Average values are widely used in statistics. Average values characterize the qualitative indicators of commercial activity: distribution costs, profit, profitability, etc.
Average - This is one of the common generalization techniques. A correct understanding of the essence of the average determines its special significance in the conditions market economy, when the average through the individual and random allows us to identify the general and necessary, to identify the trend of patterns of economic development.
average value - these are general indicators in which actions are expressed general conditions, patterns of the phenomenon being studied.
Statistical averages are calculated on the basis of mass data from correctly statistically organized mass observation (continuous and selective). However, the statistical average will be objective and typical if it is calculated from mass data for a qualitatively homogeneous population (mass phenomena). For example, if you calculate the average wages in cooperatives and state-owned enterprises, and the result is extended to the entire population, then the average is fictitious, since it was calculated based on a heterogeneous population, and such an average loses all meaning.
With the help of the average, differences in the value of a characteristic that arise for one reason or another in individual units of observation are smoothed out.
For example, the average productivity of a salesperson depends on many reasons: qualifications, length of service, age, form of service, health, etc.
Average output reflects the general property of the entire population.
The average value is a reflection of the values of the characteristic being studied, therefore, it is measured in the same dimension as this characteristic.
Each average value characterizes the population under study according to any one characteristic. In order to obtain a complete and comprehensive understanding of the population being studied according to a number of essential characteristics, in general it is necessary to have a system of average values that can describe the phenomenon from different angles.
There are different averages:
arithmetic mean;
geometric mean;
harmonic mean;
mean square;
average chronological.
Let's look at some types of averages that are most often used in statistics.
Arithmetic mean
The simple arithmetic mean (unweighted) is equal to the sum of the individual values of the attribute divided by the number of these values.
Individual values of a characteristic are called variants and are denoted by x(); the number of population units is denoted by n, the average value of the characteristic is denoted by 
. Therefore, the arithmetic simple mean is equal to:
According to the discrete distribution series data, it is clear that the same characteristic values (variants) are repeated several times. Thus, option x occurs 2 times in total, and option x 16 times, etc.
The number of identical values of a characteristic in the distribution series is called frequency or weight and is denoted by the symbol n.
Let's calculate the average salary of one worker 
in rub.:
The wage fund for each group of workers is equal to the product of options and frequency, and the sum of these products gives the total wage fund of all workers.
In accordance with this, the calculations can be presented in general form:
The resulting formula is called the weighted arithmetic mean.
As a result of processing, statistical material can be presented not only in the form of discrete distribution series, but also in the form of interval variation series with closed or open intervals.
The average for grouped data is calculated using the weighted arithmetic average formula:
In the practice of economic statistics, it is sometimes necessary to calculate the average using group averages or averages of individual parts of the population (partial averages). In such cases, group or private averages are taken as options (x), on the basis of which the overall average is calculated as an ordinary weighted arithmetic average.
Basic properties of the arithmetic mean .
The arithmetic mean has a number of properties:
1. The value of the arithmetic mean will not change from decreasing or increasing the frequency of each value of the characteristic x by n times.
If all frequencies are divided or multiplied by any number, the average value will not change.
2. The common multiplier of individual values of a characteristic can be taken beyond the sign of the average:
3. The average of the sum (difference) of two or more quantities is equal to the sum (difference) of their averages:
4. If x = c, where c is a constant value, then 
.
5. The sum of deviations of the values of attribute X from the arithmetic mean x is equal to zero:
Harmonic mean.
Along with the arithmetic mean, statistics uses the harmonic mean, the inverse of the arithmetic mean of the inverse values of the attribute. Like the arithmetic mean, it can be simple and weighted.
Characteristics of variation series, along with averages, are mode and median.
Fashion - this is the value of a characteristic (variant) that is most often repeated in the population under study. For discrete distribution series, the mode will be the value of the variant with the highest frequency.
For interval distribution series with equal intervals, the mode is determined by the formula:
Where 
- initial value of the interval containing the mode;
- the value of the modal interval;
- frequency of the modal interval;
- frequency of the interval preceding the modal one;
- frequency of the interval following the modal one.
Median - this is an option located in the middle of the variation series. If the distribution series is discrete and has an odd number of members, then the median will be the option located in the middle of the ordered series (an ordered series is the arrangement of population units in ascending or descending order).
