Variation is the difference in the values ​​of a characteristic among different units of a given population in the same period or point in time.

For example, employees of a firm differ in income, time spent on work, height, weight, favorite pastime in free time etc.

Variation occurs as a result of the fact that the individual values ​​of the attribute are formed under the combined influence of various factors (conditions), which are combined in different ways in each separate case. Thus, the value of each option is objective.

The study of variation in statistics has great importance helps to understand the essence of the phenomenon under study. It is especially relevant in the period of the formation of a mixed economy. Measuring variation, finding out its cause, identifying the influence of individual factors gives important information(for example, about the life expectancy of people, incomes and expenses of the population, the financial situation of the enterprise, etc.) for making scientifically based management decisions.

The average value gives a generalizing characteristic of the feature of the studied population, but it does not reveal the structure of the population, which is very essential for its knowledge. The average does not show how the variants of the averaged feature are located near it, whether they are concentrated near the average or deviate significantly from it. The average value of a trait in two sets may be the same, but in one case all individual values ​​differ little from it, and in the other, these differences are large, i.e. in one case, the variation of the attribute is small, and in the other, it is large; this is very important for characterizing the reliability of the average value.

The more the options for individual units of the population differ from each other, the more they differ from their average, and vice versa, the less the options differ from each other, the less they differ from the average, which in this case will more realistically represent the entire population. That is why it is impossible to confine oneself to the calculation of one average in some cases. Other indicators characterizing deviations are also needed. individual values from the overall average.

This can be shown with an example. Suppose that two teams, each consisting of three people, are doing the same work. Let the number of parts, pieces, manufactured per shift by individual workers be:

in the first brigade - 95, 100, 105 (= 100 units);

in the second brigade - 75, 100, 125 (= 100 units).

The average output per worker in both teams is the same and amounts to = 100 units, however, the fluctuation in the output of individual workers in the first team is much less than in the second.

Therefore, it becomes necessary to measure the variation of a trait in populations. For this purpose, a number of generalizing indicators are used in statistics.

• Ш The indicators of variation include: the range of variation, the average linear deviation, variance and the average standard deviation, the coefficient of variation.
• Ш The most elementary indicator of the variation of a trait is the range of variation R, which is the difference between the maximum and minimum values ​​of the trait:

In our example, the range of variation in the shift production of parts is: in the first team - R1 = 10 pcs. (i.e. 105 -- 95); in the second brigade - R2 = 50 pcs. (i.e. 125 - 75), which is 5 times more.

This indicates that, with numerical equality, the average output of the first brigade is more “stable”. The range of variation can serve as a basis for calculating possible reserves for output growth. The second brigade has more such reserves, since if all the workers reach the maximum production of parts for this brigade, it can produce 375 pieces, i.e. (3x125), and in the first - only 315 pieces, i.e. (3 x 105).

However, the range of variation shows only the extreme deviations of the trait and does not reflect the deviations of all variants in the series. When studying variation, one cannot limit oneself only to determining its scope. To analyze variation, an indicator is needed that reflects all the fluctuations of a varying trait and gives a generalized characteristic. The simplest indicator of this type is the average linear deviation

Ш The average linear deviation d is the arithmetic mean of the absolute values ​​of the deviations of individual options from their arithmetic mean (it is always assumed that the average is subtracted from the option: ().

Average linear deviation:

For ungrouped data

where n is the number of members of the series;

For grouped data

where is the sum of the frequencies of the variation series.

In formulas (5.18) and (5.19), the differences in the numerator are taken modulo, (otherwise the numerator will always be zero - algebraic sum deviations of options from their arithmetic mean). Therefore, the average linear deviation as a measure of the variation of a feature is rarely used in statistical practice (only in those cases where the summation of indicators without taking into account signs makes economic sense). With its help, for example, the composition of workers, the rhythm of production, and the turnover of foreign trade are analyzed.

The variance of a feature is middle square deviations of options from their average value, it is calculated by the formulas of simple and weighted variances (depending on the initial data):

§ simple variance for ungrouped data

§ weighted variance for the variation series

Formula (5.21) is applied if the variants have their own weights (or frequencies of the variation series).

The formula for calculating the variance (5.20) can be transformed, taking into account that

those. the variance is equal to the difference between the mean of the squares of the options and the square of their mean.

The technique for calculating the dispersion using formulas (5.20), (5.21) is rather complicated, and when large values options and frequencies can be cumbersome.

The calculation can be simplified using the properties of the variance (proven in mathematical statistics). Here are two of them:

the first - if all the values ​​of the attribute are reduced or increased by the same constant value And, then the variance will not change from this;

the second - if all the values ​​of the attribute are reduced or increased by the same number of times (i times), then the variance will decrease or increase, respectively, by i2 times. Using the second property of the variance, dividing all options by the value of the interval, we obtain the following formula for calculating the variance in variational series with at equal intervals according to the method of moments:

where is the dispersion calculated by the method of moments;

i - the value of the interval;

new (transformed) values ​​of the options (A is a conditional zero, for which it is convenient to use the middle of an interval that has highest frequency);

Moment of the second order;

The square of the moment of the first order.

The calculation of the variance by formula (5.23) is less laborious.

Dispersion is of great importance in economic analysis. In mathematical statistics important role to characterize the quality statistical estimates plays their variance. Below, in particular, we will show the decomposition of the variance into the corresponding elements, which make it possible to estimate the influence various factors, causing the variation of the trait; use of variance to build indicators of the closeness of the correlation when evaluating the results of sample observations.

• Ш The standard deviation is equal to the square root of the dispersion:
  • § for ungrouped data

§ for the variation series

The standard deviation is a generalizing characteristic of the size of the variation of a trait in the aggregate; it shows how much on average specific options deviate from their average value; is an absolute measure fluctuation of the attribute and is expressed in the same units as the variants, therefore it is economically well interpreted.

Let's denote: 1 - the presence of the feature of interest to us; 0 -- its absence; p is the proportion of units with this feature; q - the proportion of units that do not have this feature; p+q=1. Let's calculate the mean value of the alternative feature and its variance. Average value of an alternative feature

variation mean quadratic

since p + q = 1.

