Absolute values are the results of statistical observations. In statistics, unlike mathematics, all absolute values have a dimension (a unit of measurement), and can also be positive and negative.
Units absolute values reflect the properties of units of the statistical population and can be simple, reflecting 1 property (for example, the mass of cargo is measured in tons) or complex, reflecting several interrelated properties (for example, ton-kilometer or kilowatt-hour).
Units absolute values can be 3 types:
- natural- are used to calculate quantities with homogeneous properties (for example, pieces, tons, meters, etc.). Their disadvantage is that they do not allow summing dissimilar quantities.
- Conditionally natural- apply to absolute values with homogeneous properties, but exhibiting them in different ways. For example, the total mass of energy carriers (firewood, peat, coal, oil products, natural gas) is measured in toe. - tons of reference fuel, since each of its types has a different calorific value, and 29.3 mJ / kg is taken as the standard. Similarly, the total number of school notebooks is measured in US dollars. - conditional school notebooks with a size of 12 sheets. Similarly, canning products are measured in a.c.b. - conditional cans with a capacity of 1/3 liter. Similarly products detergents is reduced to a conditional fat content of 40%.
- Cost units of measurement are expressed in rubles or in another currency, representing a measure of the value of an absolute value. They make it possible to summarize even heterogeneous values, but their disadvantage is that it is necessary to take into account the inflation factor, so statistics always recalculates cost values in comparable prices.
Absolute values can be momentary or interval. Momentary absolute values show the level of the studied phenomenon or process at a certain point in time or date (for example, the amount of money in your pocket or the value of fixed assets on the first day of the month). Interval absolute values are the final accumulated result for a certain period (interval) of time (for example, salary for a month, quarter or year). Interval absolute values, unlike moment ones, allow subsequent summation.
The absolute statistic is denoted X, and their total number in the statistical population is N.
The number of quantities with the same feature value is denoted f and called frequency(recurrence, occurrence).
By themselves, absolute statistical values do not give a complete picture of the phenomenon under study, since they do not show its dynamics, structure, and the relationship between parts. For these purposes, relative statistical values are used.
The concept and types of relative values
Relative statistic is the result of the ratio of two absolute statistical values.
If absolute values with the same dimension are related, then the resulting relative value will be dimensionless (the dimension will be reduced) and is called coefficient.
Often applied artificial dimension of coefficients. It is obtained by multiplying them:
- for 100 - receive interest (%);
- per 1000 - receive ppm (‰);
- per 10000 - receive decimille(‰O).
The artificial dimension of coefficients is used, as a rule, in colloquial speech and in formulating results, but it is not used in the calculations themselves. Most often, percentages are used, in which it is customary to express the obtained values of relative values.
More often instead of the name relative statistic a shorter synonym is used - index(from lat. index- indicator, coefficient).
Depending on the types of correlated absolute values, when calculating relative values, different types of indices: dynamics, plan task, plan fulfillment, structure, coordination, comparison, intensity.
Dynamic index
Dynamic index(growth factor, growth rate) shows how many times the studied phenomenon or process has changed over time. It is calculated as the ratio of the value of the absolute value in the reporting (analyzed) period or point in time to the base (previous):
The criterion value of the index of dynamics is "1", that is: if iД>1 - there is an increase in the phenomenon in time; if iД =1 - stability; if iD
If we subtract its criterion value "1" from the dynamics index and express the resulting value as a percentage, then we get with the criterion value "1":
If T>0, then the growth of the phenomenon takes place; T=0 - stability, T In some textbooks, the dynamics index is called growth factor or growth rategrowth rate, regardless of the result obtained, which can show not only growth, but also stability or decline. Therefore, the more logical and more commonly used names are precisely And .
