The statistical totality consists of a set of units, objects or phenomena homogeneous in some respects and at the same time different in magnitude features. The value of the features of each object is determined both by the common for all units of the population, and by its individual features.

Analyzing the ordered distribution series (ranking, interval, etc.), one can notice that the elements of the statistical population are clearly concentrated around some central values. Such a concentration of individual values ​​of a feature around some central values, as a rule, takes place in all statistical distributions. The tendency of individual values ​​of the studied feature to group around the frequency distribution center is called central trend. To characterize the central trend of distribution, generalizing indicators are used, which are called average values.

Average value in statistics, they call a generalizing indicator that characterizes the typical size of a feature in a qualitatively homogeneous population under specific conditions of place and time and reflects the value of a variable feature per unit of the population. The average value is calculated in most cases by dividing the total volume of the feature by the number of units that have this feature. If, for example, the monthly wage bill and the number of workers per month are known, then the average monthly wage can be determined by dividing the wage bill by the number of workers.

The average values ​​are such indicators as the average length of the working day, week, year, the average wage category of workers, the average level of labor productivity, the average national income per capita, the average crop yield in the country, the average food consumption per capita, etc. .d.

Average values ​​are calculated from both absolute and relative values, they are named indicators and are measured in the same units of measurement as the averaged attribute. They characterize the value of the studied population with one number. The average values ​​reflect the objective and typical level of socio-economic phenomena and processes.

Each average characterizes the studied population according to one of some signs, but to characterize any population, describe its typical features and qualitative features, a system of average indicators is needed. Therefore, in the practice of domestic statistics, to study socio-economic phenomena, as a rule, it is used system of averages. So, for example, indicators of average wages are evaluated together with indicators of labor productivity (average output per unit of working time), capital-labor ratio and energy conservation, the level of mechanization and automation of work, etc.

In statistical science and practice, averages are extremely important. The method of averages is one of the most important statistical methods, and the average is one of the main categories of statistical science. The theory of averages occupies one of the central places in the theory of statistics. Average values ​​are the basis for calculating the indicators of variation (Section 5), sampling errors (Section 6), ANOVA (Section 8) and correlation analysis (Section 9).

it is also impossible to present statistics without indexes, and the latter are essentially averages. The use of the method of statistical groupings also leads to the use of average values.

As already noted, the grouping method is one of the main methods of statistics. The method of averages in combination with the method of groupings is an integral part of a scientifically developed statistical methodology. Average indicators organically complement the method of statistical groupings.

Average values ​​are used to characterize the change in phenomena over time, to calculate the average growth and growth rates. For example, a comparison of the average growth rates of labor productivity and its remuneration for a certain period (a number of years) reveals the nature of the development of the phenomenon over the studied period of time, separately labor productivity and separately wages. Comparison of the growth rates of these two phenomena gives an idea of ​​the nature and peculiarity of the ratio of growth or decrease in labor productivity relative to its payment for certain periods of time.

In all cases, when it becomes necessary to characterize by one number the totality of values ​​of a characteristic that change, its average value is used.

In the statistical population, the value of the attribute changes from object to object, that is, it varies. By averaging these values ​​and providing the level of the attribute value to each member of the population, we abstract from the individual values ​​of the attribute, thereby, as it were, replacing the series of distribution of attribute values ​​with the same value equal to the average value. However, such an abstraction is justified only if the averaging does not change the main property in relation to the given feature as a whole. This is the main property of the statistical population, associated with the individual values ​​of the trait, and which, when averaged, must be kept unchanged, is called the defining property of the average in relation to the trait under study. In other words, the average, replacing the individual values ​​of the attribute, should not change the total volume of the phenomenon, i.e. obligatory such equality: the volume of the phenomenon is equal to the product of the average value by the size of the population. For example, if from three barley yield values ​​(x, = 20.0; 23.3; 23.6 centners / ha), the average (20.0 + 23.3 + 23.6) is calculated: 3 = 22.3 centners / ha, then according to the defining property of the mean, the following equality must be observed:

As can be seen from the above example, the average yield of barley does not coincide with any of the individual ones, since in no farm the yield obtained is 22.3 c/ha. However, if we imagine that each farm received 22.3 c/ha, then the total yield will not change and will be equal to 66.9 c/ha. Consequently, the average, replacing the actual value of individual individual indicators, cannot change the size of the entire sum of the values ​​of the studied trait.

