Plan:
Introduction
The following section:
1. The nature and importance of average quantities.
2. Average arithmetic quantities and their fields of application.
3. Arithmetic properties of the average.
4. The procedure for calculating the arithmetic mean using the "conditional moment" method.
5. Average geometric quantities.
References
Introduction
In our daily life, in our life, we meet and use average quantities at every step, but we usually use the word average itself rarely in our sentences . For example, to the question of how much salary you receive, we answer soums per month. In fact, we do not receive the same amount of salary every month , and here we mean the average monthly salary.
So what is an average and what are its characteristics? What types and forms of average quantities are there? It is natural that questions arise .
In general, the mean value is the value obtained by compressing the distribution series. It lies between the major and minor terms of the series
To simplify the problem, let's assume that we are gradually compacting the statistical series by compressing it from the side . In this case, the quantitative differences between its variants become larger, and their number decreases. Due to this, the number of line options increases . If we continue the process of compaction, the final result of the array variant is represented by a single quantity. The number of variants is equal to the total number of the initial line . This amount is the average amount of this series . It lies between the largest and smallest values of the series . Here, a statistical series means a pure mathematical series, that is, a series of pure numbers. These numbers are neither conditioned nor interrelated , they have complete freedom.
Statistical series are fundamentally different from series of mathematical numbers. They are a series of indicators that describe the phenomena of the material world , and their quantitative values depend on the conditions of their occurrence.
Average is an indicator that summarizes a statistical collection
Thus, the indicators that summarize the studied statistical set by variable characteristics are called average indicators (quantities) . Average quantities as an important descriptive parameter of variation series have the following properties (7.1. definition).
Version 7.1. Important properties of average quantities .
The average amount differs from the series amounts, it deviates from them
Averaging is like a mass process where the law of large numbers applies. Variants of a certain distribution line are added (combined) to each other one after the other . As a result, large and small quantities affect each other, combine and compress each other. The differences between them disappear. As a result, the series is flattened, its variants are multiplied by the amount and expressed by a certain average level . So, the characteristic of average quantities is that they ignore the distinguishing properties of a series of elements, from which they are always abstracted.
The average must meet a number of requirements to fully and accurately perform its functions.
the average quantity to perform its functions fully and accurately, the following conditions must be present:
 the set for which the average amount is determined must be homogeneous, in sufficient volume;
of the studied collection should be sufficiently consistent with the objective distribution law specific to the event. This requirement comes from the law of large numbers.
The internal law of the series expresses the necessary interrelationships between the basic properties of the statistical set and forms the average quantity . Circumstances, on the other hand, act as random forces that cause and effect the outcome and cause series quantities to deviate from the mean. Ultimately, certain quantitative values and distributional patterns of series variants are formed as a result of the interaction and interaction of the main internal causes and random forces of external conditions . Due to the law of large numbers, under the influence of randomness, the differences between these quantities feed each other, and on the average, a tendency, a regularity, is manifested^{1}.
In fact, if we define the value of certain quantities that arise under the influence of the main internal causes as Xai and the result of random causes, then some quantities of the terms of the series
From this:or (7.1)
having negative and positive values , their sum is equal to zero, and therefore . As a result .
Therefore, the average quantities represent the general laws characteristic of statistical collections.
The average amount is like the center of gravity. It also has real meaning and significance as the point on which the weight of a body or body falls through all equally acting forces .
1.2. Types of average amounts and their calculation procedure
There are many different types and forms of averages in statistics . For example, aggregate (invisible form) mean , arithmetic mean, geometric mean, harmonic mean, quadratic mean, cubic mean, chronological mean, etc. In addition to these, distribution series also have mean quantities (variant values) that perform a function similar to the mean . These include the mode, median, and various quantiles. They are called ordinal or periodic middle terms (quantities) of the series.
average quantities depends on the goals and objectives of the investigation, the characteristics of the process and event being studied, and the nature of the data at our disposal in certain circumstances.
Arithmetic average is an average based on such a scientific rule that if some values of the symbol are replaced with it, their total sum should not change and should be distributed proportionally to the number of units of the set.
The arithmetic mean is the simplest and most widely used type of average. It is based on the total product (sum) obtained by adding certain quantitative values of the studied symbol in the units of the set and the number of units. If we look at the average arithmetic quantity from the point of view of a variation series, it is such an average value of the variant of that series that when calculating it, the total sum of the values of the variants is considered as a constant quantity and is interpreted as distributed proportionally to the number of variants. Therefore, the position of the average arithmetic quantity in the distribution line is determined by the fact that the values of some variants are equal to the twosided difference from it^{2}..
Arithmetic average has simple and weighted forms .
The simple arithmetic average is determined by adding together some quantities of the studied character (that is, the values of the series variants) and dividing the obtained sum by their number (that is, the number of series variants). (7.1)
Here: is the sum sign.
– some values of the character being studied (line variants)
N is their number (number of row options)
created 10, 12, 16, 12, 10, 14, 12, 16, 12, 16 total 150 products in one day (pieces) , then on average one worker per day (10+12 +16+....+16)/10=130/10=13 pcs.
The weighted arithmetic mean is the mean calculated by weighting the quantities being averaged by the number of times they occur in the set.
If X is n quantities of character
, , ,..., or ( )
in an appropriate manner
f1, f2, ..... , fn or fi ( )
A general expression of the arithmetic mean when observed from time to time
(7.2) becomes . This is the formula for the weighted arithmetic mean , where phi is called the weight of the mean. In the above example, if we group the workers according to the daily production volume ;
Product, pcs