For example, a car dealership sold 100 cars in January and 110 cars in February. Then the dynamics index will be iD = 110/100 = 1.1, which means an increase in car sales by a car dealership by 1.1 times or 10%
Scheduled Job Index
Scheduled Job Index is the ratio of the planned value of the absolute value to the base:
For example, a car dealership sold 100 cars in January and planned to sell 120 cars in February. Then the target target index will be ipz = 120/100 = 1.2, which means planning for sales growth of 1.2 times or 20%
Plan execution index
Plan execution index- this is the ratio of the actually obtained value of the absolute value in the reporting period to the planned one:
For example, a car dealership sold 110 cars in February when it was scheduled to sell 120 cars in February. Then the plan execution index will be ivp = 110/120 = 0.917, which means the plan is fulfilled by 91.7%, that is, the plan is underfulfilled by (100% -91.7%) = 8.3%.
Multiplying the indices of the planned task and the execution of the plan, we obtain the dynamics index:
In the previously discussed example about a car dealership, if we multiply the obtained values of the indices of the planned task and the execution of the plan, we will obtain the value of the dynamics index: 1.2 * 0.917 = 1.1.
Structure index
Structure index(share, share) is the ratio of any part of the statistical population to the sum of all its parts:
The structure index shows what proportion is a separate part of the population from the entire population.
For example, if there are 20 girls and 10 young people in the group of students under consideration, then the structure index (share) of girls will be 20/(20+10) = 0.667, that is, the share of girls in the group is 66.7%.
Coordination index
Coordination index- this is the ratio of one part of the statistical population to its other part, taken as the basis for comparison:
The coordination index shows how many times more or how many percent is one part of the statistical population compared to its other part, taken as the basis for comparison.
For example, if in a group of students of 20 girls and 10 young people, we take the number of girls as a comparison base, then the coordination index of the number of young people will be 10/20 = 0.5, that is, the number of young people is 50% of the number of girls in the group.
Comparison Index
Comparison Index- this is the ratio of the values of the same absolute value in the same period or point in time, but for different objects or territories:
Where A, B - signs of compared objects or territories.
For example, in January 2009, the number of inhabitants in Nizhny Novgorod was approximately 1280 thousand people, and in Moscow - 10527 thousand people. Let us take Moscow as object A (since it is customary to put a larger number in the numerator when calculating the comparison index), and Nizhny Novgorod as object B, then the index for comparing the number of residents of these cities will be 10527/1280 = 8.22 times, that is, in Moscow the number there are 8.22 times more residents than in Nizhny Novgorod.
Intensity index
Intensity index- this is the ratio of the values of two interconnected absolute quantities with different dimensions, related to the same object or phenomenon.
For example, a bakery shop sold 500 loaves of bread and earned 10,000 rubles from it, then the intensity index would be 10,000/500 = 20 [rubles/loaf of bread], that is, the selling price of bread was 20 rubles. for a loaf
Most fractional quantities are intensity indices.
4.1. The concept of absolute and relative magnitude in statistics
Studying mass social phenomena, statistics in its conclusions relies on numerical data obtained in specific conditions of place and time. results statistical observation registered primarily in the form of primary absolute values. Thus, the bulk of the absolute national economic indicators are recorded in the primary accounting documents. The absolute value reflects the level of development of the phenomenon.
In statistics, all absolute values are named, measured in specific units, and, unlike the mathematical concept of an absolute value, can be both positive and negative (losses, decline, losses, etc.).
Natural units of measurement can be simple (tons, pieces, meters, liters) and complex, which are a combination of several dissimilar quantities (cargo turnover railway transport expressed in ton-kilometers, electricity production in kilowatt-hours). In statistics, absolute indicators are also used, expressed in conditionally natural units of measurement (for example, various types of fuel are converted into equivalent fuel).
Value units of measurement are used, for example, to express the volume of heterogeneous products in value (monetary) form - rubles. When using cost meters, price changes over time are taken into account. This disadvantage of cost meters is overcome by the use of "constant" or "comparable" prices of the same period.
In labor units of measurement (man-days, man-hours), the total labor costs at the enterprise, the labor intensity of individual operations are taken into account.
From the point of view of a specific study, the set of absolute values can be considered as consisting of indicators individual, characterizing the size of the trait in individual units of the population, and total, characterizing the final value of the attribute for a certain part of the population.