Feature variance

Substituting q \u003d 1-p into the dispersion formula, we get

Thus, = pq -- the variance of an alternative attribute is equal to the product of the proportion of units that have the attribute and the proportion of units that do not have this attribute.

For example, if there are 4,500 men and 5,500 women per 10,000 people in a district, then

Variance of the alternative feature = pq = 0.45 * 0.55 = 0.2475.

The limiting value of the variance of the alternative feature is 0.25. It is obtained at p = 0.5.

Standard deviation of the alternative feature

If, for example, 2% of all parts are defective (p = 0.02), then 98% are good (q = 0.98), then the variance of the reject rate

0,02- 0,98 = 0,0196.

The standard deviation of the share of marriage will be:

0.14, i.e. = 14%.

When calculating the mean values ​​and variance for the interval distribution series true values feature are replaced by the central (middle) values ​​of the intervals, which differ from the average arithmetic values included in the interval. This leads to the appearance of a systematic error in the calculation of the variance. WF Sheppard found that the error in the calculation of the variance, caused by the use of grouped data, is 1/12 of the square of the interval (i.e. i2/12) both in the direction of underestimation and in the direction of overestimation of the variance.

Sheppard's correction should be applied if the distribution is close to normal, refers to a trait with a continuous nature of variation, built on a large amount of initial data (n> 500). However, based on the fact that in a number of cases both errors, acting in opposite directions, are neutralized and compensate each other, it is sometimes possible to refuse to introduce corrections.

The smaller the value of the variance and the standard deviation, the more homogeneous (quantitatively) the population and the more typical it will be average value.

In statistical practice, it often becomes necessary to compare variations of various features. For example, great interest presents a comparison of variation in workers' age and skill, length of service, and size wages, cost and profit, length of service and labor productivity, etc. For such comparisons indicators of the absolute variability of characteristics are unsuitable: it is impossible to compare the variability of work experience, expressed in years, with the variation of wages, expressed in rubles.

To carry out such comparisons, as well as comparisons of the fluctuation of the same attribute in several populations with different arithmetic mean, a relative indicator of variation is used - the coefficient of variation.

The coefficient of variation is the ratio of the standard deviation to the arithmetic mean, expressed as a percentage:

The coefficient of variation is used not only for a comparative assessment of the variation of population units, but also as a characteristic of population homogeneity. The set is considered quantitatively homogeneous if the coefficient of variation does not exceed 33%.

Let's show the calculation different ways variation indicators on the example of data on the shift output of the work teams presented interval series distributions (Table 5.7).

Calculate the average shift output, pcs.:

Calculate the output dispersion according to (5.21):

Find the standard deviation, pcs.:

Let's define the coefficient of variation, %:

Thus, this team of workers is quite homogeneous in output, since the variation of the trait is only 8%.

Now let's calculate the dispersion according to the formula (5.22) and according to the method of moments according to the formula (5.23), for the calculation we will use the data in Table. 5.7, columns 8-11.

Calculation of dispersion according to the formula (5.20):

Calculation of dispersion by the method of moments, see formula (5.21):

where A \u003d 50 is the central variant with the highest frequency;

i \u003d 20 - the value of the interval of this series;

Table 5.7

Distribution of workers in shift output of product A and calculated values ​​for calculating variation indicators

Groups of workers in shift production of products, pcs.	Number of workers	Interval x	Estimated values

As you can see, the method of calculating the variance by the method of moments is the least laborious.

Average values ​​are generalizing statistics, which give a summary (final) characteristic of mass social phenomena, since they are built on the basis of a large number individual values variable sign. To clarify the essence of the average value, it is necessary to consider the features of the formation of the values ​​of the signs of those phenomena, according to which the average value is calculated.

It is known that units of each mass phenomenon have numerous features. Whichever of these signs we take, its values ​​for individual units will be different, they change, or, as they say in statistics, vary from one unit to another. So, for example, the salary of an employee is determined by his qualifications, the nature of work, length of service and a number of other factors, and therefore varies over a very wide range. The cumulative influence of all factors determines the amount of earnings of each employee, however, we can talk about the average monthly wages of workers in different sectors of the economy. Here we operate with a typical, characteristic value of a variable attribute, related to a unit of a large population.

The average reflects that general, which is typical for all units of the studied population. At the same time, it balances the influence of all factors acting on the magnitude of the attribute of individual units of the population, as if mutually canceling them. The level (or size) of any social phenomenon is determined by the action of two groups of factors. Some of them are general and main, constantly operating, closely related to the nature of the phenomenon or process being studied, and form that typical for all units of the studied population, which is reflected in the average value. Others are individual, their action is less pronounced and is episodic, random character. They operate in reverse direction, cause differences between the quantitative characteristics of individual units of the population, trying to change the constant value of the studied characteristics. The action of individual signs is extinguished in the average value. In the cumulative influence of typical and individual factors, which is balanced and mutually canceled out in generalizing characteristics, it manifests itself in general view known from mathematical statistics fundamental law big numbers.

In the aggregate, the individual values ​​of the features merge into total weight and sort of dissolve. Hence and average value acts as "impersonal", which can deviate from the individual values ​​of features, not quantitatively coinciding with any of them. The average value reflects the general, characteristic and typical for the entire population due to the mutual cancellation in it of random, atypical differences between the signs of its individual units, since its value is determined, as it were, by the common resultant of all causes.

However, in order for the average value to reflect the most typical value of a feature, it should not be determined for any populations, but only for populations consisting of qualitatively homogeneous units. This requirement is the main condition for the scientifically based application of averages and implies close connection the method of averages and the method of groupings in the analysis of socio-economic phenomena. Therefore, the average value is a generalizing indicator that characterizes typical level variable trait per unit of a homogeneous population in specific conditions of place and time.

Determining, thus, the essence of average values, it must be emphasized that the correct calculation of any average value implies the fulfillment of the following requirements:

• qualitative homogeneity of the population on which the average value is calculated. This means that the calculation of average values ​​should be based on the grouping method, which ensures the selection of homogeneous, same-type phenomena;
• exclusion of the influence on the calculation of the average value of random, purely individual causes and factors. This is achieved when the calculation of the average is based on sufficiently massive material in which the operation of the law of large numbers is manifested, and all accidents cancel each other out;
• when calculating the average value, it is important to establish the purpose of its calculation and the so-called defining indicator-tel(property) to which it should be oriented.