Topic 3. Method of averages
Average size in statistics is a generalized characteristic of qualitatively homogeneous phenomena and processes according to some varying characteristic, which shows the level of the characteristic related to a unit of the population.
average value
abstract, because characterizes the value of a characteristic in some impersonal unit of the population.Essence average value is that through the individual and random the general and necessary are revealed, that is, the tendency and pattern in the development of mass phenomena. The characteristics that are generalized in average values are inherent in all units of the population.
Due to this, the average value has great importance to identify patterns inherent in mass phenomena and not noticeable in individual units of the population. Starting with W. Petty, averages began to be considered as the main technique of statistical analysis.
General principles application of average values:
1) a reasonable choice of the population unit for which the average value is calculated is necessary;
2) when determining the average value, one must proceed from the qualitative content of the characteristic being averaged, take into account the relationship of the characteristics being studied, as well as the data available for calculation;
3) average values should be calculated based on qualitatively homogeneous populations, which are obtained by the grouping method, which involves the calculation of a system of generalizing indicators;
4) general averages must be supported by group averages.
Depending on the nature of the primary data, the scope of application and the method of calculation in statistics, the following are distinguished: main types of medium:
1) power averages(arithmetic mean, harmonic, geometric, mean square and cubic);
2) structural (nonparametric) means(mode and median).
In statistics correct description the studied population according to a varying characteristic in each individual case gives only a very specific type of average. The question of what type of average needs to be applied in a particular case is resolved through a specific analysis of the population being studied, as well as based on the principle of meaningfulness of the results when summing or when weighing. These and other principles are expressed in statistics theory of averages.
For example, the arithmetic mean and the harmonic mean are used to characterize the average value of a varying characteristic in the population being studied. The geometric mean is used only when calculating average rates of dynamics, and the quadratic mean is used only when calculating variation indices.
Formulas for calculating average values are presented in Table 3.1.
Table 3.1 – Formulas for calculating average values
| Types of averages | Calculation formulas | |
| simple | weighted | |
| 1. Arithmetic mean |  |
|
| 2. Harmonic mean | ||
| 3. Geometric mean | ||
| 4. Mean square |  |
Designations:- quantities for which the average is calculated; - average, where the bar above indicates that averaging of individual values takes place; - frequency (repeatability of individual values of a characteristic).
Obviously, the various averages are derived from general formula for power average (3.1):
, (3.1)
when k = + 1 - arithmetic mean; k = -1 - harmonic mean; k = 0 - geometric mean; k = +2 - root mean square.
Average values can be simple or weighted. Weighted averages values are called that take into account that some variants of attribute values may have different numbers; in this regard, each option has to be multiplied by this number. The “scales” in this case are the numbers of aggregate units in different groups, i.e. Each option is “weighted” by its frequency. The frequency f is called statistical weight or average weight.
If a population with qualitatively homogeneous characteristics is studied, then the average value acts here as typical average. For example, for groups of workers in a certain industry with a fixed income level, a typical average expenditure on basic necessities is determined.
When studying a population with qualitatively heterogeneous characteristics, the atypicality of average indicators may come to the fore. These, for example, are the average indicators of produced national income per capita (different age groups). Average values generalize qualitatively heterogeneous values of characteristics or systemic spatial aggregates (international community, continent, state, region, region, etc.) or dynamic aggregates extended in time (century, decade, year, season, etc.). ). Such average values are called system averages.
Eventually correct choice of average assumes the following sequence:
a) establishing a general indicator of the population;
b) determination of a mathematical relationship of quantities for a given general indicator;
c) replacing individual values with average values;
d) calculation of the average using the appropriate equation.
3.2 Arithmetic mean and its properties and calculus techniques. Harmonic mean
Arithmetic mean– the most common type of medium size; it is calculated in cases where the volume of the averaged characteristic is formed as the sum of its values for individual units of the statistical population being studied.
The most important properties of the arithmetic mean :
1. The product of the average by the sum of frequencies is always equal to the sum of the products of variants (individual values) by frequencies.
2. If you subtract (add) any arbitrary number from each option, then the new average will decrease (increase) by the same number.