Since absolute indicators are the basis of all forms of accounting and methods of quantitative analysis, it is necessary to distinguish momentary And interval absolute values. The first show the actual presence or level of the phenomenon at a certain moment, date (for example, the availability of stocks of materials or working capital, the value of work in progress, the number of residents, etc.). The second is the final cumulative result for the period as a whole (the volume of output for a month or a year, population growth for a certain period, the value of the gross grain harvest for a year and for a five-year period, etc.).
By itself, the absolute value does not give a complete picture of the phenomenon under study, does not show its structure, the relationship between individual parts, development over time. It did not reveal correlations with other absolute indicators. These functions perform relative indicators determined on the basis of absolute values.
Relative value in statistics, it is a generalizing indicator that gives a numerical measure of the ratio of two compared absolute values. Since many absolute values are interrelated, relative values of one type can in some cases be determined through relative values of another type.
The main condition for the correct calculation of the relative value is the comparability of the compared indicators and the presence of real links between the studied phenomena. Thus, according to the method of obtaining, relative indicators are always derivative values, defined in the form of coefficients, percent, ppm, decimille, etc. However, it must be remembered that these dimensionless indicators can, in fact, be assigned a specific, and sometimes quite complex, unit of measurement. Thus, for example, relative indicators of the natural movement of the population, such as birth or death rates, calculated in ppm (‰), show the number of births or deaths per year per 1,000 people of the average annual population; the relative value of the efficiency of using working time is the amount of production per one worked man-hour, etc.
4.2. Types and relationships of relative values
Relative values form a system of interrelated statistical indicators. According to the content of expressed quantitative ratios, the following types of relative values are distinguished.
1. Relative value of task completion. It is calculated as the ratio of the level actually achieved in a given period to the planned level. So, in 1988, 6103 thousand washing machines were produced. under the plan (government order) 6481 thousand units. The relative value of the implementation of the plan was
Consequently, the planned task was underfulfilled by 5.8%.
In practice, there are two types of relative performance indicators of the plan. In the first case, actual and planned levels are compared (this is the example discussed above). In the second case, the absolute value of the increase or decrease in the indicator is set in the planned task and, accordingly, the degree of fulfillment of the plan is checked against this value. So, if it was planned to reduce the cost of a unit of production by 24.2 rubles, and the actual reduction was 27.5 rubles, then the planned target for reducing the cost was fulfilled with an increase of 27.5: 24.2 = 1.136 times, i.e. the plan was exceeded by 13.6%. The indicator of the implementation of the plan at the cost level in this case will be less than one. If the actual cost of the product was 805.8 rubles. with the planned 809.1 rubles, then the value of the plan was 805.8: 809.1 \u003d 0.996, or 99.6%. The actual level of spending one product turned out to be 0.4% lower than planned.
In analytical calculations, in the study of relationships, the assessment of the implementation of the plan by the level of the indicator is more often used. An assessment of the implementation of the plan to change the level is usually given for illustration purposes, especially if it is planned to reduce the absolute value of costs, costs by type, etc.
The relative values of the dynamics, the target and the implementation of the plan are related by the ratio i=i pl.z. × i issue
2. The relative magnitude of the dynamics. It characterizes the change in the level of development of a phenomenon in time. It is obtained as a result of dividing the level of a characteristic in a certain period or point in time by the level of the same indicator in the previous period or point in time.
Thus, according to the fuel and energy balance of the USSR, the resources in 1980 were estimated at 2171.1 million tons of fuel equivalent (reference fuel), and in 1987 - at 2629.1 million tons of fuel equivalent. The relative value of the dynamics was .
Thus, the volume of fuel and energy resources increased by 1.211 times over 7 years (growth factor, growth index, index). In percentage terms, this is 121.1% (rate growth).
In other words, over 7 years, the volume of resources increased by 21.1% (the rate growth). On average, each year the amount of resources increased compared to the previous year by , or by 2.77% (average annual growth rate or index and average annual growth rate).
3. Relative values of the structure. They characterize the shares, specific weights of the constituent elements in the total. As a rule, they are obtained in the form of a percentage:
For analytical calculations, it is preferable to use a coefficient representation, without multiplying by 100.