The determining indicator can act as the sum of the values ​​of the averaged feature, the sum of its reciprocal values, the product of its values, etc. The relationship between the defining indicator and the average value is expressed as follows: if all values ​​of the averaged attribute are replaced by the average value, then their sum or product in this case will not change the defining indicator. Based on this connection of the defining indicator with the average value, the initial value is built. ratio for direct calculation of the average value. The ability of averages to preserve the properties of statistical populations is called defining property.

The average value calculated for the population as a whole is called general average; average values ​​calculated for each group - group averages. The overall average reflects common features of the phenomenon under study, the group average characterizes the phenomenon that develops under the specific conditions of the given group.

The calculation methods can be different, therefore, in statistics, several types of average are distinguished, the main of which are the arithmetic average, the harmonic average and the geometric average.

In economic analysis, the use of averages is the main tool for evaluating results. scientific and technological progress, social events, search for reserves of economic development. At the same time, it should be remembered that overindulgence averages can lead to biased conclusions when conducting economic and statistical analysis. This is due to the fact that average values, being generalizing indicators, cancel out and ignore those differences in the quantitative characteristics of individual units of the population that really exist and may be of independent interest.

Types of averages

In statistics, various types of averages are used, which are divided into two large classes:

• power averages (harmonic mean, geometric mean, arithmetic mean, mean square, mean cubic);
• structural averages (mode, median).

To calculate power means all available characteristic values ​​must be used. Fashion and median are determined only by the distribution structure, therefore they are called structural, positional averages. Median and mode are often used as average characteristic in those populations where the calculation of the average power is impossible or impractical.

The most common type of average is the arithmetic average. Under arithmetic mean is understood as the value of the attribute that each unit of the population would have if The overall result of all values ​​of the attribute was distributed evenly among all units of the population. The calculation of this value is reduced to the summation of all values ​​of the variable attribute and dividing the resulting amount by total aggregate units. For example, five workers completed an order for the manufacture of parts, while the first produced 5 parts, the second - 7, the third - 4, the fourth - 10, the fifth - 12. Since the value of each option occurred only once in the initial data, to determine the average output of one worker should apply the simple arithmetic mean formula:

i.e., in our example, the average output of one worker is equal to

Along with the simple arithmetic mean, they study weighted arithmetic mean. For example, let's calculate average age students in a group of 20, whose ages range from 18 to 22, where xi- variants of the averaged feature, fi- frequency, which shows how many times it occurs i-th value in the aggregate (Table 5.1).

Table 5.1

Average age of students

Applying the weighted arithmetic mean formula, we get:

There is a certain rule for choosing a weighted arithmetic average: if there is a series of data on two indicators, for one of which it is necessary to calculate

the average value, and at the same time, the numerical values ​​\u200b\u200bof the denominator of its logical formula are known, and the values ​​\u200b\u200bof the numerator are unknown, but can be found as the product of these indicators, then the average value should be calculated using the arithmetic weighted average formula.

In some cases, the nature of the initial statistical data is such that the calculation of the arithmetic mean loses its meaning and the only generalizing indicator can only be another type of average value - average harmonic. At present, the computational properties of the arithmetic mean have lost their relevance in the calculation of generalizing statistical indicators due to the widespread introduction of electronic computers. big practical value acquired an average harmonic value, which is also simple and weighted. If the numerical values ​​of the numerator of the logical formula are known, and the values ​​of the denominator are unknown, but can be found as a quotient of one indicator by another, then the average value is calculated by the weighted harmonic mean formula.

For example, let it be known that the car traveled the first 210 km at a speed of 70 km/h, and the remaining 150 km at a speed of 75 km/h. It is impossible to determine the average speed of the car throughout the entire journey of 360 km using the arithmetic mean formula. Since the options are speeds on separate sections xj= 70 km/h and X2= 75 km/h, and weights (fi) are considered to be the corresponding segments of the path, then the products of options by weights will have neither physical nor economic sense. AT this case the meaning is acquired by the fractions of dividing the segments of the path by the corresponding speeds (options xi), i.e., the time spent on passing individual sections of the path (fi / xi). If the segments of the path are denoted by fi, then the entire path is expressed as Σfi, and the time spent on the entire path is expressed as Σ fi / xi , Then the average speed can be found as the quotient of the total distance divided by the total time spent:

In our example, we get:

If when using the average harmonic weight of all options (f) are equal, then instead of the weighted one, you can use simple (unweighted) harmonic mean:

where xi - individual options; n- the number of variants of the averaged feature. In the example with speed, a simple harmonic mean could be applied if the segments of the path traveled at different speeds were equal.

Any average value should be calculated so that when it replaces each variant of the averaged feature, the value of some final, generalizing indicator, which is associated with the averaged indicator, does not change. So, when replacing the actual speeds on individual sections of the path with their average value ( average speed) should not change the total distance.

The form (formula) of the average value is determined by the nature (mechanism) of the relationship of this final indicator with the averaged one, therefore the final indicator, the value of which should not change when the options are replaced by their average value, is called defining indicator. To derive the average formula, you need to compose and solve an equation using the relationship of the averaged indicator with the determining one. This equation is constructed by replacing the variants of the averaged feature (indicator) with their average value.

In addition to the arithmetic mean and the harmonic mean, other types (forms) of the mean are also used in statistics. All of them are special cases. degree average. If we calculate all types of power-law averages for the same data, then the values

they will be the same, the rule applies here majorance medium. As the exponent of the mean increases, so does the mean itself. The most commonly used in practical research calculation formulas various kinds power averages are presented in Table. 5.2.

Table 5.2

The geometric mean is applied when available. n growth factors, while the individual values ​​of the trait are, as a rule, relative values ​​of the dynamics, built in the form of chain values, as a ratio to the previous level of each level in the dynamics series. The average characterizes, therefore, average coefficient growth. geometric mean simple calculated by the formula

Formula geometric mean weighted It has next view:

The above formulas are identical, but one is applied at current coefficients or growth rates, and the second - at the absolute values ​​of the levels of the series.

root mean square used when calculating with quantities square functions, is used to measure the degree of fluctuation of the individual values ​​of a trait around the arithmetic mean in the distribution series and is calculated by the formula

Mean square weighted calculated using a different formula:

Average cubic used when calculating with quantities cubic functions and is calculated by the formula

weighted average cubic:

All the above average values ​​can be represented as a general formula:

where is the average value; - individual value; n- the number of units of the studied population; k- exponent, which determines the type of average.