10

12

14

16

Number of workers, people

2

4

1

3

From this:
piece
In intermediate series, the average amount is found by determining the group averages and the overall average from them, and they can also be calculated on the basis of relative amounts in this order.
To do this, first, for each interval group, its lower and upper limits are calculated as group averages equal to half of the sum index, and then the overall average for the entire series is determined.
the average for a series of relative quantities , then this problem can be solved correctly only if the average quantity is considered to have a logical structure like the relative quantities being averaged. For example,
Table 7.1
average relative quantity in interval variation series
by the level of contract fulfillment (percentage)
The number of enterprises
The volume of product delivery under the contract (million soums), fi
Average contract fulfillment rate %,
Group of enterprises by the level of contract fulfillment (percentage)

Number of enterprises
n _{i}

Product delivery volume under the contract (million soums), f _{i}

Average contract fulfillment rate %,





A

1

2

3

4

5

6

7

up to 80
8090
90100
100110
110120
120130
130 and above

1
3
5
9
7
5
4

20
60
100
180
140
100
80

75
85
95
105
115
125
135

1500
5100
9500
18900
16100
12500
10800

1
3
5
9
7
5
4

75
255
475
945
805
625
540

3
2
1
0
1
2
3

Jami

34

680


74400

34

3720


The lower limit of the first group is unknown, we conditionally accept it as the difference between the width of the interval of the next group (9080=10) and the upper limit of this group (80), that is, 8010=70%. As a result, the average level of contract fulfillment for this group is (70+80)/2=75%. For other groups, we calculate half of the sum of the lower and upper levels . In the latter group, the upper limit is unknown. We will tentatively take it as equal to adding the previous group's interval width on top of the lower level of this group (130%), that is, 130+10=140 % . In that case, the average level of contract fulfillment for the last group is (130+140)/2=135%. Now we need to set the weight for the average .
It is known that in order to determine the level of fulfillment of the contract, the volume of the product delivered in reality is compared with the amount specified in the contract . Therefore, the delivery of the product specified in the contract is taken as a weighted average . Information about him is given in column 2. And so:
stipulated in the contract to supply the same amount of products , then the number of enterprises can be obtained by weighting the weighted arithmetic average. In our example, the contracts for one enterprise in all groups average 20 mln. soum product delivery is planned. Therefore, the average level of performance of the general contract can be determined as follows:
Arithmetic average has a number of properties:
some values of the symbol (some values of the row variants) and their arithmetic mean levels is always equal to 0, that is: .
2. The sum of the squares of the differences between certain quantities of the sign and their arithmetic mean has a minimum value, that is, or
If each value of the symbol is divided (or multiplied) by a fixed arbitrary number (V), then the arithmetic mean is reduced (or increased) by this number of times:
constant arbitrary number (A) is subtracted or added from each value of the symbol , then the arithmetic mean value is also reduced or increased by this number.
5. If the arithmetic mean weight values are divided (or multiplied) by a constant arbitrary number (s), then the mean value does not change.
a symbol for two or more sets is equal to its total aggregated average value for the set:
invariant arbitrary number A is subtracted from the row variants and the result is divided by another arbitrary number B. As a result, a row is created from the given row. This is the arithmetic mean for the series
Then y is multiplied by the number B and the number A is added to the result. In the end, the actual arithmetic mean of the initial series is derived
In rows with equal spacing, it is recommended to take the middle value of the variant as "A", and the width of the interval as "B" .
In our example above, in column 7 of table 7.1, the values "Y" are given as A=100, B=10. So,
Geometric mean quantity
2.1. Concept of geometric mean quantity
The geometric mean is the average based on such a scientific rule that, as a result of replacing the averaged quantities, the result of the mutual multiplication of these quantities should not change and should be distributed according to the geometric progression of the collection units.
asymmetric, especially strongly skewed (or peaked, elongated) distribution series. Most events in socioeconomic life have this distribution.
geometric mean is the derivative of taking the cross product of the series terms ( ) under the root of n degree, i.e. (7.3).
Here: denotes the product of terms . For example, if a house is 5 m wide, 11.4 m long and 4 m high, what is the average length of the side of the house?
In a pronounced asymmetric distribution (if it arises from the nature of the phenomenon rather than chance), the arithmetic mean is always to some extent a "false" mean .
In such conditions, the geometric mean represents the central tendency of the distribution in a clear sense. As a result of the combination of random variability of the sign with legitimate, stable differences (for example, differences between the wages of employees of equal qualifications), an asymmetric distribution is formed, which, when transformed into a logarithmic scale, takes on a "normal" form, that is, for the logarithms of the sign, it has the quality of a normal distribution. will have^{3}.
The nature and properties of such distribution series find their exact expression in the geometric mean, because it is based on the logarithms of the terms of the series . In fact, if we logarithmize the expression (7.3): (7.4).
In our example above:
potentially,
2.2. Determining the geometric mean for relative changes
relative changes is expressed by the following formula:
(7.5)
or
(7.5a)
Here:
Ki  period (chain) growth coefficients in dynamics series , and in variation series  the ratio of each term (variant) to the preceding term (variant);
P is the sign of multiplication .
Example: Cotton productivity is expressed as follows depending on the amount of fertilizer.
Table 7.2
Cotton yield in fertilized field
Indicators