3. If each option is multiplied (divided) by some arbitrary number, then the new average will increase (decrease) by the same amount
4. If all frequencies (weights) are divided or multiplied by any number, then the arithmetic average will not change.
5. The sum of deviations of individual options from the arithmetic mean is always zero.
You can subtract an arbitrary constant value ( better value middle options or options with the highest frequency), reduce the resulting differences by a common factor (preferably by the value of the interval), and express the frequencies in particulars (as a percentage) and multiply the calculated average by a common factor and add an arbitrary constant value.
This method of calculating the arithmetic mean is called method of calculation from conditional zero.
Harmonic mean is called the inverse arithmetic mean, since this value is obtained at k = -1. Simple harmonic mean used when the weights of the characteristic values are the same. For example, you need to calculate the average speed of two cars that have traveled the same path, but with at different speeds: first - at a speed of 100 km/h, second - 90 km/h. Using the harmonic mean method, we calculate the average speed:
In statistical practice it is more often used weighted harmonic mean – for those cases when the weights (or volumes of phenomena) for each attribute are not equal, and in the initial ratio for calculating the average the numerator is known, but the denominator is unknown.
For example, when calculating the average price, we must use the ratio of the sales amount to the number of units sold. We do not know the number of units sold ( we're talking about about different goods), but the sales amounts of these different goods are known. Let's say you need to find out the average price of goods sold (Table 3.2).
Table 3.2 – Initial data
We get:
If you use the arithmetic average formula here, you can get an average price that will be unrealistic:
If, when calculating the average price by weight, we take the number of goods, then the correct result is given by the formula for the arithmetic weighted average. If we use the cost of the batches as weights, then the harmonic average gives the correct result.
That is, averageHarmonic is not a special type of average, but rather a special method of calculating the arithmetic average. In statistics, it is still customary to distinguish the harmonic mean as a separate type of mean, because with its help, the technique of calculating the arithmetic mean can be simplified and, more importantly, the nature of the available statistical material can be taken into account.
The correctness of the choice of the form of the mean (arithmetic or harmonic) can also be checked additional criterion: if absolute values are used as weights, any intermediate actions when calculating the average should give significant indicators. For example, to calculate the average price, multiply the price by the number of goods to obtain their cost. And dividing the cost of goods by their prices gives the quantity of goods.
Using the harmonic mean in statistics, the average percentage of plan completion is also determined (based on the actual implementation of the plan), the average time spent on performing operations (based on the average time spent on one operation and the total work time for individual employees), etc.
Geometric mean finds its application in determining average growth rates (average growth coefficients), when individual values of a characteristic are presented in the form of 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).
Mean square used to measure the variation of a characteristic in the aggregate (calculation of the standard deviation).
Valid in statistics rule of majority of averages:
X harm.< Х геом. < Х арифм. < Х квадр. < Х куб.
A statistical population consists of a set of units, objects or phenomena that are homogeneous in some respects and at the same time have different characteristics. The magnitude of the characteristics of each object is determined both by those common to all units of the population and by its individual characteristics.
Analyzing the ordered series of the distribution (ranking, interval, etc.), one can notice that the elements of the statistical population are clearly concentrated around certain central values. Such a concentration of individual attribute values around certain central values, as a rule, occurs in all statistical distributions. The tendency of individual values of the characteristic under study to group around the center of the frequency distribution is called central tendency. To characterize the central tendency of the distribution, generalizing indicators are used, which are called average values.
Average size in statistics they call a general indicator that characterizes typical size characteristic in a qualitatively homogeneous population under specific conditions of place and time and reflects the amount of varying characteristic per unit of population. The average value is calculated in most cases by dividing the total volume of the characteristic by the number of units possessing this characteristic. If, for example, the monthly wage fund and the number of workers per month are known, then the average monthly wage can be determined by dividing the wage fund by the number of workers.
The average values are indicators such as the average duration of a working day, week, year, average tariff category workers, average level of labor productivity, average national income per capita, average grain yield in the country, average food consumption per capita, etc.
Average values are calculated from both absolute and relative values, are named indicators and are measured in the same units of measurement as the averaged characteristic. They characterize the value of the population under study with one number. The average values reflect the objective and typical level of socio-economic phenomena and processes.