The main value of average values ​​is their generalizing function, i.e. in replacing a set of different individual values ​​of a trait with an average value that characterizes the entire set of phenomena. The property of the average to characterize not individual units, but to express the level of the attribute per each unit of the population is its distinctive ability. This feature makes the average a generalizing indicator of the level of varying features, i.e. an indicator that is abstracted from the individual values ​​of the value of the attribute in individual units of the population. But the fact that the average is abstract does not deprive it of scientific research. Abstraction is a necessary degree of any scientific research. In the average value, as in any abstraction, the dialectical unity of the individual and the general is realized. The relationship between the average and individual values ​​of the average features is an expression of the dialectical connection between the individual and the general.

The use of averages should be based on the understanding and interconnection of the dialectical categories of the general and the individual, the mass and the individual.

The average value reflects the general that is formed in each individual, single object. Due to this, the average becomes of great importance for revealing the patterns inherent in mass social phenomena and not noticeable in single phenomena.

Necessity is combined with chance in the development of phenomena. Therefore, averages are related to the law of large numbers. The essence of this relationship lies in the fact that when calculating the average value, random fluctuations with different directions, due to the operation of the law of large numbers, are mutually balanced, canceled out, and the main regularity, necessity, and the influence of general conditions characteristic of this population are clearly displayed in the average value. The average reflects the typical, real level of the studied phenomena. Estimating these levels and changing them in time and space is one of the main problems of averages. So, through the averages, for example, the pattern of increasing labor productivity, crop yields, and animal productivity is manifested. Consequently, average values ​​are generalizing indicators in which the action of general conditions, the regularity of the phenomenon under study, finds its expression.

With the help of average values, they study the change in phenomena in time and space, trends in their development, connections and dependencies between features, the effectiveness of various forms of organization of production, labor and technology, the introduction of scientific and technological progress, the identification of a new, progressive in the development of certain social and economic phenomena and processes.

Average values ​​are widely used in the statistical analysis of socio-economic phenomena, since it is in them that the patterns and trends in the development of mass social phenomena that vary both in time and space find their manifestation. So, for example, the pattern of increasing labor productivity in the economy is reflected in the growth of average production per worker employed in production, the increase in gross yields - in the growth of average crop yields, etc.

The average value gives a generalized characteristic of the phenomenon under study on only one basis, which reflects one of its most important aspects. In this regard, for a comprehensive analysis of the phenomenon under study, it is necessary to build a system of average values ​​for a number of interrelated and complementary essential features.

In order for the average to reflect what is truly typical and natural in the studied social phenomena, when calculating it, it is necessary to adhere to such conditions.

1. The sign by which the average is calculated must be significant. Otherwise, an insignificant or distorted average will be obtained.

2. The average should be calculated only for a qualitatively homogeneous population. Therefore, the direct calculation of averages should be preceded by statistical grouping, which makes it possible to divide the studied population into qualitatively homogeneous groups. In this regard, the scientific basis of the method of averages is the method of statistical groupings.

The question of the homogeneity of the population should not be decided formally in terms of the form of its distribution. It, as well as the question of the typicality of the average, must be solved on the basis of the causes and conditions that form the aggregate. The aggregate is also homogeneous, the units of which are formed under the influence of common main causes and conditions that determine the general level of this feature, characteristic of the entire aggregate.

3. The calculation of the average value should be based on the coverage of all units of a given type or a sufficiently large set of objects so that random fluctuations mutually cancel each other out and a regularity, typical and characteristic sizes of the studied trait appear.

4. The general requirement in the calculation of any kind of averages is the obligatory preservation of the total volume of the attribute in the aggregate when replacing its individual values ​​with an average value (the so-called defining property of the average).

