Absolute indicators

In the process of statistical observation, data are obtained on the significance of certain features that characterize each unit of the studied population. To characterize the population as a whole or its individual parts, data on individual units of the population is subjected to construction. By directly summing up the primary data, generalizing absolute indicators are obtained that characterize the size of the population and the volume (size) of the phenomenon under study within specific time and place boundaries.

Absolute indicators are of great cognitive and practical importance. Knowledge of the levels, sizes and volumes of absolute statistical indicators is necessary for planning, managing and analyzing the economic activity of the national economy, its branches and enterprises. In absolute terms, most of the planned targets for the development of the country's economy, meeting the needs of society in a variety of products and services are set, and control over their implementation is carried out.

With the help of absolute indicators characterizing the volume of the country's gross domestic product, gross national income, the value of fixed assets, the number of employees, the wage fund of the enterprise, production in the economy and other socio-economic phenomena.

Absolute indicators- these are quantities expressing the dimensions of social phenomena as such, without their relation to other phenomena. For example, as of January 1, 2010, the population of Ukraine was 46.0 million people, and as of January 1, 2008, there were 43,475 farms.

Absolute indicators expressing the size of social phenomena within certain boundaries of time and territory, characterizing the overall size of the statistical population.

They are named numbers, always have a certain dimension and units.

Depending on the nature of the phenomenon and the objectives of the study, absolute indicators are expressed in physical, cost, labor and conventionally natural units of measurement.

Absolute indicators can express the dimensions, volumes and levels of social phenomena at a certain moment (on the contract (ld) on January 1, 2010, the number of cows on the farm was 770 heads) and for a certain period of time (milk production on the farm in 2009 amounted to 21600 c).

According to the way of expressing the dimensions of the studied phenomena, absolute indicators are divided into individual, group and general.

Individual called such absolute indicators that express the size of the quantitative characteristics of individual units of the population. For example, the number of employees of an enterprise, the production of gross output in an agricultural firm, the profit of an enterprise, etc.

Group absolute indicators express the size of the signs or the number of units in separate parts (groups) of the marriage. they are obtained during the processing of statistical observation materials by summing up the absolute dimensions of a feature in individual population units or by counting the number of population units included in separate groups.

General called absolute indicators that express the size of the feature in all units of the population. They are the result of a summary of statistical observation data. For example, the wage fund of farms in the region, the cost of fixed production assets in the region's SOOO, the gross potato harvest in the country, etc.

The concept of relative values, their types

Absolute indicators play an important role in the system of statistical indicators. At the same time, when studying socio-economic phenomena, statistics cannot be limited to calculating only absolute indicators, since they often do not give a sufficiently complete picture of the phenomenon under study. So, for example, when comparing the absolute indicator of beef production on the farm, let's say 3600 centners with the planned level, the level of the last year or the established project, successes and shortcomings in the work of the farm become clearly visible. If the increase in production compared to the previous year was +12%, and the plan was fulfilled by 97% and in relation to the design capacity is 92%, then it becomes clear that the farm has sufficient reserves to increase beef production. Therefore, in statistical analysis, along with absolute values, there is a need to calculate derivative generalizing indicators - average and relative indicators. Average values ​​are discussed in detail in Sec. 4. Now let's dwell on the characteristics of relative indicators.

relative called indicators expressing quantitative relationships between socio-economic phenomena. they are obtained by dividing two absolute or average values. Thus, the yield of grain crops (relative indicator) is obtained by comparing two absolute indicators of gross harvest and sown area: 48,000 centners: 1200 ha = = 40.0 centners / ha.

In this case, the value with which they are compared is called the basis, or base of comparison and a comparable value - current or reporting.

When calculating relative values, it should be borne in mind that the numerator always contains an indicator that reflects the phenomenon that is being studied, i.e. compared indicator, and in the denominator - the indicator with which to compare, which is taken as the basis or base of comparison.

Relative indicators are of great analytical importance. They are calculated to obtain characteristics of various aspects of social life. With their help, they express the degree of fulfillment of plans, the efficiency and intensity of social production, labor productivity, the degree of satisfaction of the material and cultural needs of people, the structure and dynamics of production, etc.

With the help of relative indicators, like and dissimilar values ​​can be compared.

One of the most important advantages of relative indicators is that they make it possible to compare such phenomena, the absolute dimensions of which are directly incommensurable. For example, the production of gross agricultural output per 100 hectares of land, population density, the production of certain types of food per capita, etc.

Depending on the comparison base, relative indicators can be expressed in various forms: coefficients (shares), percentages (%), ppm (%0), prodecemile (%00).

If the base of comparison is taken as a unit (equated to one), then the relative value (comparison result) is called the coefficient (share) and shows how many times the value under study is greater than the base. If the value of the base or base of comparison is taken as 100%, the result of calculating the relative value will be expressed as a percentage.

To avoid hard-to-perceive small relative values, the base value is sometimes taken as 1000 or 10000 units. In cases where the comparison base is taken as 1000 (for example, when calculating demographic coefficients), the result of the comparison is expressed in ppm, and when 10000 - prodecemile. They are used in comparisons of phenomena that rarely occur to give relative values ​​a convenient form for perception. For example, instead of the number of tractors per 100 hectares of arable land 1.87, 18.7 ppm are used. per 1000 ha.

In cases where the value being compared is greater than the base, the relative indicator can be expressed either as a coefficient or as a percentage. When the compared indicator is less than the base, the relative indicator is better expressed as a percentage, if relatively small values ​​behind the numerical value are compared with large ones, the relative indicators are expressed in ppm or prodecemile. Thus, in these forms of expression, the birth rates, mortality rates, natural and mechanical population growth, marriage rates, divorces, crime, the number of people with higher education and the number of hospital beds per 10,000 people are calculated. the population, etc.

Depending on the content and cognitive significance, the following types of relative indicators are distinguished: structure, plan task, plan fulfillment, dynamics, intensity, coordination, differentiation, comparison, etc.

Relative indicators of the structure represent the ratio of the part to the whole or the proportion of the units in the total volume of the population. They characterize the structure and composition of the studied population, which makes it possible to single out the main links, elements in a complex phenomenon and focus attention on them in further analysis. they are obtained by dividing the value of each part of the population by their total. These indicators are expressed in fractions of a unit (coefficients) or percentages. Structure indicators for any attribute, which in total give 100%, constitute a structural series. An example of relative indicators of the structure can be the composition of the population of Ukraine by sex (see Table 3.2), the proportion of cows in the total number of cattle, the structure of sown areas, cost, labor costs, products, etc.