When using the same source data, the more k in general formula power mean, the larger the mean. It follows from this that there is a regular relationship between the values ​​of power means:

The average values ​​described above give a generalized idea of ​​the population under study, and from this point of view, their theoretical, applied, and cognitive significance is indisputable. But it happens that the value of the average does not coincide with any of the real existing options, therefore, in addition to the considered averages, in statistical analysis it is advisable to use the values ​​of specific options that occupy a well-defined position in an ordered (ranked) series of characteristic values. Among these quantities, the most commonly used are structural, or descriptive, average- mode (Mo) and median (Me).

Fashion- the value of the trait that is most often found in this population. With regard to the variational series, the mode is the most frequently occurring value of the ranked series, i.e., the variant with the highest frequency. Fashion can be used to determine the most visited stores, the most common price for any product. It shows the size of the feature characteristic of a significant part of the population, and is determined by the formula

where x0 is the lower limit of the interval; h- interval value; fm- interval frequency; fm_ 1 - frequency of the previous interval; fm+ 1 - frequency of the next interval.

Median the variant located in the center of the ranked row is called. The median divides the series into two equal parts in such a way that on both sides of it there is the same number of population units. At the same time, in one half of the population units, the value of the variable attribute is less than the median, in the other half it is greater than it. The median is used when examining an element whose value is greater than or equal to or simultaneously less than or equal to half of the elements of the distribution series. Median gives general idea about where the values ​​of the feature are concentrated, in other words, where their center is located.

The descriptive nature of the median is manifested in the fact that it characterizes the quantitative boundary of the values ​​of the varying attribute, which are possessed by half of the population units. The problem of finding the median for a discrete variational series is solved simply. If all units of the series are given sequence numbers, then the serial number of the median variant is defined as (n + 1) / 2 with an odd number of terms n. If the number of terms in the series is an even number, then the median will be the average value of two variants having serial numbers n/ 2 and n / 2 + 1.

When determining the median in interval variation series, the interval in which it is located (the median interval) is first determined. This interval is characterized by the fact that its accumulated sum of frequencies is equal to or exceeds half the sum of all frequencies of the series. The calculation of the median of the interval variation series is carried out according to the formula

where X0- the lower limit of the interval; h- interval value; fm- interval frequency; f- the number of members of the series;

∫m-1 - the sum of the accumulated terms of the series preceding this one.

Along with the median for more complete characteristics the structures of the studied population also use other values ​​of options that occupy a quite definite position in the ranked series. These include quartiles and deciles. Quartiles divide the series by the sum of frequencies into 4 equal parts, and deciles - into 10 equal parts. There are three quartiles and nine deciles.

The median and mode, unlike the arithmetic mean, do not cancel individual differences in the values ​​of a variable attribute and therefore are additional and very important characteristics statistical aggregate. In practice, they are often used instead of the average or along with it. It is especially expedient to calculate the median and mode in those cases when the studied population contains a certain number of units with a very large or very small value of the variable attribute. These values ​​of options, which are not very characteristic for the population, while affecting the value of the arithmetic mean, do not affect the values ​​of the median and mode, which makes the latter very valuable indicators for economic and statistical analysis.

Variation indicators

aim statistical study is revealing basic properties and patterns of the studied statistical population. In the process of consolidated data processing statistical observation are building distribution lines. There are two types of distribution series - attributive and variational, depending on whether the attribute taken as the basis of the grouping is qualitative or quantitative.

variational called distribution series built on a quantitative basis. Values quantitative traits for individual units, the aggregates are not constant, they differ more or less from each other. This difference in the value of a trait is called variations. Separate numerical values traits that occur in the studied population are called value options. The presence of variation in individual units of the population is due to the influence a large number factors on the formation of the trait level. The study of the nature and degree of variation of signs in individual units of the population is critical issue any statistical study. Variation indicators are used to describe the measure of trait variability.

Another important task statistical research is to determine the role of individual factors or their groups in the variation of certain signs of the population. To solve this problem in statistics, special methods variation studies based on the use of a scorecard that measures variation. In practice, the researcher is faced with enough large quantity options for the values ​​of the attribute, which does not give an idea of ​​the distribution of units by the value of the attribute in the aggregate. To do this, all variants of the attribute values ​​are arranged in ascending or descending order. This process is called row ranking. The ranked series immediately gives a general idea of ​​the values ​​that the feature takes in the aggregate.

The insufficiency of the average value for an exhaustive characterization of the population makes it necessary to supplement the average values ​​with indicators that make it possible to assess the typicality of these averages by measuring the fluctuation (variation) of the trait under study. The use of these indicators of variation makes it possible to make the statistical analysis more complete and meaningful, and thus to better understand the essence of the studied social phenomena.

by the most simple signs variations are minimum and maximum - is the smallest and highest value trait in the aggregate. The number of repetitions of individual variants of feature values ​​is called repetition rate. Let us denote the frequency of repetition of the feature value fi, the sum of frequencies equal to the volume of the studied population will be:

where k- number of variants of attribute values. It is convenient to replace frequencies with frequencies - w.i. Frequency- relative frequency indicator - can be expressed in fractions of a unit or percentage and allows you to compare variation series with different number observations. Formally we have:

To measure the variation of a trait, various absolute and relative performance. The absolute indicators of variation include the mean linear deviation, the range of variation, variance, standard deviation.

Span variation(R) is the difference between the maximum and minimum values ​​of the trait in the studied population: R= Xmax - Xmin. This indicator gives only the most general idea of ​​the fluctuation of the trait under study, as it shows the difference only between the extreme values ​​of the variants. It is completely unrelated to the frequencies in the variation series, i.e., to the nature of the distribution, and its dependence can give it an unstable, random character only on extreme values sign. The range of variation does not provide any information about the features of the studied populations and does not allow us to assess the degree of typicality of the obtained average values. The scope of this indicator is limited to fairly homogeneous populations, more precisely, it characterizes the variation of a trait, an indicator based on taking into account the variability of all values ​​of the trait.