The average value is a generalizing indicator that characterizes the typical level of the phenomenon. It expresses the value of the attribute, related to the unit of the population.

The average value is:

1) the most typical value of the attribute for the population;

2) the volume of the sign of the population, distributed equally among the units of the population.

The characteristic for which the average value is calculated is called “averaged” in statistics.

The average always generalizes the quantitative variation of the trait, i.e. in average values, individual differences in the units of the population due to random circumstances are canceled out. In contrast to the average, the absolute value that characterizes the level of a feature of an individual unit of the population does not allow comparing the values ​​of the feature for units belonging to different populations. So, if you need to compare the levels of remuneration of workers at two enterprises, then you cannot compare two employees of different enterprises on this basis. The wages of the workers selected for comparison may not be typical for these enterprises. If we compare the size of wage funds at the enterprises under consideration, then the number of employees is not taken into account and, therefore, it is impossible to determine where the level of wages is higher. Ultimately, only averages can be compared, i.e. How much does one worker earn on average in each company? Thus, there is a need to calculate the average value as a generalizing characteristic of the population.

It is important to note that in the process of averaging, the aggregate value of the attribute levels or its final value (in the case of calculating average levels in a time series) must remain unchanged. In other words, when calculating the average value, the volume of the trait under study should not be distorted, and the expressions made when calculating the average must necessarily make sense.

Calculating the average is one common generalization technique; the average indicator denies the general that is typical (typical) for all units of the studied population, at the same time it ignores the differences between individual units. In every phenomenon and its development there is a combination of chance and necessity. When calculating averages, due to the operation of the law of large numbers, randomness cancels each other out, balances out, so you can abstract from the insignificant features of the phenomenon, from the quantitative values ​​of the attribute in each specific case. In the ability to abstract from the randomness of individual values, fluctuations, lies the scientific value of averages as generalizing characteristics of aggregates.

In order for the average to be truly typifying, it must be calculated taking into account certain principles.

Let us dwell on some general principles for the application of averages.

1. The average should be determined for populations consisting of qualitatively homogeneous units.

2. The average should be calculated for a population consisting of a sufficiently large number of units.

3. The average should be calculated for the population, the units of which are in a normal, natural state.

4. The average should be calculated taking into account the economic content of the indicator under study.

5.2. Types of averages and methods for calculating them

Let us now consider the types of averages, the features of their calculation and areas of application. Average values ​​are divided into two large classes: power averages, structural averages.

Power-law averages include the most well-known and commonly used types, such as geometric mean, arithmetic mean, and mean square.

The mode and median are considered as structural averages.

Let us dwell on power averages. Power averages, depending on the presentation of the initial data, can be simple and weighted. simple average is calculated from ungrouped data and has the following general form:

,

where X i is the variant (value) of the averaged feature;

n is the number of options.

Weighted average is calculated by grouped data and has a general form

,

where X i is the variant (value) of the averaged feature or the middle value of the interval in which the variant is measured;

m is the exponent of the mean;

f i - frequency showing how many times the i-e value of the averaged feature occurs.

If we calculate all types of averages for the same initial data, then their values ​​will not be the same. Here the rule of majorance of averages applies: with an increase in the exponent m, the corresponding average value also increases:

In statistical practice, more often than other types of weighted averages, arithmetic and harmonic weighted averages are used.

Types of Power Means

Type of power middle	Indicator degrees (m)	Calculation formula
		Simple	weighted
harmonic
Geometric
Arithmetic
quadratic
cubic

The harmonic mean has a more complex structure than the arithmetic mean. The harmonic mean is used for calculations when the weights are not the units of the population - the carriers of the trait, but the products of these units and the values ​​of the trait (i.e. m = Xf). The average harmonic downtime should be used in cases of determining, for example, the average costs of labor, time, materials per unit of output, per part for two (three, four, etc.) enterprises, workers engaged in the manufacture of the same type of product , the same part, product.