Relative target indicator represents the ratio of the value of the indicator, which is set for the planned period, to its value achieved for the previous period or any other period taken as the basis for comparison.

Relative plan completion rate is the ratio of the actual level achieved to the planned target.

Relative indicators of dynamics characterizing the change in social phenomena over time. They are defined as the ratio of the level under study to the level taken as the base of comparison (to the previous year, or to the constant base of comparison). Relative indicators of dynamics are expressed in the form of coefficients (rates) of growth, absolute and relative growth. This type of relative indicators is discussed in more detail in Sec. 10 is specifically devoted to statistical processing and analysis of time series.

Relative indicators of dynamics, plan task and plan fulfillment are interconnected by the following equality: the relative indicator of dynamics is equal to the product of the relative indicators of the plan task and plan fulfillment.

Let's look at this relationship with an example. On the farm, there is data on the average daily growth of young cattle (g): for the base year (y0) - 420, according to the plan (upl) - 450 and actually (y1) - 465.

Relative intensity indicators characterize the ratio of oppositely named, but interconnected by a certain dependence of quantities. Relative intensity indicators are calculated by dividing the absolute value of the phenomenon under study by the absolute value, which characterizes the volume of the environment in which the phenomenon develops and spreads. The relative value shows how many units of one population account for a unit of another population. An example of relative intensity values ​​can be population density per 1 km2, gross output per 100 hectares of arable land, one hryvnia of fixed production assets, the average annual worker, the number of cows per 100 hectares of agricultural land, etc. Indicators of this kind are often called qualitative, since they reflect the most important qualitative aspects of production: the level of intensification, the armament of labor, the productivity of land and animals, cost recovery, etc.

Relative indicators of coordination characterize the ratio of different structural units of the same population (for example, the ratio between the urban and rural population, men and women, workers and employees, fixed and circulating assets, power and working machines, etc.). Relative indicators of coordination are most often expressed as the number of units of one part per 100 or 1000 units of another part.

Relative indicators of differentiation obtained as a result of comparing two structural series, one of which characterizes the ratio of parts of the population by the number of units, and the second - by the value of any sign. For example, a comparison of the share of farms in terms of number and the share of gross output, land, workers, etc. in these farms.

Relative comparison values are obtained as a result of comparing similar indicators related to different objects taken over the same period or point in time (for example, comparing hop yields in two farms for the reporting period, pig productivity in farms in two regions over five years, etc.).

One of the important conditions for the correct calculation, comparison and analysis of relative indicators is to ensure comparability of data. This means that the absolute and relative indicators taken for calculations, comparisons and analysis should: 1) refer to the same range of objects and units of observation or the same population; 2) be determined according to a single methodology, which ensures their comparison in meaning; 3) belong to one territory; 4) characterize data for the same period or point in time; 5) have the same units of measurement.

Relative indicators can be simple and composite. In statistical analysis composite ratios, which are equal to several simple indicators, it is advisable to decompose into a number of simple relative indicators that have independent significance.

Such a decomposition makes it possible to study the dependence of the relative composite indicator on its factors. In this case, the relationship itself has the form of a certain equation. Most often, the method of decomposition of composite indicators is used when studying the output per unit of production resources (land, fixed assets, labor), output per machine and worker, costs per unit area or head of animals. The decomposition schemes of indicators may vary depending on the nature of the information and the tasks of the analysis. So, for example, grain production per 1 ha of agricultural land - a relative indicator can be represented as a product of such simple indicators:

We will consider the ratio of these indicators in the following example (Table 3.11).

Let's check the relationship between the calculated indicators and draw conclusions:

base year 16.3 = 42.5-0.500.96-0.80;

reporting year 19.6 = 48.7-0.52-0.97-0.80;

reporting year to the base year 1.2024 = 1.1459-1.0400 1.0104-1.0000.

In the reporting year, 3.3 centners more grain was produced per hectare of agricultural land than in the base year, mainly due to an increase in the yield of grain crops from 42.5 to 48.7 centners per hectare, that is, by 14.59%. In part, grain production increased due to an increase in the share of grain crops in the total sown area (by 4%) and more intensive use of arable land for sowing (the share of crops in arable land increased by 1.04%).

Table 3.11. Data for the analysis of grain production per 1 and agricultural land on the farm

In economics, statistical disciplines are in priority positions. This is due to various reasons. First of all, within the framework of general economic specialties, statistical research acts as the basis for the development and improvement of analytical methods. In addition, they are an independent direction with its own subject.

Absolute and relative values

These concepts act as key elements in statistical science. They are used to determine the quantitative characteristics, the dynamics of their change. Absolute and relative values ​​reflect different characteristics, but without one, others cannot exist. The former express the quantitative dimensions of this or that phenomenon, regardless of others. It is impossible to assess the ongoing changes and deviations from them. They express the volume and level of a process or phenomenon. Absolute values ​​are always named numbers. They have a dimension or unit of measure. They can be natural, labor, monetary and so on. For example, standard hours, pieces, thousand rubles. etc. Average and relative values, on the contrary, express the ratio of several exact dimensions. It can be established for several phenomena or for one, but taken in a different volume and in a different period. These elements act as a quotient of statistical numbers, which characterizes their quantitative ratio. To determine the relative values, you need to divide one size by another, taken as the base. The latter may be planned data, actual data from previous years or another enterprise, and so on. Relative can be expressed as a percentage (if the base is taken as 100) or coefficients (if the base is one).

Classification of statistical numbers

Absolute values ​​are presented in two types:

1. Individual. They characterize the size of the trait in specific units. For example, it can be the amount of an employee's salary, a bank deposit, and so on. These dimensions are found directly in the course of statistical observation. They are recorded in the primary accounting documentation.
2. Total. Values ​​of this type reflect the total indicator of the attribute for the totality of objects. These dimensions act as the sum of the number of units (the population size) or the volume of the varying characteristic.

Units

Natural absolute values ​​can be simple. These are, for example, tons, liters, rubles, pieces, kilometers. They can be complex, characterizing a combination of several quantities. For example, statistics use ton-kilometers to establish the freight turnover of railway transport, kilowatt-hours to estimate electricity production, and so on. Conditionally natural units are also used in research. For example, the tractor park can be converted into reference machines. Value units are used to characterize a heterogeneous product in terms of money. This form, in particular, is used in assessing the income of the population, gross output. Using value units, extras take into account the dynamics of prices over time, and overcome the disadvantage due to "comparable" or "constant" prices for the same period. Labor values ​​take into account the total cost of work, the complexity of certain operations that make up the technological cycle. They are expressed in etc.