unfertilized field

Named after fertilizer

Given a little less than the norm

Given in the norm

Given more than the norm

Productivity (s/ha)

10

13

19.5

35.7

39.3

Compared to the previous level (K _{i })



1.3

1.5

1.83

1.1

fertilizer :
After potentiation _
or 140.8%.
Therefore, cotton yield increased by 1.41 times or 41% due to increasing the rate of fertilization in fertilized fields .
2.3. Determining the geometric mean of the distribution based on the relative change
mean for a variational series can also be determined by relative variation. For this, it is necessary to make the following mathematical changes to the formula (7.3) , as a result

(7.7)
Since the relative changes calculated with respect to the previous terms are chain coefficients K (similar to the relative quantities of chain dynamics) and their number (m) is one less than the number of series terms (n), n=m+1

where:  basic ( compared to the initial period or limit) growth coefficients.
Since the relative changes calculated with respect to the initial term of the series are ground coefficients (Kzam, similar to the relative quantities of dynamics with a constant basis!) and their number (m) is one less than the number of terms of the series, n=m +1 or. (7.8)
mean productivity calculated based on the data of table 7.1
was equal.
If we calculate according to formula 7.7a ,
If we logarithmize,
Let's make it potential
If we calculate the geometric mean productivity for the areas that were given fertilizer only
Let's make it potential
So , the yield in the fertilized areas is 2.44 times higher than in the unfertilized area (24.4:10). The reason why this result differs from the result of the previous calculation (1.41) is that we are talking about the result (increased productivity) obtained by increasing the fertilizer rate in the previously fertilized areas . Here, it is assumed that the productivity of the entire fertilized area will increase compared to the unfertilized area . So, in this case, the full effect of the fertilizer is being determined, while earlier the effect of the additional fertilizer was estimated.
the formula leads to the same conclusion. In order to determine the average productivity in all areas according to this formula , we determine the growth coefficients of the ground (compared to the unfertilized area): 13/10=1.3; 19.5/10=1.95; 35.7/10=3.57; 39.3/10=3.93.
As a result
If we logarithmize
Let's make it potential
average yield for the fields that were only fertilized according to the formula (7.8), it is necessary to calculate the coefficients of land change compared to the yield of the field that was initially given a little fertilizer (13 s/ha), or is 19.5/13=1.5; 35.7/13=2.746; 39.3/13=3.02.
this case.
From this
Let's make it potential
geometric mean , as a result of logarithmization of series quantities, numbers of different sizes come to one base (decimal or natural). At the same time, in this process, the differences in the quality of the phenomenon manifested in the asymmetry of distribution also come to the same basis, the initial comparative state, because they are directly expressed in the quantities of the phenomenon, in the differences between them. If we look at the qualitative changes that occur as a result of the accumulation of quantitative changes, just as when the rubber is stretched and released, it takes the initial state, the logic that when the quantities are brought back to the initial basis, the quality also returns to the initial state is the geometric mean. is based on its essence.
2.4. Mathematical properties of the geometric mean
The geometric mean , like the arithmetic mean, has a number of mathematical properties. If the positive and negative deviations of some averaged quantities from the arithmetic mean are equal to each other, then for the first property, for the geometric mean, the relative differences of variable quantities from this mean are equal to each other . Here, the relative difference refers to the ratio of the value of a specific variable to the geometric mean.
In fact, if relative differences are represented by these, then these differences are greater or less than one, depending on whether the value of the variable is greater or less than the geometric mean. By multiplying the relative differences with each other and taking into account the formula (7.3).
If we use logarithmization to convert the geometric mean into an arithmetic mean, then the logarithm of the geometric mean has all the properties of the arithmetic mean. In the process of calculating both the arithmetic mean and the geometric mean, some quantities of the studied symbol (values of the row variants (1,n)) are replaced by their average, and this replacement is performed according to a certain rule (requirement). For example , is based on the rule that the arithmetic mean is the geometric mean. Mathematically, ( 1,n )  variable quantities, their function is to find the arithmetic mean, and the geometric mean is considered a constant, that is, an unchanging quantity .
So, in mathematical terms, the average quantity ( ) is a function of variables ( ) such that [ ], when determining it, the sum of arithmetic operations performed with variables is considered a constant, that is, an unchanging quantity.
the harmonic mean, the sum of the inverse values of the variable quantities is considered as a constant quantity.
The harmonic mean is such an average quantity that, when changing the variables, the sum of their inverse values is regarded as an invariant quantity.
self evident that when determining the average for economic events, this rule must be based on the economic nature of the event, of course. Otherwise, the obtained average amount and its qualitative basis will not be equivalent to each other.
Simple harmonic mean : (7.9)
or in short:
The weighted average harmonic quantity is used when the quantities being averaged have different weights (Wi) and is calculated as follows: (7.10)
It is known that any average quantity appears from the ratio of two indicators to each other. The first indicator represents the total size of the character being studied, while the second indicator determines the number (weight, encounter rate) of the owner of this character. If certain levels of the signal are known with information representing the magnitude of the signal (ie, the rate of the ratio) , then the average is calculated using the harmonic mean formula. If the size of the symbol and the number of sets are known, but some levels are unknown, then the aggregate average formula is used, i.e. (7.11)
And finally, if the number of variants (objects) with some variants for the intervals of the set parts is known, then the arithmetic mean is used.
Therefore, before calculating the average amount, it is necessary to determine the ratio that represents its essence. Then, depending on which data is known and which is unknown, the average should be calculated using one or another formula.
For example: the following information is provided:
Table 7.3
The procedure for calculating the average salary by enterprises
Indicators