The main requirement for the formula for calculating the average value is that all stages of the calculation have a real meaningful justification; the resulting average value should replace the individual values ​​of the attribute for each object without breaking the connection between individual and summary indicators. In other words, the average value should be calculated in such a way that when each individual value of the averaged indicator is replaced by its average value, some final summary indicator connected in one way or another with the averaged indicator remains unchanged. This result is called determining since the nature of its relationship with individual values ​​determines the specific formula for calculating the average value. Let's show this rule on the example of the geometric mean.

Geometric mean formula

most often used when calculating the average value of individual relative values ​​of the dynamics.

The geometric mean is used if a sequence of chain relative values ​​of dynamics is given, indicating, for example, an increase in production compared to the level of the previous year: i 1 , i 2 , i 3 ,…, i n . Obviously, the volume of production in the last year is determined by its initial level (q 0) and subsequent growth over the years:

q n =q 0 × i 1 × i 2 ×…×i n .

Taking q n as a defining indicator and replacing the individual values ​​of the dynamics indicators with average ones, we arrive at the relation

From here

A special type of average values ​​- structural averages - is used to study the internal structure of the series of distribution of attribute values, as well as to estimate the average value (power type), if, according to the available statistical data, its calculation cannot be performed (for example, if there were no data in the considered example). and on the volume of production, and on the amount of costs by groups of enterprises).

Indicators are most often used as structural averages. fashion - the most frequently repeated feature value - and median - the value of a feature that divides the ordered sequence of its values ​​into two parts equal in number. As a result, in one half of the population units, the value of the attribute does not exceed the median level, and in the other half it is not less than it.

If the feature under study has discrete values, then there are no particular difficulties in calculating the mode and median. If the data on the values ​​of the attribute X are presented in the form of ordered intervals of its change (interval series), the calculation of the mode and median becomes somewhat more complicated. Since the median value divides the entire population into two parts equal in number, it ends up in one of the intervals of the feature X. Using interpolation, the median value is found in this median interval:

,

where X Me is the lower limit of the median interval;

h Me is its value;

(Sum m) / 2 - half of the total number of observations or half of the volume of the indicator that is used as a weighting in the formulas for calculating the average value (in absolute or relative terms);

S Me-1 is the sum of observations (or the volume of the weighting feature) accumulated before the beginning of the median interval;

m Me is the number of observations or the volume of the weighting feature in the median interval (also in absolute or relative terms).

When calculating the modal value of a feature according to the data of the interval series, it is necessary to pay attention to the fact that the intervals are the same, since the indicator of the frequency of feature values ​​X depends on this. For an interval series with equal intervals, the mode value is determined as

,

where X Mo is the lower value of the modal interval;

m Mo is the number of observations or the volume of the weighting feature in the modal interval (in absolute or relative terms);

m Mo-1 - the same for the interval preceding the modal;

m Mo+1 - the same for the interval following the modal;

h is the value of the interval of change of the trait in groups.

TASK 1

The following data are available for the group of industrial enterprises for the reporting year

№ enterprises	Production volume, million rubles		Average number of employees, pers.	Profit, thousand rubles
	197,7	10,0		13,5
		22,8	1500	136,2
	465,5	18,4	1412	97,6
	296,2	12,6	1200	44,4
	584,1	22,0	1485	146,0
	480,0	119,0	1420	110,4
	57805	21,6	1390	138,7
	204,7			30,6
	466,8	19,4	1375	111,8
	292,2	113,6	1200	49,6
	423,1	17,6	1365	105,8
	192,6			30,7
	360,5	14,0	1290	64,8
	280,3	10,2		33,3

It is required to perform a grouping of enterprises for the exchange of products, taking the following intervals:

up to 200 million rubles


from 200 to 400 million rubles


1. from 400 to 600 million rubles
   

   For each group and for all together, determine the number of enterprises, the volume of production, the average number of employees, the average output per employee. The grouping results should be presented in the form of a statistical table. Formulate a conclusion.
   