Relative values

The main condition for their calculation is the comparability of units and the existence of a real connection between the phenomena under study. The value with which the comparison is carried out (the denominator in a fraction) acts, as a rule, as the base or base of the ratio. Depending on its choice, the result can be expressed in different fractions of a unit. It can be tenths, hundredths (percent), thousandths (10th part of% - ppm), ten thousandths (hundredth of% - decimille). Comparable units can be either the same or different. In the second case, their names are formed from the units used (c/ha, rub./person, etc.).

Types of relative values

Several types of these units are used in statistics. So, there is a relative value:

1. structures.
2. Planned task.
3. intensity.
4. Speakers.
5. coordination.
6. Comparisons.
7. Degrees of economic development.

The relative value of the task expresses the ratio of what is planned for the upcoming period to what has actually developed for the current period. The plan unit is calculated in the same way. The relative size of the structure is a characteristic of the share of specific parts of the population under study in its total volume. Their calculation is carried out by dividing the number in individual parts by their total number (or volume). These units are expressed as percentages or simple multiples. For example, this is how the proportion of the urban population is calculated.

Dynamics

The relative value reflects in this case the ratio of the level of the object in a particular period to its status in the past tense. In other words, it is characterized by a change in a phenomenon over a period of time. The relative value characterizing the dynamics is called The choice of the base in the calculation is carried out depending on the purpose of the study.

Intensity

The relative value can reflect the degree of development of a phenomenon in a particular environment. In this case, we talk about intensity. Their calculation is carried out by comparing opposite quantities that are related to each other. They are set, as a rule, based on 1000, 100 and so on units of the study population. For example, per 100 hectares of land, per thousand people, and so on. These indicators of relative values ​​are named numbers. For example, this is how population density is calculated. It is expressed as the average number of citizens per square meter. km of territory. The characteristics of the degree of economic development serve as a subtype of such units. These, for example, include such types of relative values ​​as the level of GNP, GDP, VID, and so on. per capita. These characteristics play an important role in the analysis of the economic situation in the country.

Coordination

The value of relative values ​​can characterize the proportionality of the individual elements of the whole to each other. The calculation is carried out by dividing one part by another. Relative quantities in this case act as a subtype of units of intensity. The difference lies in the fact that they reflect the level of distribution of heterogeneous parts of the same population. The base can be one or another sign, depending on the goal. In this regard, for the same whole, several relative values ​​of coordination can be calculated.

Mapping

Relative comparison values ​​are units that are partial divisions of similar statistical features that act as characteristics for different objects, but refer to the same moment or period. For example, the ratio of the cost of a particular type of product produced by two enterprises, labor productivity for different industries, and so on is calculated.

Economic evaluation

In this study, absolute and relative units are actively used. The former are used to establish the ratio of reserves and expenses with sources of financing and evaluate the enterprise in terms of financial stability. Relative indicators reflect the structure of funds with the state of fixed and working capital. Economic evaluation uses horizontal analysis. The most generalizing absolute value that characterizes the financial stability of the company is the lack or excess of sources of financing costs and reserves. The calculation is made by subtraction. The result is the difference in the size of the sources (minus non-current assets), the means of which stocks are formed, and their number. The key elements in this are the following statistical units:

1. Own current assets.
2. General indicator of planned sources.
3. Long-term borrowed and own funds.

Deterministic factorial research

This analysis is a specific technique for studying the impact of factors whose interaction with the results has a functional character. This study is conducted by creation and evaluation. Relative indicators are widely used in this analysis. In most cases, factor analysis uses multiplicative models. For example, profit can be expressed as the product of the quantity of goods and the unit cost. Part of the analysis in this case is carried out in 2 ways:

1. implies a chain substitution. The change in the result due to the factor is calculated as the product of the deviation of the studied trait by the base of another according to the selected sequence.
2. The relative difference method is used to measure the impact of factors on the increase in the result. It is used when there are previously calculated percentage deviations in the source data.

Time Series

They represent a change in the numerical indicators of social phenomena over time. One of the most important directions in this analysis is the study of the development of events for specific periods. Among them:

Conclusion

Undoubtedly, relative values ​​have a high scientific value. However, in practice they cannot be used in isolation. They are always in relationship with absolute indicators, expressing the ratio of the latter. If this is not taken into account, then it is impossible to accurately characterize the phenomena under study. Using relative values, you need to show what specific absolute units are hidden behind them. Otherwise, you can draw wrong conclusions. Only the complex use of relative and absolute values ​​can act as the most important means of information and analysis in the study of various phenomena occurring in socio-economic life. In general, the transition to the calculation of deviations makes it possible to compare the economic potential and the result of the activities of enterprises that differ significantly in terms of the amount of resources used or other characteristics. Relative values, in addition, can smooth out some processes (force majeure, inflation, and others) that can distort absolute units in financial statements.

Light, reaching the boundary between two media, abruptly changes its direction. Some of it will return to its original environment, as light reflection is observed. Let us find out how the relative refractive index differs from the absolute value. In the presence of a transparent second medium, a partial passage of light through the boundary of the available media will be observed. The beam will change its original direction, that is, it will be refracted.

optical phenomena

Due to partial refraction, there is an apparent change in the shape of various objects, their size, position. In order to understand how the relative refractive index differs from the absolute index, consider simple experiments. Place a small object at the bottom of an opaque empty glass. We arrange the glass so that its edge, as well as the center of the coin, are on the same straight line.

Without changing your position, pour water into a glass. When its level rises visually, the bottom of the glass, along with the coin, will rise. The coin, which was initially only partially visible, is now visible in its entirety.

Let's try to place a pencil in a glass of water. When viewed from the side, the effect of dividing the pencil into two parts is created. Such an experiment can easily be explained by the appearance of refraction of light. It is possible to calculate the absolute and relative refractive index, knowing from which medium the transition occurs to which.

How does refraction happen?

The transition relates the magnitude of the angle of incidence and refraction. It was established in the eighteenth century, when Huygens' principle was used in experiments.

From the results of various experiments, it is possible to confirm the formulation of the law: the incident and refracted beam, as well as the perpendicular drawn to the point of incidence, are located in the same plane.

Refractive index

Absolute indices and relative refractive indices are associated with the transition from one medium to another. The physical meaning of the refractive index can be derived from the Huygens principle. To calculate it, find the ratio of the speed of light in media, at the interface of which refraction is observed.

Absolute indices and relative refractive indices differ from each other. An indicator that is determined relative to vacuum is considered absolute. It is calculated as the ratio of the sine of the angle of incidence to the sine of the refracted angle when the beam passes from vacuum to a certain medium.

A medium that has a lower absolute refractive index is considered to be a less optically dense medium. The ratio of indicators does not exceed one.

