Arithmetic mean sign. Determining the mean, variance and shape of the distribution. Descriptive statistics. Can it happen that the arithmetic mean becomes equal to the geometric mean?

The topic of arithmetic mean and geometric mean is included in the mathematics program for grades 6-7. Since the paragraph is quite easy to understand, it is quickly completed, and by the end academic year schoolchildren forget him. But knowledge of basic statistics is needed for passing the Unified State Exam, and also for international exams SAT. Yes and for everyday life developed analytical thinking never hurts.

How to calculate the arithmetic mean and geometric mean of numbers

Let's say there is a series of numbers: 11, 4, and 3. The arithmetic mean is the sum of all numbers divided by the number of given numbers. That is, in the case of the numbers 11, 4, 3, the answer will be 6. How do you get 6?

Solution: (11 + 4 + 3) / 3 = 6

The denominator must contain a number equal to the number of numbers whose average needs to be found. The sum is divisible by 3, since there are three terms.

Now we need to figure out the geometric mean. Let's say there is a series of numbers: 4, 2 and 8.

Average geometric numbers is called the product of all given numbers, located under the root with a degree equal to the number of given numbers. That is, in the case of numbers 4, 2 and 8, the answer will be 4. This is how it turned out:

Solution: ∛(4 × 2 × 8) = 4

In both options, we got whole answers, since special numbers were taken for the example. This does not always happen. In most cases, the answer has to be rounded or left at the root. For example, for the numbers 11, 7 and 20, the arithmetic mean is ≈ 12.67, and the geometric mean is ∛1540. And for the numbers 6 and 5, the answers will be 5.5 and √30, respectively.

Can it happen that the arithmetic mean becomes equal to the geometric mean?

Of course it can. But only in two cases. If there is a series of numbers consisting only of either ones or zeros. It is also noteworthy that the answer does not depend on their number.

Proof with units: (1 + 1 + 1) / 3 = 3 / 3 = 1 (arithmetic mean).

∛(1 × 1 × 1) = ∛1 = 1(geometric mean).

Proof with zeros: (0 + 0) / 2=0 (arithmetic mean).

√(0 × 0) = 0 (geometric mean).

There is no other option and cannot be.

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. Average values are closely related to the law large numbers. The essence of this dependence is that with a large number of observations, random deviations from the 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 average.

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, feature variants 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.

Basics

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 averages, 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 - special case Kolmogorov averages for ϕ (x) = x α (\displaystyle \phi (x)=x^(\alpha )) . Averages of different degrees are connected by inequality about averages. The most common special cases:
    1. arithmetic mean (α = 1 (\displaystyle \alpha =1));
    2. mean square (α = 2 (\displaystyle \alpha =2));
    3. harmonic mean (α = − 1 (\displaystyle \alpha =-1));
    4. 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)
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.
average value random variable- the same as mathematical expectation random variable. Essentially, it is the average value of its distribution function.

What symbol denotes the arithmetic mean?

Let's say the sum is capital epsilon...

Xenia

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.
At large quantities measurements, positive and negative random errors occur equally often. According to multiple measurements of any physical quantity you can determine its arithmetic mean value. 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 resulting 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:
n
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 the 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 average arithmetic 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 are proposed in ancient Greece 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 every time different times, 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 put, 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. Who had good teacher in mathematics, it was possible to 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 good people They told me 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.

Obtained by adding all members of a number series and dividing the sum by the number of members. For example, the arithmetic value of 7, 20, 152 and 305 is 484/4 = 121. However, the average value does not allow us to judge the spread of numbers. compare: geometric mean.

Business. Dictionary. - M.: "INFRA-M", Publishing House "Ves Mir". Graham Betts, Barry Brindley, S. Williams and others. General edition: Doctor of Economics Osadchaya I.M.. 1998 .

See what "ARITHMETICAL AVERAGE" is in other dictionaries:

- (arithmetic mean) The sum of N numbers x1 x2,...,xN divided by N, which is expressed by the formula (Σixi)/N. The arithmetic mean can be calculated for any finite sequence N numbers, where they can be positive, zero or... ... Economic dictionary

- (arithmetic mean) The average value obtained by adding all the terms of a number series and dividing the sum by the number of terms, for example, the arithmetic mean of 7, 20, 107 and 350 is 484/4 = 121. However, the average value does not allow one to judge ... ... Financial Dictionary

arithmetic mean- aritmetinis vidurkis statusas T sritis fizika atitikmenys: engl. arithmetic average; arithmetic mean; arithmetical mean vok. arithmetischer Mittelwert, m; arithmetisches Mittel, n rus. arithmetic mean, n; arithmetic mean value, n … Fizikos terminų žodynas

arithmetic mean (value of the result of geodetic measurements)- 3.7.2 arithmetic mean (value of the result of geodetic measurements) An estimate of the value of a geodetic quantity from multiple equal-precision measurements, obtained by the formula where is the result of an individual measurement, n is the number of measurements. Source …

This term has other meanings, see average meaning. In mathematics and statistics, the arithmetic mean is one of the most common measures of central tendency, representing the sum of all observed values divided by their ... ... Wikipedia

arithmetic mean- 3.1 arithmetic mean; arithmetic mean / average: The sum of values divided by their number. [ISO 3534 1:1993, 2.26] Source... Dictionary-reference book of terms of normative and technical documentation

The average value, which characterizes any group of observations, is calculated by adding the numbers from this series and then dividing the resulting sum by the number of summed numbers. If one or more numbers included in the group... ... Medical terms

NUMBER AVERAGE, ARITHMETIC MEAN- (arithmetic mean) average value characterizing any group of observations; is calculated by adding the numbers from this series and then dividing the resulting sum by the number of summed numbers. If one or more numbers... ... Explanatory dictionary of medicine

- (average) One number representing a series of numbers; average value. See: arithmetic mean; geometric mean; median. Business. Explanatory dictionary. M.: INFRA M, Publishing house... ... Dictionary of business terms

- (average) 1. One number representing a series of numbers; average value. See: arithmetic mean; geometric mean; median. 2. Method of dividing losses in property insurance... Financial Dictionary

This term has other meanings, see average meaning.