   DECISION
   

   Let's make a grouping of enterprises for the exchange of products, the calculation of the number of enterprises, the volume of production, the average number of employees according to the formula of a simple average. The results of grouping and calculations are summarized in a table.
   

   Groups by production volume	№ enterprises	Production volume, million rubles	Average annual cost of fixed assets, million rubles	average sleep juicy number of employees, pers.	Profit, thousand rubles	Average output per worker
   1 group up to 200 million rubles	1,8,12	197,7 204,7 192,6	10,0 9,4 8,8	900 817	13,5 30,6 30,7
   			28,2	2567	74,8	0,23
   Middle level		198,3			24,9
   2 group from 200 to 400 million rubles	4,10,13,14	196,2 292,2 360,5 280,3	12,6 113,6 14,0 10,2	1200 1200 1290	44,4 49,6 64,8 33,3
   		1129,2	150,4	4590	192,1	0,25
   Middle level		282,3	37,6	1530	64,0
   3 group from 400 to 600 million	2,3,5,6,7,9,11	592 465,5 584,1 480,0 578,5 466,8 423,1	22,8 18,4 22,0 119,0 21,6 19,4 17,6	1500 1412 1485 1420 1390 1375 1365	136,2 97,6 146,0 110,4 138,7 111,8 105,8
   		3590	240,8	9974	846,5	0,36
   Middle level		512,9	34,4	1421	120,9
   Total in aggregate		5314,2	419,4	17131	1113,4	0,31
   Aggregate average		379,6	59,9	1223,6	79,5

   Conclusion. Thus, in the aggregate under consideration, the largest number of enterprises in terms of output fell into the third group - seven, or half of the enterprises. The value of the average annual value of fixed assets is also in this group, as well as the large value of the average number of employees - 9974 people, the enterprises of the first group are the least profitable.
   

   TASK 2
   

   We have the following data on the enterprises of the company
   

   Number of the enterprise belonging to the company	I quarter		II quarter
   	Output, thousand rubles		Worked by working man-days	Average output per worker per day, rub.
   	59390,13

The average value is the most valuable from an analytical point of view and the universal form of expression of statistical indicators. The most common average - the arithmetic average - has a number of mathematical properties that can be used in its calculation. At the same time, when calculating a specific average, it is always advisable to rely on its logical formula, which is the ratio of the volume of the attribute to the volume of the population. For each mean, there is only one true reference ratio, which, depending on the data available, may require different forms of means. However, in all cases where the nature of the averaged value implies the presence of weights, it is impossible to use their unweighted formulas instead of the weighted average formulas.

The average value is the most characteristic value of the attribute for the population and the size of the attribute of the population distributed in equal shares between the units of the population.

The characteristic for which the average value is calculated is called averaged .

The average value is an indicator calculated by comparing absolute or relative values. The average value is

The average value reflects the influence of all factors influencing the phenomenon under study, and is the resultant for them. In other words, repaying individual deviations and eliminating the influence of cases, the average value, reflecting the general measure of the results of this action, acts as a general pattern of the phenomenon under study.

Conditions for the use of averages:

Ø homogeneity of the studied population. If some elements of the population subject to the influence of a random factor have significantly different values ​​of the studied trait from the rest, then these elements will affect the size of the average for this population. In this case, the average will not express the most typical value of the feature for the population. If the phenomenon under study is heterogeneous, it is required to break it down into groups containing homogeneous elements. In this case, group averages are calculated - group averages expressing the most characteristic value of the phenomenon in each group, and then the overall average value for all elements is calculated, characterizing the phenomenon as a whole. It is calculated as the average of the group averages, weighted by the number of population elements included in each group;

Ø a sufficient number of units in the aggregate;

Ø the maximum and minimum values ​​of the trait in the studied population.

Average value (indicator)- this is a generalized quantitative characteristic of a trait in a systematic population under specific conditions of place and time.