Absolute indicators and relative refractive indices are included in special tables, depending on the type of medium. The law of light refraction makes it possible to calculate the path of rays in various optical devices, for example, in a triangular prism made of glass or other transparent materials.

Statistics indicators

Absolute scores and relative scores are associated with statistical calculations. They are a quantitative characteristic of a social and economic phenomenon and process in the case of qualitative certainty. Its essence lies in the fact that the indicator is interconnected with the internal content of the analyzed process or phenomenon, its essence. Thanks to these indicators, you can determine what and how can be calculated. The absolute and relative change of indicators is carried out using certain methods.

For example, to calculate the volume of industrial goods, you first need to establish the types of activities of the enterprise, determine the results of the work, and only after that carry out calculations of output.

Set of parameters

In statistics, rather complex phenomena and processes are considered, so they cannot be analyzed using a single parameter. Only in the system absolute and relative indicators, variations of other values ​​allow obtaining a reliable result.

The system of indicators in statistics is a set of interconnected values ​​that have different levels, aimed at solving a particular statistical problem. Absolute and relative statistical indicators are considered in unity, they allow you to study marketable and shipped products, the volume of goods sold, and calculate the cost of delivery.

Kinds

There are two types of them:

• indicator - category;
• specific statistic.

The second type is characterized by the size, magnitude of the analyzed process or phenomenon at a certain point in time. It implies a clear binding to a certain territory. For example, one can talk about the specific value of industrial production assets, indicating the place and time of settlement.

Analysis of absolute and relative indicators allows us to evaluate the turnover of catering enterprises, public trade.

Features of absolute indicators

As the statistical observation is carried out, information appears about the values ​​of certain features that characterize any unit of the population. To carry it out individually or as a whole, information is subjected to a summary, then generalized indicators are obtained, which reflect the results of a quantitative analysis of the phenomenon or object under study.

Arguing over how the relative indicator differs from the absolute indicator, we highlight the accuracy. A statistical absolute indicator is considered as a value that reflects the physical parameters, cost and time characteristics of socio-economic processes and various phenomena.

Absolute and relative financial indicators help to conduct research not only within a small company, but also on the scale of industrial giants.

Individual indicators in statistics are obtained during observations, weighing, measurements, calculations, and evaluation of quantitative characteristics. In some situations, absolute individual indicators have a difference character. For example, a comparison is made between the number of official unemployed at the beginning and at the end of the calendar year, the company's revenue on weekdays and holidays is estimated.

Volumetric summary indicators that characterize the volume of the population or feature for the analyzed object can be obtained as a result of grouping various individual values ​​or from a summary.

For conversion into conventional units of measurement, special coefficients are used, which are calculated as the ratio of the consumer properties of the analyzed product to its reference indicator.

Absolute and relative indicators of dynamics in a market economy are aimed at conducting a monetary assessment of socio-economic processes and phenomena.

For example, an important indicator for any country is the calculation of the gross domestic product. Labor units of measurement, which allow taking into account the labor intensity of specific operations, as well as the total labor costs at the enterprise, are calculated in man-hours and man-days.

Relative indicators

Despite the importance of absolute indicators in cognitive and practical human activity, it is necessary to systematically conduct a comparative comparison. Relative and absolute averages together give an accurate picture of a particular process or phenomenon.

A relative indicator is a value that results from dividing two absolute indicators. It fully reflects the relationship between the quantitative characteristics of phenomena and analyzed processes.

Based on the above classification, it is possible to compare indicators of the same name that refer to different periods, objects, territories. As a result of such a comparison, you can get a percentage, compare the result with the maximum and minimum baseline.

The relative indicator of dynamics is the ratio of the level of the analyzed phenomenon and process for a specific period of time to the level of a similar process or phenomenon in the past.

This value demonstrates how many times the considered level will exceed the baseline, or determine the percentage of indicators. The option concerning the multiple ratio is called the multiple, and when it is multiplied by one hundred percent, the growth rate is obtained.

Features of the calculations

Regardless of the financial and economic form of activity, all companies are engaged in operational and strategic planning, make comparisons between the results obtained and the planned plans. For such activities, the relative indicators of the plan are used, as well as the parameters of its implementation.

The relative indicator of the structure is defined as the ratio of the structural parts of the analyzed object and a single whole. It is expressed as a percentage or shares.

The base of comparison is that part that has the maximum share or is considered a priority from a social, economic, or other point of view.

Relative intensity indicators determine the possibility of propagation of the analyzed phenomenon or process in a certain environment. It is used in cases where the absolute value is not enough to formulate clear conclusions about the size, saturation, density, and extent of distribution. For example, you can calculate the birth rate, calculate the population density.

The relative indicator of comparison is the ratio of similar absolute indicators that characterize various enterprises, firms, countries, regions, regions.

Average values

Such values ​​are used at the stage of processing and systematization of the initial statistical results. The need to identify average values ​​is explained by the fact that different units of the analyzed populations of even one attribute have some differences.

Average values ​​are indicators that summarize and systematize certain features or groups of characteristics.

When choosing homogeneous qualitative characteristics, the calculation of the average value is carried out by adding all the values, dividing them by the number of measurements.

For example, for employees in a certain industry who have a fixed salary, you can calculate the average income level. If desired, you can also analyze the amount of necessary expenses that go to pay for housing, food, and essential goods.

If a study is conducted with heterogeneous qualitative characteristics, in this case it is advisable to compare with indicators for the region, territory, country, region. Such a variant of processing the results involves the use of system averages, that is, there are no restrictions in the calculations within the same team or enterprise.

It is these statistical calculations that are most often used to compare the quality of life of the population, their level of income, and gross domestic product.

The value of average indicators is called average values, generalizing formulas are used to calculate them. The average value can be replaced by a large number of individual parameters of the analyzed trait, studying the properties that will be inherent in all parts of this population. This will make it possible to avoid accidents, to find common patterns that are due to common causes.

Conclusion

All statistical indicators perform an accounting function. Both employees and managers who study data on current system indicators use objective information.

Otherwise, there is a high risk that serious errors will be made in statistical calculations, which will negatively affect the economic condition of not only the enterprise, but also certain categories of citizens.

Of particular importance are statistical indicators for the formation of a human view of a particular situation or problem. For example, the population of the country wants to know about the average level of material income of citizens, is interested in the birth rate, unemployment rate.

Statistical indicators: absolute and relative values

1. Statistical indicators, their types.
2. Absolute value.
3. Relative values.

Statistical indicators, their types

Each unit of the statistical population can be characterized using statistical indicators. statistic – this is a quantitative-qualitative generalizing characteristic of some property of a group of units or an aggregate as a whole, and this is what distinguishes it from a trait. For example: the average salary in Ukraine is a statistical indicator, and the salary of a particular person is a sign.

