
Types of groups.
Depending on the tasks solved with the help of groupings, the following types are distinguished:
Typological
Structural
Analytical
The main task of the typology is to classify socio-economic phenomena by identifying groups that are homogeneous to qualitative relations.
Qualitative homogeneity is understood in the sense that, with respect to the property being studied, all units of the population obey the same law of development. For example: grouping of enterprises from economic sectors.
An absolute value is an indicator that expresses the size of a socio-economic phenomenon.
In statistics, a relative value is an indicator that expresses the quantitative relationship between phenomena. It is obtained by dividing one absolute value by another absolute value. The quantity with which we make comparisons is called basis or comparison base.
Absolute quantities are always named quantities.
Relative values are expressed in coefficients, percentages, ppm, etc.
The relative value shows how many times, or by what percentage, the compared value is greater or less than the comparison base.
In statistics, there are 8 types of relative quantities:
Averages are one of the most common summary statistics. They aim to characterize with one number a statistical population consisting of a minority of units. Averages are closely related to the law of large numbers. The essence of this dependence lies in the fact that with a large number of observations, random deviations from general statistics cancel each other out and, on average, a statistical pattern appears more clearly.
Using the method average The following main tasks are solved:
1. Characteristics of the level of development of phenomena.
2. Comparison of two or more levels.
3. Study of the interrelations of socio-economic phenomena.
4. Analysis of the location of socio-economic phenomena in space.
To solve these problems, statistical methodology has developed various types of averages.
To clarify the method for calculating the arithmetic mean, we use the following notation:
X - arithmetic sign
X (X1, X2, ... X3) - variants of a certain characteristic
n - number of population units
Average value of attribute
Depending on the source data, the arithmetic mean can be calculated in two ways:
1. If the statistical observation data are not grouped, or the grouped options have the same frequencies, then the simple arithmetic mean is calculated:
2. If the frequencies grouped in the data are different, then the weighted arithmetic mean is calculated:
Number (frequency) of options
Sum of frequencies
The arithmetic mean is calculated differently in discrete and interval variation series.
In discrete series, variants of a feature are multiplied by frequencies, these products are summed, and the resulting sum of products is divided by the sum of frequencies.
Let's consider an example of calculating the arithmetic mean in a discrete series:
In interval series, the value of a characteristic is given, as is known, in the form of intervals, therefore, before calculating the arithmetic mean, you need to move from an interval series to a discrete one.
The middle of the corresponding intervals is used as the Xi options. They are defined as half the sum of the lower and upper bounds.
If an interval does not have a lower limit, then its middle is determined as the difference between the upper limit and half the value of the following intervals. In the absence of upper limits, the middle of the interval is determined as the sum of the lower limit and half the value of the previous interval. After the transition to a discrete series, further calculations occur according to the method discussed above.
If weight fi are given not in absolute terms, but in relative terms, then the formula for calculating the arithmetic mean will be as follows:
pi - relative values of the structure, showing what percentage the frequencies of the variants are in the sum of all frequencies.
If the relative values of the structure are specified not in percentages, but in shares, then the arithmetic mean will be calculated using the formula:
Average value
Average value- numerical characteristics of a set of numbers or functions (in mathematics); - a certain number between the smallest and largest of their values.
Basic information
The starting point for the development of the theory of averages was the study of proportions by the school of Pythagoras. At the same time, no strict distinction was made between the concepts of average size and proportion. A significant impetus to the development of the theory of proportions from an arithmetic point of view was given by Greek mathematicians - Nicomachus of Geras (late 1st - early 2nd century AD) and Pappus of Alexandria (3rd century AD). The first stage in the development of the concept of average is the stage when the average began to be considered the central member of a continuous proportion. But the concept of average as the central value of a progression does not make it possible to derive the concept of average in relation to a sequence of n terms, regardless of the order in which they follow each other. For this purpose it is necessary to resort to a formal generalization of averages. The next stage is the transition from continuous proportions to progressions - arithmetic, geometric and harmonic ( English).
In the history of statistics, for the first time, the widespread use of averages is associated with the name of the English scientist W. Petty. W. Petty was one of the first to try to give the average value a statistical meaning, linking it with economic categories. But Petty did not describe the concept of average size or distinguish it. A. Quetelet is considered to be the founder of the theory of averages. He was one of the first to consistently develop the theory of average values, trying to provide a mathematical basis for it. A. Quetelet distinguished two types of averages - actual averages and arithmetic averages. Actually, the average represents a thing, a number, that actually exists. Actually, averages or statistical averages should be derived from phenomena of the same quality, identical in their internal meaning. Arithmetic averages are numbers that give the closest possible idea of many numbers, different, although homogeneous.