Arithmetic mean(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").

To denote the arithmetic mean of the entire population it is used greek letterμ. 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.

It has been proven in elementary algebra 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:

x 1 + x 2 + x 3 3 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3))(3)).)

For four numbers, you need to add them and divide by 4:

x 1 + x 2 + x 3 + x 4 4 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3)+x_(4))(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 statistics

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 the residents, will surprisingly yield large number because of 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 Investment

If the numbers multiply, 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 statistics

When 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.

Basics

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:
    1. arithmetic mean (α = 1 (\displaystyle \alpha =1));
    2. mean square (α = 2 (\displaystyle \alpha =2));
    3. harmonic mean (α = − 1 (\displaystyle \alpha =-1));
    4. 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)
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 symbol 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 became 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 put everything under 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.

Most of all in eq. In practice, we have to use the arithmetic mean, which can be calculated as the simple and weighted arithmetic mean.

Arithmetic average (SA)-n The most common type of average. It is used in cases where the volume of a varying characteristic for the entire population is the sum of the values of the characteristics of its individual units. Social phenomena are characterized by the additivity (totality) of the volumes of a varying characteristic; this determines the scope of application of SA and explains its prevalence as a general indicator, for example: the general salary fund is the sum of the salaries of all employees.

To calculate SA, you need to divide the sum of all feature values by their number. SA is used in 2 forms.

Let's first consider a simple arithmetic average.

1-CA simple (initial, defining form) is equal to the simple sum of the individual values of the characteristic being averaged, divided by the total number of these values (used when there are ungrouped index values of the characteristic):

The calculations made can be generalized into the following formula:

(1)

Where - the average value of the varying characteristic, i.e., the simple arithmetic average;

means summation, i.e. the addition of individual characteristics;

x- individual values of a varying characteristic, which are called variants;

n - number of units of the population

Example 1, it is required to find the average output of one worker (mechanic), if it is known how many parts each of 15 workers produced, i.e. given a series of ind. attribute values, pcs.: 21; 20; 20; 19; 21; 19; 18; 22; 19; 20; 21; 20; 18; 19; 20.

Simple SA is calculated using formula (1), pcs.:

Example2. Let's calculate SA based on conditional data for 20 stores included in the trading company (Table 1). Table.1

Distribution of stores of the trading company "Vesna" by sales area, sq. M

Store no.		Store no.

To calculate the average store area ( ) it is necessary to add up the areas of all stores and divide the resulting result by the number of stores:

Thus, the average store area for this group of retail enterprises is 71 sq.m.

Therefore, to determine a simple SA, you need to divide the sum of all values of a given characteristic by the number of units possessing this characteristic.

Where f 1 , f 2 , … ,f n – weight (frequency of repetition of identical signs);

– the sum of the products of the magnitude of features and their frequencies;

– the total number of population units.

- SA weighted - With The middle of options that are repeated a different number of times, or, as they say, have different weights. The weights are the number of units in different groups aggregates (identical options are combined into a group). SA weighted – average of grouped values x 1 , x 2 , .., x n, – calculated:

(2)

Where X- options;

f- frequency (weight).

Weighted SA is the quotient of dividing the sum of the products of options and their corresponding frequencies by the sum of all frequencies. Frequencies ( f) appearing in the SA formula are usually called scales, as a result of which the SA calculated taking into account the weights is called weighted.

We will illustrate the technique of calculating weighted SA using example 1 discussed above. To do this, we will group the initial data and place them in the table.

The average of the grouped data is determined as follows: first, the options are multiplied by the frequencies, then the products are added and the resulting sum is divided by the sum of the frequencies.

According to formula (2), the weighted SA is equal, pcs.:

Distribution of workers for parts production

The data presented in the previous example 2 can be combined into homogeneous groups, which are presented in table. Table

Distribution of Vesna stores by sales area, sq. m

Thus, the result was the same. However, this will already be a weighted arithmetic mean value.

In the previous example, we calculated the arithmetic average provided that the absolute frequencies (number of stores) are known. However, in a number of cases, absolute frequencies are absent, but relative frequencies are known, or, as they are commonly called, frequencies that show the proportion or the proportion of frequencies in the entire set.

When calculating SA weighted use frequencies allows you to simplify calculations when the frequency is expressed in large, multi-digit numbers. The calculation is made in the same way, however, since the average value turns out to be increased by 100 times, the result should be divided by 100.

Then the formula for the arithmetic weighted average will look like:

Where d– frequency, i.e. the share of each frequency in the total sum of all frequencies.

(3)

In our example 2, we first determine the share of stores by group in the total number of stores of the Vesna company. So, for the first group the specific gravity corresponds to 10%
. We get the following data Table3