A statistical indicator is a generalizing characteristic of the object under study, which combines its qualitative and quantitative certainty. quality the content of the indicator depends on the essence of the object under study (phenomenon, process) and is reflected in its name (number of goods sold, daily revenue, annual profit, etc.). quantitative the side of the phenomenon is represented by the number and its meter. The connecting link between qualitative content and numerical expression is indicator model, which reveals the statistical structure of the indicator, establishes, what, where, when, how to be measured. It substantiates units of measurement and computational operations. The indicator model reflects the rules for its construction and calculation.

Indicators are classified:

1. According to the method of calculation for:
- primary, are determined by summarizing and grouping data and are presented in the form of absolute values;
- derivatives, are calculated on the basis of primary or secondary indicators and are in the form of averages or relative values.

2. On the basis of time for:
- interval, characterize the state of the object (phenomenon, process) for a certain time (day, month, year). For example, the volume of products sold for the year, the production facilities of the enterprise put into operation during the quarter, the shift output of a worker, etc.;
- moment, characterize the phenomenon at a certain point in time. For example, the attendance of employees at the beginning of the shift, the availability of free taxis at the time of ordering, the state of the company's balance sheet accounts at the beginning and end of the year (quarter), working capital balances at the beginning of the month, etc.

3. According to the relationship of the object under study, pairs of mutually inverse (direct and inverse) statistical indicators are distinguished, which exist in parallel and characterize the same phenomenon. Straight the indicator increases with the increase of the phenomenon, o fraternal, on the contrary, decreases. For example, production output per unit of time is a direct indicator, and time spent per unit of output is an inverse indicator.

Absolute and relative values ​​can be expressed in statistical terms.

Absolute value

In statistics, absolute indicators are called total indicators that characterize either the size of a sign for individual units of the population (for example: the size of the salary of an individual employee) or the final value of the sign for a set of objects (wage fund of an enterprise). The absolute values ​​are named numbers, i.e. having a unit of measure. Depending on the specific task of the study and the nature of the phenomenon, natural, labor and value (cash) units.

Cost meters allow you to evaluate the activity of heterogeneous objects. For example: the volume of production of a machine-building plant is measured in units of output; the volume of work of a cargo ATP - in tons, ton-kilometers; passenger ATP - in passengers, passenger-kilometers; taxi fleet in paid kilometers. The indicators of the volume of production of the above enterprises are expressed in various natural units of measurement and therefore they are incomparable. If it is necessary to compare these enterprises, the results of their work should be considered in terms of value, i.e. in income.
AT labor units of measurement (man-day, man-hour) take into account labor costs at the enterprise or the labor intensity of individual operations of the technological cycle.
If there is a need to bring together several varieties of products for the same consumer purpose, the volume of such a phenomenon is expressed in conditionally natural units. Recalculation in conventional units is carried out using special reduction coefficients. For example, the fuel balance is compiled in tons of reference fuel. The standard is coal, the calorific value of which is 7000 cal per 1 kg. Caloric reduction coefficients for Donetsk coal - 0.9; natural gas - 1.2, etc.

When solving a certain range of analytical problems, absolute values ​​are presented in the form of balances, in which the indicators are grouped according to the sources of formation and directions of use. Dynamic balances are also widely used, which are compiled according to the scheme:
(balances at the beginning of the period) + (receipts) - (expenses) = (balances at the end of the period).

According to the ratio of absolute values ​​presented in the form of balances, the balance of processes is assessed. For example, the balance of income and expenditures of the population, the balance of export-import operations, etc.

Relative values

Relative value in statistics, it is a generalizing indicator, which is a quotient of the division of two absolute indicators and gives a numerical measure of the relationship between them. In this case, the numerator of the fraction is the value that is being compared, and the denominator is the value that is being compared. The latter is called base or basis of comparison. If the base of comparison is taken as one, then the relative value is expressed in the form of a coefficient and shows how many times the compared value is greater or less than the base. So, if we compare the number of students of the fourth (21 people) and second (49 people) courses of the specialty "Accounting and Audit", then we get a relative value in the form of a coefficient (49:21 = 2.33), which shows that students of the second course 2.33 times more. The comparison base can be 100, 1000, 10000 or 100000 units. Then the relative value is expressed respectively in percent (%), ppm (0/00), prodecimals (0/000) and procentimes 0/0000).

The choice of one form or another of the relative value depends on its absolute value. If the compared value is more than the base of comparison by 2 times or more, then the form of the coefficient is usually chosen (as in the example above). If the relative value is close to one, as a rule, it is expressed as a percentage, if it is very small, then in ppm, etc. For example, 0.0025 can be expressed as 0.25% or 2.5 0/00, or 250/000.
In accordance with the analytical function, the following types of relative values ​​are distinguished: the relative values ​​of dynamics, the planned task, the fulfillment of the planned task, structure, comparison, intensity, coordination.
Relative values ​​of dynamics() characterize the change in the level of a phenomenon over time, are calculated by dividing the level of a feature in the analyzed period or point in time by the level of the same feature in the previous period or point in time. Relative values ​​can be basic, when one year is taken as the comparison base, and chain values ​​- the previous year is taken as the comparison base.
For example, the production of electricity from nuclear power plants in Ukraine is characterized by the following data.

Then a) the basic relative values ​​of the dynamics of electricity production:
; ; ;.
b) chain relative values ​​of production dynamics):
; ; .

Relative value of the planned task() is calculated as the ratio of the level planned for the upcoming period to the level actually formed in the current period. For example, the volume of production in 2003 amounted to 100,000 pieces. conditional products, for 2004 it is planned to produce 110,000 pcs. products. Then

Relative value of the execution of the planned target() represents the ratio of the level actually achieved in this period to the planned one. Example: in 2004, it was planned to produce 110,000 units. conditional products, actually produced 105,000 pcs.

There is the following dependence between the relative values ​​of the dynamics, the planned task and the fulfillment of the plan

Example. It was envisaged to increase production by 5%, the actual growth was 7.5%. It is necessary to determine the degree of fulfillment of the planned task.