Each average characterizes the population under study according to one particular characteristic, 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. For example, indicators of average wages are assessed together with indicators of labor productivity (average output per unit of working time), capital-labor ratio and energy production, 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 measures of variation (Section 5), sampling errors (Section 6), variance (Section 8) and correlation analysis(section 9).
It is also impossible to imagine statistics without indices, and the latter essentially represent average values. The use of the statistical grouping method 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 grouping method is component scientifically developed statistical methodology. Average indicators organically complement the method of statistical groupings.
Average values are used to characterize changes in phenomena over time, to calculate average growth rates and increments. For example, a comparison of the average growth rates of labor productivity and wages for a certain period (a number of years) reveals the nature of the development of the phenomenon over the period of time being studied, separately labor productivity and separately wages. A comparison of the growth rates of these two phenomena gives an idea of the nature and peculiarity of the relationship between the growth or decline of labor productivity relative to its payment for certain periods of time.
In all cases when it becomes necessary to characterize with one number a set of values of a characteristic that changes, its average value is used.
In a statistical aggregate, the value of a characteristic changes from object to object, that is, it varies. By averaging these values and providing the level value of the attribute to each member of the population, we abstract from the individual values of the attribute, thereby, as it were, replacing the series of distributions of attribute values with the same value equal to the average value. However, such an abstraction is legitimate only if the averaging does not change the basic property in relation to the given feature as a whole. This basic property of a statistical population, associated with individual values of a characteristic, and which, when averaging, must be kept unchanged, is called the defining property of the average in relation to the characteristic under study. In other words, the average, replacing the individual values of the attribute, should not change the overall volume of the phenomenon, i.e. This equality is mandatory: the volume of the phenomenon is equal to the product of the average value and the size of the population. For example, if from three values of barley yield (x, = 20.0; 23.3; 23.6 c/ha), the average is calculated (20.0 + 23.3 + 23.6): 3 = 22.3 c/ ha, then according to the defining property of the average the following equality must be observed:
As can be seen from the above example, the average barley yield does not coincide with any of the individual ones, since not a single farm yielded 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 values of the characteristic being studied.
The main significance of average values lies in their generalizing function, i.e. in replacing many different individual values of a characteristic with an average value that characterizes the entire set of phenomena. The ability of the average to characterize not individual units, but to express the level of a characteristic per each unit of the population is its distinctive ability. This feature makes the average a generalizing indicator of the level of varying characteristics, i.e. an indicator that abstracts from the individual values of the value of a characteristic 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 averaged characteristic serves as an expression of the dialectical connection between the individual and the general.
The use of averages should be based on the understanding and interrelation of the dialectical categories of general and individual, mass and individual.
The average value reflects what is common in each individual, individual object. Thanks to this, the average becomes of great importance for identifying patterns inherent in mass social phenomena and not noticeable in individual phenomena.
In the development of phenomena, necessity is combined with chance. Therefore, average values are related to the law large numbers. The essence of this connection is that when calculating the average value, random fluctuations that have different directions, due to the law of large numbers, are mutually balanced, canceled out, and the average value clearly displays the basic pattern, necessity, and influence of general conditions characteristic of a given population. The average reflects the typical, real level of the phenomena being studied. Estimating these levels and changing them in time and space is one of the main tasks of averages. Thus, through averages, for example, the pattern of increasing labor productivity, crop yields, and animal productivity is manifested. Consequently, average values represent general indicators in which the effect of general conditions and the pattern of the phenomenon being studied are expressed.
Using average values, we study changes in phenomena in time and space, trends in their development, connections and dependencies between characteristics, effectiveness various forms organization of production, labor and technology, implementation scientific and technological progress, identification of new, progressive in the development of certain socio-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 harvests - in the growth of average crop yields, etc.
The average value gives a generalized characteristic of the phenomenon under study based on only one characteristic, 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 social phenomena being studied, when calculating it, it is necessary to adhere to the following conditions.
1. The criterion by which the average is calculated must be significant. Otherwise, an insignificant or distorted average will be obtained.
2. The average must be calculated only for a qualitatively homogeneous population. Therefore, the direct calculation of averages must be preceded by statistical grouping, which makes it possible to divide the population under study 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 a population should not be decided formally by the form of its distribution. This, like the question of the typicality of the average, must be resolved based on the causes and conditions that form the totality. The totality is also homogeneous, the units of which are formed under the influence of common main causes and conditions that determine general level of a given characteristic, characteristic of the entire population.