The set of relative values of the structure shows the structure of the phenomenon under study.
Consider, for example, the structure of the formation and distribution of fuel and energy resources (FER) of Russia in the form of a fuel and energy balance (FEB) (Table 4.1.).
Table 4.1
Sources of formation of fuel and energy resources of Russia
From Table. 4.1. it can be seen that the main part of the resources is formed due to the extraction of fuel. Approximately 8–9% of annual resources were available at the beginning of the year in the form of reserves.
5. Relative values of coordination (RVC). They characterize the ratio of the parts of this population to one of them, taken as the basis for comparison. HVAC show how many times one part of the population is greater than the other, or how many units of one part account for 1, 10, 100, 1000, ... units of the other part. Relative values of coordination can be calculated both by absolute indicators and by indicators of the structure.
Thus, taking the supply of fuel resources for export in 1987 as a comparison base, we will see that for each conditional ton of export deliveries there are 2.342 times more resources consumed domestically for energy production, and 2.363 times more resources intended for production. technological purposes. End-of-year balances are 57.8% compared to annual export deliveries
(9,20: 15,91 = 242: 418,3 = 0,578).
According to the relative values of coordination, it is possible to restore the initial relative indicators of the structure if we calculate the ratio of the relative value of the coordination of a given part (RFC) to the sum of all RFCs (including the one that is taken as the base of comparison):
For example, the share of export deliveries is
1: (2.342 + 2.364 + 1 + 0.578) = 0.1591 or 15.9%.
6. Relative values of comparison (RBC). They characterize the comparative sizes of the same-name absolute values relating to the same period or point in time, but to different objects or territories. Through these indicators, the powers are compared various kinds equipment, labor productivity of individual workers, production of this type of products by different enterprises, regions, countries. For example, in the production of oil and gas in 1985, the USSR surpassed the USA: in oil - 1.36 times, in gas - 1.24 times. The level of electricity production (billion kWh) in the USSR was 1544:2650 = 0.583, or 58.3%, of the US level.
With known growth coefficients (indices of dynamics) and the initial ratio of levels, one can find the condition for equality of levels in the upcoming period t:
.
Hence OBC a / b \u003d Y a / Y b \u003d (i a / i b) t,
those. 
.
The found value of t shows after what period of time the level of the studied phenomenon at object A will be equal to the level of the same phenomenon at object B.
In particular, with an average annual growth rate of electricity production in the USA of 4.5% and in the USSR of 6.9% (according to data for 1961-1985)
.
Comparing the indicators of the dynamics of different phenomena, one more type of relative comparison values is obtained - lead (lag) coefficients in terms of growth or growth. So, if labor productivity at the enterprise increased by 12%, and the wage fund increased by 7.5%, then the labor productivity lead coefficient in terms of growth rates will be 112: 107.5 \u003d 1.042; the growth rate lead coefficient is 12: 7.5 = 1.60.
7. Relative intensity values. They characterize the degree of distribution or development of a given phenomenon in a particular environment. They represent the ratio of the absolute level of one indicator, characteristic of the studied environment, to another absolute indicator, also inherent in this environment and, as a rule, being a factor sign for the first indicator. Thus, in the study of demographic processes, indicators of fertility, mortality, natural increase, etc. are calculated. as the ratio of the number of births (deaths) or the size of population growth per year to the average annual population of a given territory per 1000 people. If the values obtained are very small, then they make a calculation for 10,000 people. So, as of 1987, we have in the country as a whole K birth. = 19.8 ‰, K natural growth = 9.9 ‰. Including in Novosibirsk = 15.2 ‰, K sm. = 9.1 ‰, K marriage rate = 10.9 ‰, K razv. = 5.2 ‰, etc.
Relative intensity values are, for example, indicators of output per unit of working time, costs per unit of output, labor intensity, efficiency of use production assets etc., since they are obtained by comparing opposite quantities related to the same phenomenon and the same period or point in time. The method of calculating the relative intensity values is used to determine the average levels (average output, average labor costs, average cost of products, average price, etc.). Therefore, it is widely believed that relative intensity values are one way of expressing average values.