Each type of average can appear either in the form of a simple or in the form of a weighted average. The correct choice of the middle form follows from the material nature of the object of study. Simple average formulas are used if the individual values of the characteristic being averaged are not repeated. When in practical research individual values of the characteristic being studied occur several times in units of the population under study, then the frequency of repetitions of individual values of the characteristic is present in the calculation formulas of power averages. In this case, they are called weighted average formulas.
Hierarchy of averages in mathematics
- The average value of a function is a concept defined in many ways.
- More specifically, but based on arbitrary functions, Kolmogorov means are determined for a set of numbers.
- power average is a special case of Kolmogorov averages with ϕ (x) = x α (\displaystyle \phi (x)=x^(\alpha )) . Averages of different degrees are connected by inequality about averages. The most common special cases:
- arithmetic mean (α = 1 (\displaystyle \alpha =1));
- mean square (α = 2 (\displaystyle \alpha =2));
- harmonic mean (α = − 1 (\displaystyle \alpha =-1));
- by continuity as α → 0 (\displaystyle \alpha \to 0) the geometric mean is defined further, which is also the Kolmogorov mean for ϕ (x) = log x (\displaystyle \phi (x)=\log x)
- power average is a special case of Kolmogorov averages with ϕ (x) = x α (\displaystyle \phi (x)=x^(\alpha )) . Averages of different degrees are connected by inequality about averages. The most common special cases:
- More specifically, but based on arbitrary functions, Kolmogorov means are determined for a set of numbers.
- Weighted average is a generalization of the average to the case of an arbitrary linear combination:
- Weighted arithmetic mean.
- Weighted geometric mean.
- Weighted harmonic mean.
- average chronological - generalizes the values of a characteristic for the same unit or population as a whole, changing over time.
- logarithmic mean, determined by the formula a ¯ = a 1 − a 2 ln (a 1 / a 2) (\textstyle (\bar (a))=(\frac (a_(1)-a_(2))(\ ln(a_(1)/a_(2))))), used in heat engineering
- the logarithmic average, determined in electrical insulation in accordance with GOST 27905.4-88, is defined as l o g a ¯ = log a 1 + l o g a 2 + . . . + . . . l o g a n a 1 + a 2 + . . . + a n (\textstyle log(\bar (a))=(\frac (\log a_(1)+loga_(2)+...+...loga_(n))(a_(1)+a_( 2)+...+a_(n)))) (logarithm to any base)
In probability theory and statistics
Main article: Distribution center indicators- nonparametric means - mode, median.
- the average value of a random variable is the same as the mathematical expectation of a random variable. Essentially, it is the average value of its distribution function.
What sign denotes the arithmetic mean?
Let's say the sum is capital epsilon...
Ksenia
The arithmetic mean is the limit around which the individual values of the observed and studied characteristics are grouped. The arithmetic mean is the quotient of dividing the sum of the values of a particular characteristic by the number of elements in the population. In statistics, the arithmetic mean is usually denoted through individual values of a characteristic (or particular results of an experiment) - through x1, x2, x3, etc., and the total number of characteristics (or the number of experiments) - n.
With a large number of measurements, positive and negative random errors occur equally often. From repeated measurements of any physical quantity, its arithmetic mean value can be determined. Repeated measurements also make it possible to establish the accuracy of the measurement, both for the final result and for individual measurements, i.e., to find the boundaries within which the obtained result of the measured value lies.
With n measurements of a certain quantity, we obtain n different values. The closest to the true value of the measured value will be the arithmetic mean of all measurements.
If we denote individual measurements by a\, az, a3, ..an, then the arithmetic mean value of the measured value is determined by the formula:
P
n - at + ag + - + D„_\1 a,-
A _ ------------------
=Y-^
^J P
The values of individual measurements differ from the arithmetic mean value a0 by the following values:
The absolute values of the differences (Da^Dag,...) between the arithmetic mean value of the measured quantity and the value of individual measurements are called absolute errors of individual measurements. The arithmetic mean of the absolute errors of all measurements, which is necessary to determine the relative measurement error and record the final result, is calculated by the formula:
^-. (2)
This error is called the average absolute measurement error. By accepting one sign of absolute errors, we thereby deliberately take the largest possible error.What is the arithmetic mean? How to find the arithmetic mean?
Formula for arithmetic average?
Alex-89
The arithmetic mean of several numbers is the sum of these numbers divided by their number.
x av - arithmetic mean
S - sum of numbers
n - number of numbers.
For example, we need to find the arithmetic mean of the numbers 3, 4, 5 and 6.
To do this, we need to add them up and divide the resulting sum by 4:
(3 + 4 + 5 + 6) : 4 = 18: 4 = 4,5.
Alsou - sh
As a mathematician, I am interested in questions on this subject.
I'll start with the history of the issue. Average values have been thought about since ancient times. Arithmetic mean, geometric mean, harmonic mean. These concepts were proposed in ancient Greece by the Pythagoreans.