Thus, the target was exceeded by 2.38%.
Relative values ​​of the structure show the specific gravity (share) of individual parts in the totality. They are calculated by dividing the number of units in individual parts by the total number of units in the population. The relative values ​​of the structure are called shares, their sum is 1 or 100%. On the use of shares, a comparative analysis of the composition of populations of different sizes, an assessment of structural shifts over time is based. The difference between shares is called percentage points.
Relative comparison values() are indicators that are the quotient of the division of the absolute values ​​of the same name, belonging to different aggregates, but to the same period or moment. For example, as of January 1, 1996, 2,630,000 people lived in Kyiv, and 1,555,000 in Kharkov. people Then shows that in Kyiv the population is 69% more than in Kharkov, and shows that in Kharkov the population is 41% less than in Kyiv. (The absolute values ​​of the same name are the urban population, the aggregates are different cities).
Relative intensity values- show the degree of distribution or the level of development of a particular phenomenon in a particular environment. They are calculated by comparing opposite quantities. An example is population density, which is determined by dividing the population by the area of ​​the territory where it lives, or labor productivity. These indicators are usually defined in terms of 100, 1000, etc. units of the studied population.
Relative values ​​of coordination characterize the relationship between the individual parts of one whole. Calculated by dividing one part by another.
Example. As of January 1, 1996, the urban population of Ukraine amounted to 34.8 million people, the rural population - 16.5 million people.
When studying the urban population, they calculate . The obtained value shows that the urban population is more than the rural population by 2 times or by 110%.
If we take the number of the rural population as the base of comparison, then the relative indicator of coordination is equal to . This means that in Ukraine in 1996 the rural population was 53% less than the urban population. (Whole: population of Ukraine, parts: urban and rural population.)

Topic 5. Absolute and relative values. Mean values ​​and indicators of variation

1. Absolute values

2. Relative values

3. The essence of the average in statistics, types and forms of averages

4. Arithmetic mean and conditions for its application

5. Harmonic mean and conditions for its application

6. Structural averages

7. Types of variation indicators

Target : introduce the concept of "average value"; consider the types of averages and methods for their calculation; properties of the arithmetic mean; variation indicators.

After studying, you will be able : correctly determine the average values ​​and indicators of variation.

Information sources:

1. Statistics: Textbook / Ed. V.G. Ionina. - M.: INFRA-M, 2008.

2. Course in the theory of statistics: Textbook / Ed. V.N. Salina, E.Yu. Churikov. – M.: Finance and Statistics, 2006.

3. Godin A.M. Statistics: Textbook. – M.: Dashkov i K’, 2008.

4. Galkina V.A. Statistics: Textbook: M.: RGAZU, 2002.

5. Gromyko G.L. The theory of statistics. Workshop. - M.: INFRA-M, 2008.

6. Theory of Statistics: Textbook / Ed. R.A. Shmoylova M.: Finance and Statistics, 2007.

Absolute and relative values ​​are generalizing statistics. indicators that characterize the quantitative side of social phenomena. There are two types of generalizing indicators: absolute and relative values.

1. Absolute values

Absolute statistics are of great theoretical and practical importance. They are individual and total. As generalizing indicators, absolute values ​​are always total values ​​that can be population size indicators(number of enterprises, number of workers, number of students) and indicators of the volume of features(wages of workers, volume of output of goods and services, etc.).

Absolute values ​​are named numbers that have a certain dimension and units of measurement. They characterize indicators at a certain point in time or over a period. At the moment time, absolute values ​​show the state of the phenomenon (population, students, universities, enterprises); for the period - the results of the process (volume of production of goods and services, turnover, etc.). In the first case, the absolute values ​​are moment indicators, in the second - interval. Such a division of absolute values ​​is of great importance when calculating average levels in time series.

Depending on the reasons and goals, natural, conditionally natural, monetary and labor units of measurement are used in statistics. Natural units of measurement can be simple(for example, tons - transported cargo) and constituent(for example, ton-kilometers - freight turnover).

In international practice, the following physical units of measurement are used: meters, kilometers, miles, liters, barrels, pieces, kilograms, etc.

Conditionally natural meters are used in cases where a product has several varieties. Then the total volume can be determined based on the consumer properties of all varieties of the product. So, soap of different varieties is converted into conditional soap with a 40% content of fatty acids; canned food of various volumes is transferred to conditional cans with a volume of 353.4 cm3; various types of organic fuel - into conventional fuel with a calorific value of 29.3 mJ/kg (7000 kcal/kg). Conversion into conditionally natural units of measurement is carried out on the basis of special coefficients calculated as the ratio of the consumer properties of individual varieties of the product to its reference value.

At measures 1.3.1. During the reporting period, the company produced the following types of soap and detergents:

Determine the total amount of products produced by the enterprise in conditionally natural units of measurement. Soap of 40% fat content is taken as a conventional unit of measurement.

Decision

Let us calculate the conversion coefficients. If the conditional unit of measurement is 40% fat soap, then this value is taken equal to one. Then we calculate the coefficients of conversion into conditional soap as follows: laundry soap 60% fat: 60: 40 = 1.5; toilet soap 80% fat: 80: 40 = 2.0; laundry detergent 10% fat: 10: 40 = 0.25.

Table 1.3.1

Total production of soap and detergents by types for the reporting period

The total production of soap and detergents in 40% terms amounted to 3750 kg.

A special place is given to cost units of measurement, which make it possible to give a monetary assessment of socio-economic indicators (output of goods and services, gross domestic product (GDP), gross national product (GNP), etc.).

Labor units of measure (man-days, man-hours) make it possible to take into account both the total costs at the enterprise and the labor intensity of individual operations of the technological process.

In practice, in the absence of the necessary information, absolute values ​​are obtained by calculation, for example, on the basis of balance linking:

3n + P \u003d R + 3k

where Зн - stock at the beginning of the period;

P - receipt for the period;

P - consumption for the period;

Зк - stock at the end of the period.

R \u003d Zn + P - Zk

The total volume of the trait can also be calculated from the data on the mean value and population size. So, if the average number of students in a group is 25 people, the number of groups of students in a given specialty is 12, then the total number of students studying in this specialty is 300 people. (25 ´ 12).

Absolute statistical values ​​are widely used in the analysis and forecasting of the state and development of the phenomena of social life.

On the basis of absolute values, relative values ​​are calculated.

2. Relative values

Relative values(indicators) characterize the quantitative ratio of the compared absolute values. They are obtained by comparing two indicators. The numerator of the ratio is the compared value, it is called current or reporting magnitude, the denominator of the ratio is called base of comparison or basis of comparison. As a rule, the comparison base is taken equal to 1, 100, 1000, 10000. If the base is 1, then the relative value shows how many times the current value is greater than the base value, or what fraction of the base value it is, and is expressed in coefficients. If the base of comparison is 100, then the relative value is expressed as a percentage (%), if the base of comparison is 1000 - in ppm (%0), 10000 - in decimilles (%00).