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 are mutually equal to each other and a pattern appears, typical and characteristic sizes of the characteristic being studied.
4. General requirement When calculating any type of average values, it is mandatory to maintain unchanged the total volume of the attribute in the aggregate when replacing its individual values with the average value (the so-called defining property of the average).
Topic 5. Average values as statistical indicators
The concept of average value. Scope of averages in statistical research
Average values are used at the stage of processing and summarizing the obtained primary statistical data. The need to determine average values is due to the fact that, as a rule, individual values of the same characteristic for different units of the populations under study are not the same.
Average size called an indicator that characterizes the generalized value of a characteristic or group of characteristics in the population under study.
If a population with qualitatively homogeneous characteristics is studied, then the average value acts here as typical average. For example, for groups of workers in a certain industry with a fixed income level, the typical average expenditure on basic necessities is determined, i.e. the typical average generalizes qualitatively homogeneous values of the attribute in a given population, which is the share of expenses among workers of this group on essential goods.
When studying a population with qualitatively heterogeneous characteristics, the atypicality of average indicators may come to the fore. These, for example, are the average indicators of produced national income per capita (different age groups), average indicators of grain yields throughout Russia (regions of different climatic zones and different grain crops), average indicators of the birth rate of the population for all regions of the country, average temperatures for a certain period, etc. Here, average values generalize qualitatively heterogeneous values of characteristics or systemic spatial aggregates (international community, continent, state, region, region, etc.) or dynamic aggregates extended over time (century, decade, year, season, etc.) . Such average values are called system averages.
Thus, the significance of average values lies in their generalizing function. The average value replaces big number individual values of a characteristic, revealing common properties inherent in all units of the population. This, in turn, allows us to avoid random causes and identify general patterns due to common causes.
Types of average values and methods of their calculation
At the stage of statistical processing, a variety of research problems can be set, for the solution of which it is necessary to select the appropriate average. In this case, it is necessary to be guided by the following rule: the quantities that represent the numerator and denominator of the average must be logically related to each other.
power averages;
structural averages.
Let us introduce the following conventions:
The quantities for which the average is calculated;
Average, where the bar above indicates that averaging of individual values takes place;
Frequency (repeatability of individual characteristic values).
Various averages are derived from the general power average formula:
(5.1)
when k = 1 - arithmetic mean; k = -1 - harmonic mean; k = 0 - geometric mean; k = -2 - root mean square.
Average values can be simple or weighted. Weighted averages These are values that take into account that some variants of attribute values may have different numbers, and therefore each option has to be multiplied by this number. In other words, the “scales” are the numbers of aggregate units in different groups, i.e. Each option is “weighted” by its frequency. The frequency f is called statistical weight or average weight.
Arithmetic mean- the most common type of average. It is used when the calculation is carried out on ungrouped statistical data, where you need to obtain the average term. The arithmetic mean is the average value of a characteristic, upon obtaining which the total volume of the characteristic in the aggregate remains unchanged.
The formula for the arithmetic mean (simple) has the form
where n is the population size.
For example, the average salary of an enterprise’s employees is calculated as the arithmetic average:
The determining indicators here are the salary of each employee and the number of employees of the enterprise. When calculating the average, the total amount of wages remained the same, but distributed equally among all employees. For example, you need to calculate the average salary of workers in a small company employing 8 people:
When calculating average values, individual values of the characteristic that is averaged can be repeated, so the average value is calculated using grouped data. In this case we are talking about using arithmetic average weighted, which has the form
(5.3)
So, we need to calculate the average stock price of some joint stock company at stock exchange trading. It is known that the transactions were carried out within 5 days (5 transactions), the number of shares sold at the sales rate was distributed as follows:
1 - 800 ak. - 1010 rub.
2 - 650 ak. - 990 rub.
3 - 700 ak. - 1015 rub.
4 - 550 ak. - 900 rub.
5 - 850 ak. - 1150 rub.
The initial ratio for determining the average price of shares is the ratio of the total amount of transactions (TVA) to the number of shares sold (KPA):
OSS = 1010·800+990·650+1015·700+900·550+1150·850= 3,634,500;
KPA = 800+650+700+550+850=3550.