To characterize the variation of a trait, it is necessary to generalize the deviations of all values ​​from any value typical for the population under study. Such indicators

variations, such as the mean linear deviation, variance and standard deviation, are based on the consideration of deviations of the values ​​of the attribute of individual units of the population from the arithmetic mean.

Average linear deviation is the arithmetic mean of the absolute values ​​of the deviations of individual options from their arithmetic mean:

The absolute value (modulus) of the variant deviation from the arithmetic mean; f- frequency.

The first formula is applied if each of the options occurs in the aggregate only once, and the second - in series with unequal frequencies.

There is another way to average the deviations of options from the arithmetic mean. This method, which is very common in statistics, is reduced to calculating the squared deviations of options from the mean value with their subsequent averaging. In this case, we get a new indicator of variation - the variance.

Dispersion(σ 2) - the average of the squared deviations of the variants of the trait values ​​from their average value:

The second formula is used if the variants have their own weights (or frequencies of the variation series).

In economic and statistical analysis, it is customary to evaluate the variation of an attribute most often using the standard deviation. Standard deviation(σ) is the square root of the variance:

The mean linear and mean square deviations show how much the value of the attribute fluctuates on average for the units of the population under study, and are expressed in the same units as the variants.

In statistical practice, it often becomes necessary to compare the variation of various features. For example, it is of great interest to compare variations in the age of personnel and their qualifications, length of service and wages, etc. For such comparisons, indicators of the absolute variability of signs - the average linear and standard deviation - are not suitable. It is impossible, in fact, to compare the fluctuation of work experience, expressed in years, with the fluctuation of wages, expressed in rubles and kopecks.

When comparing the variability of various traits in the aggregate, it is convenient to use relative indicators of variation. These indicators are calculated as the ratio of absolute indicators to the arithmetic mean (or median). Using as absolute indicator variations, the range of variation, the average linear deviation, the standard deviation, get the relative fluctuation indicators:

The most commonly used indicator of relative volatility, characterizing the homogeneity of the population. The set is considered homogeneous if the coefficient of variation does not exceed 33% for distributions close to normal.

General theory statistics: lecture notes Nina Vladimirovna Konik

LECTURE №5. Mean values ​​and indicators of variation

1. Average values ​​and general principles their calculations

Average values ​​refer to generalizing statistical indicators that give a summary (final) characteristic of mass social phenomena, since they are built on the basis of a large number of individual values ​​of a varying attribute. To clarify the essence of the average value, it is necessary to consider the features of the formation of the values ​​of the signs of those phenomena, according to which the average value is calculated.

It is known that the units of each mass phenomenon have numerous features. Whichever of these signs is taken, its values ​​for individual units will be different, they change, or, as they say in statistics, vary from one unit to another. So, for example, the salary of an employee is determined by his qualifications, the nature of work, length of service and a number of other factors, and therefore varies over a very wide range. The cumulative influence of all factors determines the amount of earnings of each employee. Nevertheless, we can talk about the average monthly wages of workers in different sectors of the economy. Here we operate with a typical, characteristic value of a variable attribute, related to a unit of a large population.

The average value reflects the general that is characteristic of all units of the studied population. At the same time, it balances the influence of all factors acting on the magnitude of the attribute of individual units of the population, as if mutually canceling them. The level (or size) of any social phenomenon is determined by the action of two groups of factors. Some of them are general and main, constantly operating, closely related to the nature of the phenomenon or process being studied, and form that typical for all units of the studied population, which is reflected in the average value. Others are individual, their action is less pronounced and is episodic, random. They act in the opposite direction, cause differences between the quantitative characteristics of individual units of the population, seeking to change the constant value of the characteristics being studied. The action of individual signs is extinguished in the average value. In the combined influence of typical and individual factors, which are balanced and mutually canceled out in generalizing characteristics, the fundamental law of large numbers known from mathematical statistics is manifested in a general form.

In the aggregate, the individual values ​​of the signs merge into a common mass and, as it were, dissolve. Hence, the average acts as an "impersonal" value, which can deviate from the individual values ​​of features, not quantitatively coinciding with any of them. Thus, the average reflects the general, characteristic and typical for the entire population due to the mutual cancellation in it of random, atypical differences between the signs of its individual units, since its value is determined, as it were, by the common resultant of all causes.

However, in order for the average to reflect the most typical value of a feature, it should not be determined for any populations, but only for populations consisting of qualitatively homogeneous units. This requirement is the main condition for the scientifically based application of averages and implies a close relationship between the method of averages and the method of groupings in the analysis of socio-economic phenomena.

Hence, average value- this is a general indicator that characterizes the typical level of a variable trait per unit of a homogeneous population in specific conditions of place and time.

Defining the essence of averages in this way, it must be emphasized that the correct calculation of any average implies the fulfillment of the following requirements:

1) qualitative homogeneity of the population for which the average is calculated. The calculation of the average for phenomena of different quality (various types) contradicts the very essence of the average, since the development of such phenomena is subject to different, rather than general, laws and causes. This means that the calculation of average values ​​should be based on the grouping method, which ensures the selection of homogeneous, same-type phenomena;

2) exclusion of the influence on the calculation of the average value of random, purely individual causes and factors. This is achieved when the calculation of the average is based on sufficiently massive material, in which the operation of the law of large numbers is manifested, and all accidents cancel each other out;

3) when calculating the average value, it is important to establish the purpose of its calculation and the so-called defining indicator (property) on which it should be oriented. The determining indicator can act as the sum of the values ​​of the averaged attribute, the sum of its reciprocal values, the product of its values, etc. The relationship between the defining indicator and the average is expressed as follows: if all values ​​of the averaged attribute are replaced by their average value, then the sum or product case, the defining indicator will not be changed. On the basis of this connection of the determining indicator with the average value, an initial quantitative ratio is built for the direct calculation of the average value. The ability of averages to preserve the properties of statistical populations is called the defining property.