And now the question that interests us. What is meant by arithmetic mean of several numbers:
So, to find the arithmetic mean of numbers, you need to add all the numbers and divide the resulting sum by the number of terms.
The formula is:
Example. Find the arithmetic mean of the numbers: 100, 175, 325.
Let's use the formula for finding the arithmetic mean of three numbers (that is, instead of n there will be 3; you need to add up all 3 numbers and divide the resulting sum by their number, i.e. by 3). We have: x=(100+175+325)/3=600/3=200.
Answer: 200.
Arithmetic is considered the most elementary branch of mathematics and studies simple operations with numbers. Therefore, the arithmetic mean is also very easy to find. Let's start with a definition. The arithmetic mean is a value that shows which number is closest to the truth after several successive operations of the same type. For example, when running a hundred meters, a person shows a different time each time, but the average value will be within, for example, 12 seconds. Finding the arithmetic mean in this way comes down to sequentially summing all the numbers in a certain series (race results) and dividing this sum by the number of these races (attempts, numbers). In formula form it looks like this:
Sarif = (Х1+Х2+..+Хn)/n
The arithmetic mean is the average number between several numbers.
For example, between the numbers 2 and 4, the average number is 3.
The formula for finding the arithmetic mean is:
You need to add up all the numbers and divide by the number of these numbers:
For example, we have 3 numbers: 2, 5 and 8.
Finding the arithmetic mean:
X=(2+5+8)/3=15/3=5
The scope of application of the arithmetic mean is quite wide.
For example, knowing the coordinates of two points on a segment, you can find the coordinates of the middle of this segment.
For example, the coordinates of the segment: (X1,Y1,Z1)-(X2,Y2,Z2).
Let us denote the middle of this segment by coordinates X3,Y3,Z3.
We separately find the midpoint for each coordinate:
Beautiful glade
The arithmetic mean is numbers added together and divided by their number, the resulting answer is the arithmetic mean.
For example: Katya put 50 rubles in the piggy bank, Maxim 100 rubles, and Sasha put 150 rubles in the piggy bank. 50 + 100 + 150 = 300 rubles in the piggy bank, now we divide this amount by three (three people put money in). So 300: 3 = 100 rubles. These 100 rubles will be the arithmetically average, each of them put in the piggy bank.
There is such a simple example: one person eats meat, another person eats cabbage, and the arithmetically average they both eat cabbage rolls.
The average salary is calculated in the same way...
The arithmetic mean is the average of the given...
Those. Simply, we have a number of sticks of different lengths and want to find out their average value..
It is logical that for this we bring them together, getting a long stick, and then divide it into the required number of parts..
Here comes the arithmetic mean...
This is how the formula is derived: Sa=(S(1)+..S(n))/n..
Birdie2014
The arithmetic mean is the sum of all values and divided by their number.
For example the numbers 2, 3, 5, 6. You need to add them up 2+ 3+ 5 + 6 = 16
We divide 16 by 4 and get the answer 4.
4 is the arithmetic mean of these numbers.
Azamatik
The arithmetic mean is the sum of numbers divided by the number of these same numbers. And finding the arithmetic mean is very simple.
As follows from the definition, we must take the numbers, add them and divide by their number.
Let's give an example: we are given the numbers 1, 3, 5, 7 and we need to find the arithmetic mean of these numbers.
- first add these numbers (1+3+5+7) and get 16
- We need to divide the resulting result by 4 (quantity): 16/4 and get the result 4.
So, the arithmetic mean of the numbers 1, 3, 5 and 7 is 4.
Arithmetic mean - the average value among the given indicators.
It is found by dividing the sum of all indicators by their number.
For example, I have 5 apples weighing 200, 250, 180, 220 and 230 grams.
We find the average weight of 1 apple as follows:
- we are looking for the total weight of all apples (the sum of all indicators) - it is equal to 1080 grams,
- divide the total weight by the number of apples 1080:5 = 216 grams. This is the arithmetic mean.
This is the most commonly used indicator in statistics.
Green cheburechek
We know this from school. Anyone who had a good math teacher could remember this simple action the first time.
When finding the arithmetic mean, you need to add up all the available numbers and divide by their number.
For example, I bought 1 kg of apples, 2 kg of bananas, 3 kg of oranges and 1 kg of kiwi at the store. How many kilograms of fruit did I buy on average?
7/4= 1.8 kilograms. This will be the arithmetic mean.
Byemon epu
I remember taking the final test in mathematics
So there it was necessary to find the arithmetic mean.
It’s good that kind people suggested what to do, otherwise there would be trouble.
For example, we have 4 numbers.
Add up the numbers and divide by their number (in this case 4)
For example the numbers 2,6,1,1. Add 2+6+1+1 and divide by 4 = 2.5
As you can see, nothing complicated. So the arithmetic mean is the average of all numbers.