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Types of absolute values
Types and relationships of relative values
The concept of absolute and relative magnitude in statistics
Studying mass social phenomena, statistics in its conclusions relies on numerical data obtained in specific conditions of place and time. The results of statistical observation are recorded primarily in the form of primary absolute values. The absolute value reflects the level of development of the phenomenon.
Absolute statistics show the volume, size, levels of various socio-economic phenomena and processes. They reflect the levels in physical terms of volume, weight, etc. In general, absolute statistics are named numbers. They always have a certain dimension and units of measure. The latter determine the essence of the absolute value.
natural- such units that reflect the size of objects, things in physical measures (weight, volume, area, etc.).
Cash (value)- are used to characterize many economic indicators in value terms.
Labor– used to determine labor costs (man-hour, man-day)
Conditionally natural units that are used to bring together several varieties use values(t. c. t \u003d 29.3 MJ / kg; soap 40% fat).
Individual- reflect the size of quantitative traits in individual units of the studied population.
General- express the size, the magnitude of the quantitative characteristics of the entire studied population as a whole.
Absolute values reflect the availability of certain resources, this is the basis of material accounting. They most objectively reflect the development of the economy.
Absolute values are the basis for calculating various relative statistics.
Relative value in statistics- this is a generalizing indicator that gives a numerical measure of the ratio of two compared absolute values. Since many absolute values are interrelated, relative values of one type can in some cases be determined through relative values of another type.
Denominator (base of comparison, base) is the value against which the comparison is made.
Comparable (reporting, current) value - is the value being compared.
The relative value shows how many times the compared value is greater or less than the base value, or what proportion the first is in relation to the second. In some cases, the relative value shows how many units of one value fall on the unit of another.
Important property– the relative value abstracts the differences in absolute values and makes it possible to compare such phenomena, the absolute dimensions of which are not directly comparable.
The main condition for the correct calculation of the relative value is the comparability of the compared indicators and the presence of real links between the studied phenomena. Thus, according to the method of obtaining, relative indicators are always derivative values, defined in the form of coefficients, percent, ppm, decimille, etc. However, it must be remembered that these dimensionless indicators can, in fact, be assigned a specific, and sometimes quite complex, unit of measurement. Thus, for example, relative indicators of the natural movement of the population, such as birth or death rates, calculated in ppm (‰), show the number of births or deaths per year per 1,000 people of the average annual population; the relative value of the efficiency of using working time is the amount of production per one worked man-hour, etc.
Relative statistics is called a generalizing characteristic, expressed as a numerical measure of the ratio of two compared absolute values. These indicators are used to study the structure of the phenomenon under study, to compare its level of development with the level of development of another phenomenon, to assess the changes occurring in the phenomenon under study, etc.
A relative statistic is obtained by dividing one absolute value by another.
IN general view the relative statistic formula would look like this:
Relative indicators can be expressed as coefficients, percentages, ppm and decimille.
If the base of comparison is taken as one, then the relative indicator is expressed in the form of a coefficient. If the base of comparison is taken as one hundred units, then the relative indicator is expressed as a percentage. If the base of comparison is taken as a thousand units, then the relative indicator is expressed in ppm (a tenth of a percent), if for ten thousand - in a decimill (hundredth of a percent).
Speakers;
plan and execution of the plan;
structures;
coordination;
Intensity and level economic development;
Comparisons.
Relative indicator of dynamics characterizes the change in the phenomenon under study over time and represents the ratio of indicators characterizing the phenomenon in the current period and the previous (or base) period.
OPD =
The indicator calculated in this way is called the coefficient of growth (decrease). It shows how many times the indicator of the current period is more (less) than the indicator of the previous (base) period. Expressed as a percentage, the relative indicator of dynamics is called the rate of growth (decrease).
T p \u003d (y i / y i-1) * 100%
T p \u003d (y i / y o) * 100%
Example: the population of the Russian Federation according to the 2002 census. amounted to 145181.9 thousand people, according to the 1989 census. - 147021.9 thousand people Determine the coefficient and rate of growth (decrease).