The average calculated for the population as a whole is called the general average, the averages calculated for each group are called group averages. The general average reflects the general features of the phenomenon under study, the group average gives a characteristic of the size of the phenomenon that develops under the specific conditions of this group.

Methods of calculation can be different, and in connection with this, several types of average are distinguished in statistics, the main of which are the arithmetic average, the harmonic average and the geometric average.

In economic analysis, the use of averages is an effective tool for evaluating the results of scientific and technological progress, social measures, and finding hidden and unused reserves for economic development.

At the same time, it should be remembered that excessive focus on averages can lead to biased conclusions when conducting economic and statistical analysis. This is due to the fact that average values, being generalizing indicators, cancel out and ignore those differences in the quantitative characteristics of individual units of the population that really exist and may be of independent interest.

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According to sample survey the grouping of depositors according to the size of the deposit in the Sberbank of the city was made:

Define:

1) range of variation;

2) the average size contribution;

3) average linear deviation;

4) dispersion;

5) standard deviation;

6) coefficient of variation of contributions.

Decision:

This distribution series contains open intervals. In such series, the value of the interval of the first group is conventionally assumed to be equal to the value of the interval of the next, and the value of the interval of the last group is equal to the value of the interval of the previous one.

The interval value of the second group is 200, therefore, the value of the first group is also 200. The interval value of the penultimate group is 200, which means that the last interval will also have a value equal to 200.

1) Define the range of variation as the difference between the largest and the smallest value sign:

The range of variation in the size of the contribution is 1000 rubles.

2) The average size of the contribution is determined by the formula of the arithmetic weighted average.

Let's preliminarily define discrete quantity feature in each interval. To do this, using the simple arithmetic mean formula, we find the midpoints of the intervals.

The average value of the first interval will be equal to:

the second - 500, etc.

Let's put the results of calculations in the table:

Deposit amount, rub.	Number of contributors, f	The middle of the interval, x	xf
200-400	32	300	9600
400-600	56	500	28000
600-800	120	700	84000
800-1000	104	900	93600
1000-1200	88	1100	96800
Total	400	-	312000

The average deposit in the city's Sberbank will be 780 rubles:

3) The average linear deviation is the arithmetic average of the absolute deviations of the individual values ​​of the attribute from the total average:

The procedure for calculating the average linear deviation in the interval distribution series is as follows:

1. The arithmetic weighted average is calculated, as shown in paragraph 2).

2. The absolute deviations of the variant from the mean are determined:

3. The obtained deviations are multiplied by the frequencies:

4. The sum of weighted deviations is found without taking into account the sign:

5. The sum of the weighted deviations is divided by the sum of the frequencies:

It is convenient to use the table of calculated data:

Deposit amount, rub.	Number of contributors, f	The middle of the interval, x
200-400	32	300	-480	480	15360
400-600	56	500	-280	280	15680
600-800	120	700	-80	80	9600
800-1000	104	900	120	120	12480
1000-1200	88	1100	320	320	28160
Total	400	-	-	-	81280

The average linear deviation of the size of the deposit of Sberbank clients is 203.2 rubles.

4) Dispersion is the arithmetic mean of the squared deviations of each feature value from the arithmetic mean.

Calculation of dispersion in interval series distribution is made according to the formula:

The procedure for calculating the variance in this case is as follows:

1. Determine the arithmetic weighted average, as shown in paragraph 2).

2. Find deviations from the mean:

3. Squaring the deviation of each option from the mean:

4. Multiply squared deviations by weights (frequencies):

5. Summarize the received works:

6. The resulting amount is divided by the sum of the weights (frequencies):

Let's put the calculations in a table:

Deposit amount, rub.	Number of contributors, f	The middle of the interval, x
200-400	32	300	-480	230400	7372800
400-600	56	500	-280	78400	4390400
600-800	120	700	-80	6400	768000
800-1000	104	900	120	14400	1497600
1000-1200	88	1100	320	102400	9011200
Total	400	-	-	-	23040000

The concept of average is well known to most people. Usually, the average value is perceived as a reflection of the common characteristic in the values ​​of many units. Such, for example, are the average age of a resident of a country, the average size of a family in a region, and the average profit of an enterprise.

Really, average value - this is a generalized assessment of a feature for a set of objects, which reflects its characteristic value. characteristic value fixes the typical value of a feature, which expresses the uniqueness of a given group of objects and its difference from the values ​​of the feature in other groups.

For example, the average wage of workers in different types activities in 2015 in Russia amounted to, thousand rubles. :

• Agriculture - 19,5;
• mining - 63.7;
• manufacturing industries - 31.8;
• construction - 29.9.

AT different levels payment, i.e. in different average wages of an employee, the features of the organization of labor in different types of activity and, ultimately, the social recognition of a particular work are manifested.

In the above example, averages are given, which are calculated for groups consisting of objects of the same type of activity and which in this sense can be called homogeneous. Similar medium called group. They are interesting in that they are associated with specific objects and the conditions of their existence. When group averages are calculated, then under the same working conditions, for example, there is a mutual cancellation of the influence of random causes on wages. At the same time, when calculating group average the influence of special, specific conditions is intensified, since they act constantly and in one direction. The group average reflects the features of homogeneous objects and cancels randomness. It is precisely for these reasons that group averages find wide practical application.

When it comes to the overall average but a set that includes several homogeneous groups, then when it is calculated, the effect of not only random, but also group features. Thus, in 2015, the total average salary of a person employed in the country's economy amounted to 34 thousand rubles. It does not reflect the features of remuneration in different types of activities, but only shows general level wages employed in ECONOMICS.

Let's compare the average wages of employees of different types of activities in 2010 and 2015. in the Russian economy (Table 6.1).

Table 6.1

Average salary in different types of activity and its changes,

Source: Russia in numbers. 2016. Tab. 7.7.

In the rate of change of averages by type of activity, i.e. in group averages, particular patterns of changes in wages are manifested: in the range from 1.41 to 1.82 times. Comparing the change in the general average, we establish the general pattern of changes in the level of wages in the country's economy: an increase of 1.62 times.

Comprehensive analysis involves the joint use of general and group averages: this allows you to characterize general patterns development and features of their manifestation in specific conditions.