unfertilized field

Named after fertilizer

Given a little less than the norm

Given in the norm

Given more than the norm

Productivity (s/ha)

10

13

19.5

35.7

39.3

Compared to the previous level (K _{i })



1.3

1.5

1.83

1.1

Calculate the average wages for January, February, March, and the first quarter for a set of businesses .
It is known that in order to calculate the average salary, it is necessary to divide the salary fund by the number of employees. In January, the rate of the ratio and the individual degrees of the sign are given. But the denominator of the ratio or the number of workers is unknown. So, to calculate the average quantity, we need to apply the harmonic mean formula according to our condition^{4}..
The average salary is soum.
In February, the denominator of the ratio and the individual degrees of the sign are given. But the pace of the ratio or the payroll is unknown. In such cases, it is necessary to use the arithmetic weighted average formula to calculate the average amount according to our above condition :
Average salary (February) sum m.
In March, both the rate and the denominator of the ratio are given. No extra work is required to calculate the average .
Average wage (March) = Wage fund / Number of workers = =257500/2000 = 128.75 soums.
Average salary (1st quarter)* = (215000+237500+257500)/ (2000+2000+2000) = 710000 / 6000 = 118.33 soums.
mean squared deviation and indicators based on it.
Mean squared _
The mean square means such an average that, when determining it, the sum of squares must be kept unchanged when replacing the sign quantities with their square mean.
If, in the process of replacing certain quantities of the sign with the average, it is necessary to keep the sum of their squares unchanged, then this average is called a quadratic average, i.e. (7.12)
Cubic average _
Similarly, if, according to the conditions of the problem, it is necessary to replace them with the average, ensuring that the sum of the cubes of certain quantities of the sign remains unchanged, then the cubic average is used: (7.13)
Graded averages _
If, when determining the average , it is necessary to ensure that the sum of the klevel values of the sign quantities remains unchanged, then we have the klevel average, i.e. (5.14)
or logarithmically (5.14a)
considered above belong to the type of general level averages and differ from each other by the level indicator. For example, kq1 means the arithmetic mean, kq2 means the quadratic mean, kq3 means the cubic mean, k=0 means the geometric mean, k=1 means the harmonic mean. we will have an average.
The larger the rank indicator , the larger the average quantity (if the quantities being averaged are variable, of course).
If the peak quantities of the symbol are equal to each other, that is, a constant quantity, then all averages are equal to this constant.
Thus, there is the following reciprocal ratio of types of averages, which is called the rule of majorant of averages .
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