Consequently, the population decreased by 1.3%.
Relative indicator of the plan (forecast) (OPP) and the implementation of the plan (OPVP) are used by all subjects of financial and economic activity, carrying out current and strategic planning and calculated by the formula:
The relative indicator of the plan characterizes the intensity of the plan task, and the relative indicator of the plan's implementation characterizes the degree of its implementation.
Example: the actual turnover of the company for 2008. amounted to 2 billion rubles. Market analysis showed that in 2009. it is possible to bring the turnover up to 2.6 billion rubles. Actual turnover for 2009 amounted to 2.5 billion rubles. Define AKI and OPVP.
OPP==130% or 1.3 times
OPVP==96%
Calculations show that the target for 2009 is 1.3 times higher than the actual level for 2008, but the plan for 2009 was only 96% fulfilled.
Relative indicators of the structure(OPS) characterize the shares (specific gravity) constituent parts aggregate in its total volume. They characterize the structure of the population and its structure.
OPS=(*100%)
OPVs are usually expressed in the form of coefficients or percentages. The sum of the coefficients should be 1, and the sum of the percentages should be 100%, since the specific weights are reduced to a common base.
The set of relative values of the structure shows the structure of the set.
Relative indicators of coordination(OPK) characterize the ratio of parts of a given statistical population to one of them, taken as the basis for comparison. They show how many times one part of the population is larger than the other, or how many units of one part of the population fall into one, ten, one hundred, etc. units of another population.
For the base of comparison, the part that has the largest share is selected, or is a priority in this population.
Relative indicators of the intensity and level of economic development(OPI) characterize the degree of distribution or the level of development of the studied phenomena or processes in a certain environment. They are formed as a result of comparing oppositely named, but in a certain way interconnected quantities.
This indicator is calculated per hundred, thousand, ten thousand, etc. units of the studied population and is used in cases where it is impossible to determine the scale of the phenomenon by the value of the absolute indicator. For example, when studying demographic processes, indicators of fertility, mortality, natural increase (decrease) of the population are calculated as the ratio of the number of births (deaths) or the amount of natural increase per year to the average annual population of a given territory per 1,000 or 10,000 people.
K r =‰
K m=‰
to natural growth =‰
Relative indicators of the level of economic development characterize the efficiency of resource use and production efficiency. These are indicators of production output, costs per unit of output, efficiency in the use of production assets, etc.
Relative Comparison Index OPS r characterizes the comparative sizes of the same-named absolute indicators relating to different objects or territories, but for the same period of time.
They are obtained as private ones from the division of the absolute indicators of the same name characterizing different objects belonging to the same period or point in time.
OPS r=
These indicators can be used to compare labor productivity in different countries compare prices of different products and economic indicators for various businesses.
Statistics studies the quantitative side of mass phenomena and processes with the help of statistical values, which are divided into absolute and relative values.
Absolute values characterize sizes in specific conditions of time and place. They characterize the entire population.
Units of measurement of absolute values:
1) natural, reflecting the natural properties of the phenomenon, is a physical measure of weight, length, etc. The main disadvantage of natural units of measurement is that it is impossible to sum up various natural absolute values;
2) conditionally natural(used for the purpose of summing consumer products of different shapes);
3) combined. They are obtained by multiplying or dividing two natural units of measurement;
4) value (cash). Eliminate the shortcomings of the previous units of measurement, allow you to evaluate heterogeneous products.
However, absolute values do not provide a comprehensive description of the phenomena and processes under study and are not always suitable for comparison. This necessitates the use of relative values, which are used in comparisons, comparisons and play the role of a ratio measure.
Relative quantities are abstract statistical quantities expressing the quantitative ratio of two quantities.
Types of relative values: 1) relative dynamics- this is the ratio of the actual value of the indicator in the reporting period (y1) to its actual value in the base, previous period (y0):
ATS = Y 1 / Y 0 × 100%.