The average is calculated in two steps. At the first stage, it is generalization individual values ​​of the studied features, in a set consisting of P units: (x-). At the second stage, the result distributed between these many P units: (x,) + P - X.

When generalizing the values ​​of the attribute y P objects of the set (x,) there is a mutual cancellation of the influence of random causes and the action of non-random systematic factors is enhanced. When distributing the generalized value of a feature between P units of the set (x; -) P its average typical value x y of one abstract unit is determined. As a result, we have either a group average over a group of homogeneous objects: (x; )-n P\u003d x, or the general average for the entire set under study (x,) -r- P= x.

There are several ways to calculate averages, which differ in the order of generalization and distribution.

Arithmetic mean summarizes individual values x f by summation, and the uniform distribution - by dividing the sum of q, by the number

units involved in the calculation:

The frequent use of the arithmetic mean is due to its special properties, which make its calculation simpler and the result easily verifiable.

The sum of the deviations of the attribute values ​​from the arithmetic mean is zero:

If the characteristic values X, change to the number L, then the arithmetic

the average will change by the same number:

If the characteristic values X, increase in BUT times, then the arithmetic mean will increase by BUT once:

If the characteristic values Xj reduce in BUT times, then the arithmetic mean will also decrease by

Average harmonic used in cases where the calculation is performed on the values ​​of a feature that is associated with the feature being studied inverse relationship, i.e. provided that V determined by the characteristic values

For example, the output per worker:

The indicator of the labor intensity of a unit of production:

The indicators of production and labor intensity are inversely related: . Therefore, when calculating the average output according to the values ​​of labor input, one should use the harmonic average

root mean square is used in cases when, when summarizing the values ​​of the attribute A/, it is necessary to avoid a zero result, since the squares calculate the average: , and from the resulting

average take the square root:

Most often, the quadratic mean is used to calculate variation rates and estimates of differences in set structures.

The geometric mean summarizes the feature values ​​by calculating

their works: , and from the result is extracted

root P th degree:

The most logically justified is the use of the geometric mean when calculating the average growth rate from chain growth rates:

The different order of calculation of averages explains different meanings result. The property of majorance of mean values ​​establishes the dependence of the value of the mean on the exponent of its degree: the higher the exponent of the mean, the greater its value. Each of the averages considered is a kind of power average (Table 6.2).

Table 6.2

Forms of averages

Medium shape	Calculation formula	Average exponent (s)
quadratic
Arithmetic
Geometric
harmonic

As an illustration of the majorance property, let us perform according to the population data federal districts RF calculation of different averages (Table 6.3).

The above example confirms that with an increase in the degree of the average: from the smallest - for harmonic, to the largest - for quadratic, the value of the average increases. The majorance property of means can be represented as inequalities: V

From the property of majorance, the conclusion follows that the choice of the method for calculating the average cannot be arbitrary. It should be based on the semantic content of the source data and on the conditions of application. specific form average.

It is known that the geometric mean is used to generalize the growth rate, and the quadratic mean is used in cases where the sum of the attribute values ​​is zero. Therefore, the most popular practices are the arithmetic and harmonic forms of the averages.

By special rules averages are calculated from the absolute and relative values ​​of the studied characteristics. Consider the features of calculating the averages using the example of data for the federal districts of the Russian Federation for 2014 (Table 6.4).

In table. 6.4 the following signs and their designations are used.

Number of people employed in the economy federal district, million people Р,.

Employed as a percentage of the total population of the federal district, % - С,.

Accounts for the turnover of retail trade per year on average per inhabitant of the federal district, thousand rubles. - T g

Accounts for investments on average per one employed in the economy of the federal district, thousand rubles. - R r

Table 63

Calculation average population of the population of the federal districts of the Russian Federation using various average

Federal	population population
Central
Northwestern
North Caucasian
Volga
Ural

Federal	Population as of 01.01.2016
Siberian
Far Eastern
Crimean
				I 196 529 418.1
Square mean (see formula (6.1))
Arithmetic mean (see formula (6.2))
Geometric mean (see formula (6.3))
Harmonic mean (see formula (6.4))

Source: Russia in numbers. 2016. Tab. 1.3.

The peculiarity of the absolute values ​​of the attribute is that they are directly related to the unit of the population and determine its absolute size. For example, for the federal district as a unit of the set, the absolute values ​​will be the population, the number of employees, the cost of manufactured products, the cost of fixed capital, profit from the sale of products, etc. The given signs relate directly to the federal district, are called primary and by their values ​​it is possible to determine the dimensions of each object under study. When processing the absolute values ​​of these features, the size of each unit is accurately taken into account and therefore there are no restrictions on generalizing their values ​​by direct summation. The average, in the calculation of which the values ​​of a single attribute are processed, is called simple. For example, a simple average is used to calculate the average number of people employed in the economy of one federal district (Table 6.4).

Table 6.4

Calculation of average values economic indicators according to federal

districts of the Russian Federation, 2014

federal district	Number of people employed in the economy, million people	Number of employed % total population	Accounts for retail trade turnover per year on average per inhabitant, thousand rubles.	Accounts for investments on average per one employed in the economy, thousand rubles.
Central
Northwest! i
Se vsro - Ka in kazs k and y
Volga
Ural
Siberian
Far Eastern
Mean

Source: Russia in numbers. 2016. Tab. 1.3.

Note: "x" sign means that this cell is not to be filled.

The calculation is performed according to the following formula:

In the economy of the federal district, on average, 8.5 million people were employed in 2014.

The average of the relative values ​​are determined but in a more complex scheme. The peculiarity of relative values ​​is that they are not directly related to the size of the units being studied, and without this consideration, the calculation of the exact average is usually impossible. In such cases, additional values ​​of characteristics should be included in the calculation, which reflect the absolute dimensions of each of the units under study. In the calculation of the average, in addition to the studied one, an additional characteristic or the weight, so the average is called weighted. When calculating a weighted average, an absolute characteristic or primary attribute always acts as a weight. Weight allows you to take into account the absolute dimensions of each unit and ensures the calculation of the exact value of the average.

In the given example, the characteristics C, G, and are relative, so the direct summation of their values ​​is not allowed. To determine the scheme for calculating their average values, we establish the procedure for calculating their individual values.