Comparable values ​​can be of the same name and different names. If the values ​​of the same name are compared, then they are expressed in coefficients, percentages and ppm. When comparing different values, the names of relative values ​​are formed from the names of the compared values: population density of the country - people / km2; productivity - c / ha, etc.

Depending on the tasks, content and cognitive significance of the expressed quantitative ratios, the following types of relative indicators are distinguished:

1) the planned task (contractual obligations);

2) fulfillment of the plan (contractual obligations);

3) speakers;

4)structure;

5) intensity and level of economic development; 6) coordination;

7) comparisons.

Relative target indicator(OPPZ). All enterprises of any form of ownership carry out, to one degree or another, both current and long-term planning. To do this, calculate the OPPZ by the ratio of the level planned for the coming period (P) to the level of the indicator achieved in the previous period (Pho):

OPPP \u003d (P / Fo) ´ 100.

Example 1.3.2. IV quarter. In 2006, the output of goods and services amounted to 90 million rubles, and in the 1st quarter. In 2007, the output of goods and services is planned in the amount of 108 million rubles.

Determine the relative value of the planned task.

Decision

OPPV = (108 / 90) ´ 100 = 120%.

Thus, in the I quarter. In 2007, it is planned to increase the output of goods and services by 20%.

Relative Plan Completion Rate (RPI).

Enterprises not only carry out planning, but also compare the actual results of work with those planned earlier. For this purpose, the relative indicator of the implementation of the plan is calculated by the ratio of the level actually achieved in the current period (F1) to the level of the planned indicator for the same period (P):

OPVP \u003d (F1 / P) ´ 100.

Example 1.3.3. The release of goods and services in the I quarter. 2008 amounted to 116.1 million rubles. with a plan of 108.0 million rubles.

Determine the degree of fulfillment of the plan for the production of goods and services in the I quarter. 2008

Decision

OPVP \u003d (11.6 / 108.0) ´ 100 \u003d 107.5%.

The plan for the release of goods and services was fulfilled by 107.5%, i.е. overfulfillment of the plan was 7.5%.

Relative indicators of dynamics (OPD).

These indicators characterize the change in the levels of any economic phenomenon over time and are obtained by dividing the level of a feature for a certain period or point in time by the level of the same indicator in the previous period or point in time. Relative values ​​of dynamics, or, as they are called, rates of growth, can be expressed in coefficients or percent and are defined using a comparison base variable - chain and constant base of comparison - basic .

Relative structure indicators (d).

They characterize the composition of the population under study, the proportions, specific weights of the elements of the population in the overall total and represent the ratio of a part of the units of the population ( f 1 ) to the total number of population units (Σ fi):

d = ( fi / Σ fi) 100,

where d- the proportion of parts of the population.

Example 1.3.4. The following data are available (Table 1.3.2).

Table 1.3.2

Retail trade turnover of the Russian Federation in 2006 (million rubles)

Calculate the relative value of the structure of the retail trade turnover of the Russian Federation by quarters and for 2006

Decision

We calculate the relative values ​​of the structure of retail turnover for each quarter and for the year as a whole.

The calculated relative values ​​of the structure are presented in Table. 1.3.3.

Table 1.3.3

Structure of retail trade turnover in the Russian Federation in 2006

Table data. 1.3.3 indicate that in the second half of 2006 in the Russian Federation there was an increase in the share of sales of non-food products.

Relative indicators of the intensity and level of economic development. The indicators characterize the degree of saturation or development of a given phenomenon in a certain environment, they are named and can be expressed in multiple ratios, percentages, ppm and other forms.

Example 1 3.5. The average annual population of the Russian Federation in 2006 was 143.55 million people, the number of births was 1397.0 thousand people.

Define the number of births per 1000 people. population (relative value of intensity characterizing the birth rate).

Decision

For every 1000 people population in 2006 in the Russian Federation was born 9.7 people.

One of the indicators of the level of economic development of the country is the indicator of gross domestic product per capita.

Example 1.3.6. The production of gross domestic product (GDP) in the Russian Federation in 2006 at current prices amounted to 10,863.4 billion rubles. The average annual population in 2006 was 143.55 million people.

Determine production of gross domestic product per capita.

Decision

GDP per capita \u003d 10863.4 / 143.55 \u003d 75,677 rubles.

Consequently, per capita GDP production in 2006 amounted to 75,677 rubles.

Relative indicators of coordination (OPK).

The indicators characterize the relationship of the parts of the studied 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 account for 1, 10, 100, 1000 units of the other part. These relative values ​​can be calculated both in absolute terms and in terms of structure indicators.

Example 1.3.7. The following data are available on the economically active population of the Russian Federation as of the end of November 2006:

Calculate how many unemployed account for 1000 employed in the economy of the Russian Federation.

Decision

GPC \u003d (6.1 / 65.8) ´ 1000 \u003d 92.7 people.

Consequently, for every 1,000 people employed in the Russian economy, there were 92.7 people. unemployed.

Relative indicators of comparison (RPE).

Indicators characterize the relationship of absolute or relative indicators of the same name, corresponding to the same period or point in time, but related to different objects or territories.

3. The essence of the average in statistics, types and forms of averages

Average in statistics- a generalizing characteristic of a set of similar phenomena according to some quantitatively varying attribute, which determines the level of the attribute per unit of the population.

Types of averages

The following notations are used in the presented formulas:

x- sign values;

- the average value of the feature;

Σ - summation sign;

P - multiplication sign;

f(frequency) and M(frequency multiplied by feature values) - weights for calculating the weighted average:

N and f - the number of population units;

M- the total volume of the variable feature.

If the averages are calculated using the same data, then the given types of averages in terms of their numerical values ​​stand in the following row:

xh < xg < ха < х q ,

illustrating the so-called rule of majorance of means.

One of the tasks of determining the average is the correct choice of the type of average.

When choosing the type of average, it is necessary to take into account the economic content of individual characteristics, which must also be preserved in the final average. At the same time, any intermediate actions, including the final result, must be economically significant.

4. Arithmetic mean and conditions for its application

The arithmetic mean is used in cases where the volume of a variable feature of the entire population is formed as the sum of the values ​​of this feature in its individual units.

The formulas and calculation technique are as follows:

simple arithmetic mean (unweighted)

weighted average

Example 1.3.8. According to Table. 1.6.2, repeated below, we calculate the average work experience of workers using the arithmetic simple formula (unweighted)

Table 1.6.2

Work experience of employees and their average monthly output of products

The use of the arithmetic mean is explained by the fact that the volume of the variable sign for the entire population - the total number of years worked by employees (51 years), is formed as the sum of the length of service of each employee.