In statistics, the following forms (types) of averages are used, called power and structural:

Ø arithmetic mean(simple and weighted);

simple

Department of Statistics

COURSE WORK

THEORY OF STATISTICS

On the topic: Averages

Completed by: Group number: STP - 72

Yunusova Gulnazia Chamilevna

Checked by: Earring Lyudmila Konstantinovna

Introduction

1. The essence of averages, general principles of application

2. Types of averages and their scope

2.1 Power averages

2.1.1 Arithmetic mean

2.1.2 Harmonic mean

2.1.3 Geometric mean

2.1.4 RMS

2.2. Structural averages

2.2.1 Median

3. Basic methodological requirements for the correct calculation of averages

Conclusion

List of used literature

Introduction

The history of the practical application of averages goes back tens of centuries. The main purpose of calculating the average was to study the proportions between quantities. The importance of calculating averages has increased in connection with the development of probability theory and mathematical statistics. The solution of many theoretical and practical problems would be impossible without calculating the average and assessing the fluctuation of the individual values ​​of the attribute.

Scientists of different directions sought to define the average. For example, the outstanding French mathematician O. L. Cauchy (1789 - 1857) believed that the average of several values ​​is a new value, which is between the smallest and largest of the considered values.

However, the Belgian statistician A. Quetelet (1796 - 1874) should be considered the creator of the theory of averages. He made an attempt to determine the nature of average values ​​and the regularities that are manifested in them. According to Quetelet, permanent causes act in the same way (permanently) on every phenomenon under study. It is they who make these phenomena similar to each other, create common patterns for all of them.

A consequence of the teachings of A. Quetelet about common and individual causes was the allocation of average values ​​as the main method of statistical analysis. He emphasized that statistical averages are not just a measure of mathematical measurement, but a category of objective reality. He identified a typical, really existing average with a true value, deviations from which can only be random.

A vivid expression of the stated view of the average is his theory of the "average person", i.e. a person of average height, weight, strength, average chest volume, lung capacity, average visual acuity and normal complexion. Averages characterize the "true" type of a person, all deviations from this type indicate ugliness or illness.

The views of A. Quetelet were further developed in the works of the German statistician V. Lexis (1837 - 1914).

Another version of the idealist theory of averages is based on the philosophy of Machism. Its founder was the English statistician A. Bowley (1869 - 1957). In averages, he saw the simplest way to describe the quantitative characteristics of a phenomenon. In defining the meaning of averages, or, as he puts it, "their function", Bowley brings to the fore the Machian principle of thinking. Thus, he wrote that the function of averages is clear: it consists in expressing a complex group with the help of a few prime numbers. The mind cannot immediately grasp the magnitudes of millions of statistics; they must be grouped, simplified, averaged.

A. Quetelet's follower was the Italian statistician C. Gini (1884-1965), the author of the large monograph "Average Values". K.Gini criticized the definition of the average given by the Soviet statistician A.Ya. . Boyarsky, and formulated his own: “The average of several values ​​is the result of actions performed on these values ​​according to a certain rule, and is either one of these values, which is not more and not less than all the others (the average real or effective), or some a new value intermediate between the smallest and the largest of the given values ​​(counting average).

In this course work, we will consider in detail the main problems of the theory of averages. In the first chapter, we will reveal the essence of averages and general principles of application. In the second chapter, we will consider the types of averages and the scope of their application using specific examples. The third chapter will consider the main methodological requirements for calculating averages.

1. The essence of averages, general principles of application

Averages are one of the most common summary statistics. They aim to characterize by one number a statistical population consisting of a minority of units. Average values ​​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 the general statistics cancel each other out and, on average, a statistical regularity is more clearly manifested.

The average value is a generalizing indicator that characterizes the typical level of the phenomenon in specific conditions of place and time. It expresses the level of the characteristic, typical for each unit of the population.

The average is an objective characteristic only for homogeneous phenomena. Averages for heterogeneous populations are called sweeping and can only be used in combination with partial averages of homogeneous populations.

The average is used in statistical studies to assess the current level of a phenomenon, to compare several populations on the same basis with each other, to study the dynamics of the development of the phenomenon under study over time, to study the relationship of phenomena.

Averages are widely used in various planned, forecast, financial calculations.