The calculation of the percentage of employed people from the total population is carried out according to the following formula: AT calculation formula

the population is unknown by the condition of the problem. For determining

we express its values ​​in terms of the number of employed P, and known values percent of the employed of the total population C,:

or

To determine the population in millions of people, it is necessary to divide the number of people employed in the economy, P, by their share in the total population C,. Therefore, it is necessary to convert the values ​​of C from percent to fractions of a unit:

Let's calculate the unknown value of the population in the additional calculated column (Table 6.5, column 2).

With known values ​​of the number of employed P, and the number of total

of the population, the calculation of the percentage of employees in letter form has the form

Overall average With is calculated according to the same scheme as the individual values ​​of the characteristic C,-. The only difference is that when calculating the overall average With the final values ​​of the compared characteristics are used: the number of employees, million people and the total population, million people That is, the calculation of the overall average With but eight

federal districts is performed according to the formula

Calculation of average values ​​of relative characteristics for the Russian economy in 2014

Table 6.5

federal district	Average annual number of people employed in the economy, million people	population % of the total population	Total population, million people	Accounts for retail trade turnover per year on average per inhabitant, thousand ov6.	Retail trade turnover for the year, billion rubles	Accounts for investment on average per employee, thousand rubles.	Investments in the economy for the year, billion rubles
			R g 100%		r g t g t%
Central
Northwestern
North Caucasian
Volga
Ural
Siberian
Far Eastern
Arithmetic mean
Average harmonic

Compiled and calculated according to: Russia in figures. 2016. Tab. 1.3.

In the Economics of Russia in 2014, the share of the employed population averaged 47.2% of the total population. The calculation was made according to harmonic weighted average, in which the weight was the primary sign P t - the number of people employed in the economy.

Similar reasoning underlies the calculation of the average values ​​of two other relative characteristics: average cost retail trade turnover per inhabitant, T thousand rubles, and the average cost of investments per employee, R thousand roubles.

Individual values ​​of the cost of retail trade turnover per inhabitant, thousand rubles, are calculated as a result of comparing the retail trade turnover for the year, billion rubles, with the total population, million people:

According to the condition of the problem, the value of the retail trade turnover is unknown. Therefore, we express the unknown values ​​of the retail trade turnover through the known values ​​of the total population and the values ​​given in the condition of the problem T g The desired retail trade turnover (commodity turnover) is the product of the total population and the value of commodity turnover per inhabitant:

The value of the retail trade turnover is measured in billion rubles, since when calculating it, the number of inhabitants in million people is multiplied by the turnover per inhabitant in thousand rubles.

Let's determine the unknown values ​​of the retail trade turnover for the year in gr. 5 tab. 6.5.

Calculation of the overall average value of retail trade turnover per inhabitant, thousand rubles, T, we will execute according to the total values ​​of the turnover amount

retail trade, billion rubles, , and the total number of

population, million people, . The calculation formula has the form

In 2014, per inhabitant in Russian Federation accounted for an average of 181.5 thousand rubles. retail turnover. In the calculation, an arithmetic weighted average was used, and the weights are absolute values total strength population:

To calculate the cost of investments per employee, it is necessary to compare the cost of investments, billion rubles, with the number of people employed in the economy, million people:

According to the condition, the cost of investments is unknown, therefore, to calculate its values, it is necessary to express investments through known values ​​of the number of employees pj and through the values ​​of investments per one employed /?, given in the condition of the problem:

Count unknown value total amount we will make investments in gr. 7 tab. 6.5.

The calculated values ​​of the total amount of investments make it possible to determine the individual values ​​of investments per employee according to the formula

For the Russian Federation as a whole, the average value of investments per one employed To we calculate as the ratio of the amount of investment for the year? /? R to the total number of employees

In 2014, investments per employee amounted to an average of 198.8 thousand rubles. In the calculation, the weighted arithmetic mean was used, the weights are the absolute values ​​of the number of employees.

The final step in calculating the averages is to check the correctness of the result. Logical verification is based on the analysis of the scheme for calculating the individual values ​​of the characteristic and on the determination of the meaning of the feature-weight. Counting control establishes whether the average is in the interval from the minimum to maximum value studied trait. If the condition X mjn then the calculation of the average is correct. If this condition is not met, then there are errors in the calculation that need to be identified and corrected.

In our example (see Table 6.5), for all values ​​of the calculated averages, this condition is met:

simple arithmetic R= 8.5, 3.3 P

weighted harmonic With= 47.2, 36.3 C 53.2;

weighted arithmetic T = 181.5, 134.7 T

weighted arithmetic R= 198.8, 142.9R383.3.

This means that no calculation errors were made in determining the averages, and the use of weighted averages to calculate averages from relative values allowed us to take into account the size of the units under study - the federal districts of the Russian Federation.

Summing up, we recall the basic rules for constructing averages.

Based on the absolute values ​​of the attribute, we can calculate a simple average. As a rule, in most cases, the arithmetic average is used. For example, calculation R.

By relative values, the calculation is performed but by a weighted average, in which the weights are the absolute values ​​of the primary feature, related in meaning to the feature under study. For example, calculation S, T and R.

As a weight, the values ​​of the attribute are used, in relation to which the values ​​are calculated. relative values secondary sign. The weight can be displayed quite simply, as, for example, when calculating With and R, where the number of employed is used as the weight R g But it can also have a complex display, as, for example, when calculating Г, for which the weight

was the total population. No matter how it is displayed

sign-weight, it should always be an absolute assessment of the object under study.

The choice of the mean form in most cases is limited to arithmetic or harmonic, since quadratic and geometric are used only in strictly defined cases.

The arithmetic form of the mean is used in cases where the condition of the task in question does not contain the values ​​of a feature that is directly related to the feature under study, i.e. when there is no information about its numerator in the calculation formula of individual values. An example would be the calculation of P, T and R.

If there is no data on the denominator of the ratio in the calculation formula, then the harmonic mean is used. In this case, the feature being studied is related to the unknown feature by an inverse relationship, as, for example, when calculating WITH.

Correctly performed calculations make it possible to obtain accurate average values ​​that reflect the characteristic value of a feature and are of interest in solving analytical and predictive problems.

• See: Russia in numbers. 2016. Tab. 7.7.