The calculation of the arithmetic mean according to the data of the distribution series has its own characteristics. Let us illustrate these features according to the grouping data in Table. 1.3.5.

arithmetic mean variation

Table 1.3.5

Calculation of the average work experience of workers based on a distribution series

In this case, you should use the weighted arithmetic mean formula, since the interval values ​​of the feature occur more than once, and these repetition numbers (frequencies) are not the same.

The specific values ​​of the attribute that should be directly involved in the calculations are the middle (centers) of the intervals (but not the average values ​​in the intervals!), And the weights are the frequencies:

This result differs from that obtained on the basis of the simple arithmetic mean. This is explained by the fact that in the calculation based on the distribution series, we do not have initial individual data, but only information about the value of the middle (center) of the interval.

5. Harmonic mean and conditions for its application

The formulas and technique for calculating the harmonic mean are as follows:

simple harmonic mean

weighted harmonic mean

General approach to selection the correctness of the form the middle one is set out in subsection 1.3.3.

In this case, we give an additional condition for applying the weighted harmonic average (since weighted averages are used more often in the practice of calculations).

The weighted harmonic mean is used when the weights are not frequencies. f, and the products of these frequencies by the values ​​of the feature: M = xf .

Example 1.3.9. The following data are available (Table 1.3.6).

Table 1.3.6

The wages of workers in the shops of the enterprise

Calculate the average wage of workers in the enterprise as a whole.

Decision

The average wages of workers by shop can be calculated by dividing the wage bill by the number of workers. This approach should be retained when calculating the overall average, i.e. in the numerator of the fraction, it is necessary to present the total wage fund for all workshops, and in the denominator - the total number of workers. However, the payroll by shop (M) is the product of average wages times the number of workers f. The payroll fund is the only possible co-meter in this case - the weight when calculating the average.

Both of these circumstances determine the use of the harmonic average, and taking into account the fact that wages in individual workshops are received by groups of workers that are not the same in size, the weighted harmonic average should be used. Then

At the same time, 783,000 rubles. - General wage fund for the enterprise, 250 people. - the total number of employees (50 and 200 people - the number for each workshop separately).

If the weights in the calculation of the average for individual units of the population are the same, then the harmonic weighted average turns into the average harmonic simple:

( M is taken out of brackets because it is a common factor). Let's illustrate the calculation on a conditional example.

Example 1.3.10. Unit price BUT, sold in the first outlet, amounted to 20 rubles, in the second - 30 rubles. What is the average selling price of a product if the revenue from sales of the product in retail outlets is the same?

Decision

Since the weights in calculating the average are the proceeds from the sale (turnover), and the proceeds themselves are the product of the price X on the quantity of goods sold /, the calculations were carried out according to the harmonic weighted average, the equality of the weights makes it possible to carry out calculations according to the formula of the average harmonic simple:

6. Structural averages

Along with the calculation of the arithmetic mean and the harmonic mean for the variational distribution series, the distributions are calculated structural averages- mode, median.

Fashion- this is the value of the trait (variant), which is most often found in the study population and has the highest frequency.

median the value of the attribute (variant) is called, which is in the middle of the variation series and divides the series in half.

In the interval variational series, the mode is calculated by the formula

where xMo- the minimum border of the modal interval;

The value of the modal interval;

Modal interval frequency;

The frequency of the interval preceding the modal;

the frequency of the interval following the modal.

The median for the interval distribution series is calculated by the formula

where is the lower limit of the median interval;

The value of the median interval;

The sum of the accumulated frequencies preceding the median;

The frequency of the median interval.

To characterize the structure of the variation series, in addition to the median, quartiles are calculated, which divide the series by the sum of frequencies into four equal parts, quintiles into five equal parts, deciles into ten equal parts and percentiles into one hundred equal parts.

Example 1.3.11. The following data are available (Table 1.3.7).

Table 1.3.7

Monthly wages of workers of a group of small enterprises in one of the regions

Calculate average wage, fashion, and median wages of small business workers.

Decision

According to the condition of the problem, there is an interval series of distribution of workers, therefore, the average wage is calculated according to the formula of the arithmetic weighted average (first, we determine the middle of each interval, i.e.

Consequently, the average monthly wage of workers in small enterprises is 4,775 rubles. Next, calculate the mode and median:

Consequently, half of the workers have an average monthly wage less than 4,667 rubles, and half - more than this amount.

7. Types of variation indicators

Indicators of variation are a numerical measure of the level of fluctuation of a feature. At the same time, according to the size of the variation indicator, a conclusion is made about the typicality, reliability of the average value found for a given population, and about the homogeneity of the population itself.

The most important types of variation indicators:

1) range of variation [ R ]

R = xmax - xmin

2) average linear deviation

3) dispersion [σ2]

4) standard deviation [σ]

5) coefficient of variation [ v ]

The range of variation takes into account only the extreme values ​​of the trait and does not take into account all intermediate ones.

The dispersion has no units.

Equal values ​​of standard deviations calculated for different populations do not allow us to conclude that the degree of variation is the same.

Coefficients of variation allow you to compare the degree of variation of a feature of different populations.

By itself, the coefficient of variation, if its value does not exceed 33-35%, allows us to conclude that the trait is relatively low, about the typicality, reliability of the average value, and about the homogeneity of the population. If it is more than 33-35%, then all the above conclusions should be reversed.

Let us illustrate the calculation of the variation indicators.

Example 1.3.12. There is a distribution series (Table 1.3.8).

Table 1.3.8

Seniority distribution

Define:

1) range of variation;

2) dispersion;

3) standard deviation;

4) coefficient of variation.

Decision

1) The range of variation is the difference between the maximum and minimum values ​​of the attribute: R = 10-1 = 9 years. notice, that R it is better to find from the original ungrouped data, which we have already done when calculating the size of the interval.

Other indicators will require more labor-intensive calculations. Let's determine the indicators of variation in the length of service of workers (Table 1.3.9).

Table 1.3.9

Calculation of indicators of variation in the length of service of workers

=54.5 / 11 = 5.0 years

xf = 54.5 found earlier (see example 1.3.8).

2) The dispersion is:

=50,75 / 11 = 4,6

3) The standard deviation is:

4) The coefficient of variation is:

= (2.1 / 5.0) ´100 = 42.0%.

Analysis of the obtained data suggests that the length of service of the enterprise's employees differs from the average length of service ( = 5.0) by an average of 2.1 years, or by 42.0%. The value of the coefficient of variation exceeds 33%, therefore, the variation in the length of service is large, the found average length of service does not represent the entire set of workers, is not its typical, reliable characteristic, and the set itself has no reason to consider it homogeneous in terms of length of service.