The main value of average values ​​is their generalizing function, i.e. replacement of a set of different individual values ​​of a feature by an average value that characterizes the entire set of phenomena. Everyone knows the features of the development of modern people, which are manifested, among other things, in the higher growth of sons in comparison with fathers, daughters in comparison with mothers at the same age. But how to measure this phenomenon?

In different families, there are very different ratios of the growth of the older and younger generations. Not every son is higher than his father, and not every daughter is higher than his mother. But if we measure the average height of many thousands of people, then by the average height of sons and fathers, daughters and mothers, one can accurately establish both the very fact of acceleration and the typical average increase in height in one generation.

For the production of the same quantity of goods of a certain type and quality, different producers (factories, firms) spend an unequal amount of labor and material resources. But the market averages these costs, and the cost of goods is determined by the average consumption of resources for production.

The weather in a certain point of the globe on the same day in different years can be very different. For example, in St. Petersburg on March 31, the air temperature for more than a hundred years of observations ranged from -20.1° in 1883 to +12.24° in 1920. Approximately the same fluctuations occur on other days of the year. According to such individual weather data in any arbitrary year, it is impossible to get an idea of ​​the climate of St. Petersburg. Climate characteristics are the average weather characteristics over a long period - air temperature, humidity, wind speed, amount of precipitation, number of hours of sunshine per week, month and whole year, etc.

If the average value generalizes qualitatively homogeneous values ​​of a trait, then it is a typical characteristic of a trait in a given population. So, we can talk about measuring the typical growth of Russian girls born in 1973 when they reach the age of 20. A typical characteristic will be the average milk yield from Black-and-White cows in the first year of lactation at a feeding rate of 12.5 feed units per day.

However, it is wrong to reduce the role of average values ​​only to the characteristics of typical values ​​of features in populations that are homogeneous in terms of this feature. In practice, much more often modern statistics use average values ​​that generalize obviously heterogeneous phenomena, such as, for example, the yield of all grain crops throughout Russia. Or consider such an average as the average consumption of meat per capita: after all, among this population there are children under one year old who do not consume meat at all, and vegetarians, and northerners, and southerners, miners, athletes and pensioners. Even more clear is the atypicality of such an average indicator as the average national income produced per capita.

The average per capita national income, the average grain yield throughout the country, the average consumption of various food products - these are the characteristics of the state as a single economic system, these are the so-called system averages.

System averages can characterize both spatial or object systems that exist simultaneously (state, industry, region, planet Earth, etc.) and dynamic systems extended in time (year, decade, season, etc.).

An example of a system average characterizing a period of time is the average air temperature in St. Petersburg for 1992, equal to +6.3°. This average summarizes the extremely heterogeneous temperatures of frosty winter days and nights, hot summer days, spring and autumn. 1992 was a warm year, its average temperature is not typical for St. Petersburg. As a typical average annual air temperature in the city, one should use the long-term average, say, for 30 years from 1963 to 1992, which is equal to +5.05°. This average is a typical average, since it generalizes homogeneous quantities; average annual temperatures of the same geographical point, varying over 30 years from +2.90° in 1976 to +7.44° in 1989

At the stage of statistical processing, a variety of research tasks can be set, for the solution of which it is necessary to choose the appropriate average. In this case, it is necessary to be guided by the following rule: the values ​​\u200b\u200bthat represent the numerator and denominator of the average must be logically related to each other.

• power averages;
• structural averages.

Let us introduce the following notation:

The values ​​for which the average is calculated;

Average, where the line above indicates that the averaging of individual values ​​takes place;

Frequency (repeatability of individual trait values).

Various means are derived from the general power mean formula:

(5.1)

for k = 1 - arithmetic mean; k = -1 - harmonic mean; k = 0 - geometric mean; k = -2 - root mean square.

Averages are either simple or weighted.

weighted averages are called quantities that take into account that some variants of the values ​​of the attribute may have different numbers, and therefore each variant has to be multiplied by this number. In other words, the "weights" are the numbers of population units in different groups, i.e. each option is "weighted" by its frequency. The frequency f is called the statistical weight or weighing average.