In this case, the average stock price was equal to
It is necessary to know the properties of the arithmetic average, which is very important both for its use and for its calculation. We can distinguish three main properties that most determined the widespread use of the arithmetic average in statistical and economic calculations.
Property one (zero): the sum of positive deviations of individual values of a characteristic from its average value is equal to the sum of negative deviations. This is a very important property, since it shows that any deviations (both + and -) caused by random reasons will be mutually canceled out.
Proof:
Property two (minimum): the sum of squared deviations of individual values of a characteristic from the arithmetic mean is less than from any other number (a), i.e. there is a minimum number.
Proof.
Let's compile the sum of squared deviations from 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)
Consequently, the extremum of the sum of squared deviations is achieved at . This extremum is a minimum, since a function cannot have a maximum.
Property three: the arithmetic mean of a constant value is equal to this constant: for 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 computer technology:
if the individual value of the attribute 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 attribute 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.
Harmonic mean. This average is called the inverse arithmetic average because this value is used when k = -1.
Simple harmonic mean is used when the weights of the attribute values are the same. Its formula can be derived from the basic formula by substituting k = -1:
For example, we need to calculate the average speed of two cars that covered the same path, but at different speeds: the first at a speed of 100 km/h, the second at 90 km/h. Using the harmonic mean method, we calculate the average speed:
In statistical practice, the harmonic weighted one is more often used, the formula of which has the form
This formula is used in cases where the weights (or volumes of phenomena) for each attribute are not equal. In the initial relationship for calculating the average, the numerator is known, but the denominator is unknown.
This chapter describes the purpose of average values, discusses their main types and forms, and calculation methods. When studying the presented material, it is necessary to understand the requirements for constructing average values, since compliance with them allows you to use these values as typical characteristics of attribute values for a set of homogeneous units.
Forms and types of averages
average value is a generalized characteristic of the level of attribute values, which is obtained per unit of the population. Unlike the relative value, which is a measure of the ratio of indicators, the average value serves as a measure of the characteristic per unit of the population.
The most important property of the average value is that it reflects what is common to all units of the population under study.
The attribute values of individual units of the population fluctuate in one direction or another under the influence of many factors, some of which may be significant or random. For example, interest rates on bank loans are determined by the initial factors for all credit institutions (the level of reserve requirements and the base interest rate on loans provided to commercial banks by the central bank, etc.), as well as the characteristics of each specific transaction, depending on the risk inherent in a given loan , its size and repayment period, costs of processing a loan and monitoring its repayment, etc.
The average value summarizes the individual values of a characteristic and reflects the influence of general conditions that are most characteristic of a given population in specific conditions of place and time. The essence of the average lies in the fact that it cancels out the deviations of the characteristic values of individual units of the population caused by the action of random factors, and takes into account the changes caused by the action of the main factors. The average value will reflect the typical level of a trait in a given population of units when it is calculated from a qualitatively homogeneous population. In this regard, the average method is used in combination with the grouping method.
Average values characterizing the population as a whole are called general, and averages, reflecting the characteristics of a group or subgroup, - group.
The combination of general and group averages allows for comparisons across time and space and significantly expands the boundaries of statistical analysis. For example, when summing up the results of the 2002 census, it was found that for Russia, like for most European countries, characterized by an aging population. Compared to the 1989 census average age of the country's residents increased by three years and amounted to 37.7 years, men - 35.2 years, women - 40.0 years (according to 1989 data, these figures were 34.7, 31.9 and 37.2 years, respectively). According to Rosstat, life expectancy at birth in 2011 for men was 63 years, for women – 75.6 years.
Each average reflects the peculiarity of the population being studied according to one characteristic. To make practical decisions, as a rule, it is necessary to characterize the population according to several characteristics. In this case, a system of averages is used.
For example, in order to achieve the required level of profitability of operations at an acceptable level of risk in banking activities, average interest rates on loans issued are set taking into account average interest rates on deposits and other financial instruments.
The form, type and method of calculating the average value depend on the stated purpose of the study, the type and relationship of the characteristics being studied, as well as on the nature of the initial data. Averages fall into two main categories:
- 1) power averages;
- 2) structural averages.
The average formula is determined by the value of the power of the average applied. With increasing exponent k the average value increases accordingly.