Relative values of the dynamics characterize the change of the phenomenon in time. In statistics, these indicators are called growth rates; 2) relative values of plan fulfillment- this is the ratio of the actual value of the indicator (y1) to its planned value (upl) of the same period:
OVVP = Y 1 / Y pl × 100%.
This relative value shows the degree of implementation of the plan as a percentage; 3) the relative value of the execution of the planned target- this is the ratio of the planned value of the indicator (sIL) to the actually achieved value in the previous period, i.e. in the base (y0):
OVPZ = Y pl / Y 0 × 100%.
Shows by how many percent the planned target is higher (lower) than actually achieved in the base period. This value is called the planned growth rate;
4) relative size of the structure- shows the composition of the phenomenon, expressed in the form of a share or specific gravity. The share (d) is the ratio of the part to the whole, i.e. the ratio of the constituent parts of the aggregate to its total volume. Specific gravity is the share expressed as a percentage. Relative values of the structure are used in statistics to characterize structural shifts;
5) relative amount of coordination- shows the ratio of the parts of the whole, i.e. the ratio of successively all parts to one of them, taken as the base. The smallest value is taken as the base. The relative value of coordination shows how many units of a given part of the whole fall on its other part, taken as the basis of comparison;
6) relative intensity value is the ratio of two opposite quantities related to each other. Characterizes the degree of development of a phenomenon in a particular environment;
7) relative comparison value is the ratio of similar quantities characterizing different objects of study for the same period. Shows how many times the numerator is greater (less) than the denominator.
The essence of averages. Types and forms of average values. Variants and frequencies
The method of averages is one of the most important methods in statistics because averages are widely used in analysis, in practice, in establishing patterns, trends, relationships, and for many other purposes. The essence of average values is that they characterize the level of the trait under study by one number. A distinctive feature of averages is that they are general indicators.
average value- this is a generalizing indicator that expresses the typical level (size) of a variable trait per unit of the population (qualitatively homogeneous).
The average value reflects the total that is hidden in each unit of the population. She captures common features, common patterns that manifest themselves by virtue of the law of large numbers. Speaking of averages, they mean that they characterize the entire population as a whole, however, along with the average, it is necessary to provide data on individual units of the population.
Problems solved using the method of averages:
1) characteristics of the level of development of the phenomenon under study;
2) comparison of two or more levels of the studied populations;
3) characteristics of changes in the level of the phenomenon in time;
4) identification and characterization of relationships between the studied populations.
P principles for constructing averages:
1) average values can be calculated only for qualitatively homogeneous aggregates;
2) averages should not be abstract, that is, only quantitative indicators. They should give a qualitative-quantitative characteristic of the phenomenon under study. Therefore, in statistics, the average value is not an abstract, abstract number, but a very specific indicator related to some phenomenon, place, time;
3) the choice of the population unit, in relation to which the average value is calculated, must be theoretically justified.
The following main types of averages are distinguished: arithmetic average; average harmonic; root mean square; geometric mean.
For the correct calculation of averages, it is necessary to introduce concepts such as variants and frequencies.
As a result, summaries and groupings get statistical series, i.e., series of digital indicators. According to their content, such rows divided into distribution lines And rows of dynamics .
The distribution series characterize the distribution of population units according to any one attribute, the varieties of which are ordered in a certain way. There are two types of distribution series - attribute and variation series.
Attribute rows are formed as a result of grouping data according to qualitative characteristics (for example, the distribution of the population by sex). There are as many groups in these series as there are variants of a qualitative trait.
Variation series- this is an ordered series of values of a varying quantitative attribute and the number of units that have a given value of the attribute (for example, the distribution of workers by wages).
In the variation series of distribution, the following elements are distinguished:
1) options(x or x1, x2 ... xn) is a series of numerical values of a quantitative attribute (for example, experience, wage, age). Options can be either absolute or relative values;
2) frequencies(m: m1, m2 ... mn) are numbers showing how many times the corresponding options are repeated (for example, the number of workers). Frequencies are usually denoted by an absolute number; if, according to the condition, the frequencies are expressed as percentages of the total or shares, then they are called relative frequencies (or) frequencies f:
f = m / Σ m .
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