It is known that transactions were carried out within 5 days (5 transactions), the number of shares sold at the sales rate was distributed as follows:

1 - 800 ac. - 1010 rubles

2 - 650 ac. - 990 rub.

3 - 700 ak. - 1015 rubles.

4 - 550 ac. - 900 rub.

5 - 850 ak. - 1150 rubles.

The initial ratio for determining the average share price is the ratio of the total amount of transactions (TCA) to the number of shares sold (KPA):

OSS = 1010 800 + 990 650 + 1015 700+900 550+1150 850= 3 634 500;

CPA = 800+650+700+550+850=3550.

In this case, the average price of the shares was equal to:

It is necessary to know the properties of the arithmetic mean, which is very important both for its use and for its calculation. There are three main properties that most of all led to the widespread use of the arithmetic mean in statistical and economic calculations.

Property one (zero): the sum of positive deviations of individual values ​​of a trait from its mean value is equal to the sum of negative deviations. This is a very important property, since it shows that any deviations (both with + and with -) due to random causes will be mutually canceled.

Proof:

Property two (minimum): the sum of the squared deviations of the individual values ​​of the trait from the arithmetic mean is less than from any other number (a), i.e. is the minimum number.

Proof.

Compose the sum of the squared deviations from the variable a:

(5.4)

To find the extremum of this function, it is necessary to equate its derivative with respect to a to zero:

From here we get:

(5.5)

Therefore, the extremum of the sum of squared deviations is reached at . This extremum is the minimum, since the function cannot have a maximum.

Property three: the arithmetic mean of a constant is equal to this constant: at a = const.

In addition to these three most important properties of the arithmetic mean, there are so-called design properties, which are gradually losing their significance due to the use of electronic computers:

• if the individual value of the sign of each unit is multiplied or divided by a constant number, then the arithmetic mean will increase or decrease by the same amount;
• the arithmetic mean will not change if the weight (frequency) of each feature value is divided by a constant number;
• if the individual values ​​of the attribute of each unit are reduced or increased by the same amount, then the arithmetic mean will decrease or increase by the same amount.

Average harmonic. This average is called the reciprocal arithmetic average, since this value is used when k = -1.

Simple harmonic mean is used when the weights of the characteristic values ​​are the same. Its formula can be derived from the base formula by substituting k = -1:

For example, we need to calculate the average speed of two cars that have traveled the same path, but at different speeds: the first at 100 km/h, the second at 90 km/h.

Using the harmonic mean method, we calculate the average speed:

In statistical practice, harmonic weighted is more often used, the formula of which is:

This formula is used in cases where the weights (or volumes of phenomena) for each attribute are not equal. In the original ratio, the numerator is known to calculate the average, but the denominator is unknown.

For example, when calculating the average price, we must use the ratio of the amount sold to the number of units sold. We do not know the number of units sold (we are talking about different goods), but we know the sums of sales of these different goods.

Suppose you want to find out the average price of goods sold:

We get

If you use the arithmetic mean formula here, you can get an average price that will be unrealistic:

Geometric mean. Most often, the geometric mean finds its application in determining the average growth rate (average growth rates), when the individual values ​​of the trait are presented as relative values. It is also used if it is necessary to find the average between the minimum and maximum values ​​of a characteristic (for example, between 100 and 1000000). There are formulas for simple and weighted geometric mean.

For a simple geometric mean:

For a weighted geometric mean:

RMS. The main scope of its application is the measurement of the variation of a trait in the population (calculation of the standard deviation).

Simple root mean square formula:

Weighted mean square formula:

(5.11)

As a result, we can say that the successful solution of the problems of statistical research depends on the correct choice of the type of average value in each specific case.

The choice of the average assumes the following sequence:

a) the establishment of a generalizing indicator of the population;

b) determination of a mathematical ratio of values ​​for a given generalizing indicator;

c) replacement of individual values ​​by average values;

d) calculation of the average using the corresponding equation.