Moscow State University of Printing Arts. Series of dynamics, their meaning. Types of dynamics series: instant and interval. Time series of absolute and relative values, mean values The components of the series of dynamics are not
Changes in socio-economic phenomena over time are studied by statistics by constructing and analyzing time series.
Rows of dynamics- these are the values of statistical indicators, which are presented in a certain chronological order.
Each time series contains two components:
1) time period indicators(years, quarters, months, days or dates);
2) indicators characterizing the object under study for time periods or on the corresponding dates, which are called the levels of the series.
By time distinguish moment and interval series of dynamics.
In moment series, the levels express the state of the phenomenon at a critical moment in time- the beginning of the month, quarter, year, etc. For example, population size, number of employees, etc. Such rows, each subsequent level fully or partially contains the value of the previous level, therefore the levels cannot be summed up, so how this leads to re-counting.
In interval - levels reflect the state of the phenomenon for a certain period of time- day, month, year, etc. These are the ranks indicators of production volume, sales volume by months of the year, the number of man-days worked, etc.
By presentation of levels distinguish series of absolute, relative and average values.
Absolute level change - in this case it can be called absolute growth - it is the difference between the compared level and the level of the earlier period, taken as the comparison base. If this base is the immediately previous level, the indicator is called chain, if the base is taken, for example, the initial level, the indicator is called basic. Formulas for absolute level change:
If the absolute change is negative, it should be called an absolute reduction.
Acceleration - it is the difference between the absolute change over a given period and the absolute change over the previous period of the same duration:
The absolute acceleration indicator is used only in the chain version, but not in the basic one. A negative value of the acceleration indicates a slowdown in growth or an acceleration in the decline in the levels of the series.
Growth rate Ki is defined as the ratio of a given level to the previous or baseline level; it shows the relative rate of change in the series. If the growth rate is expressed as a percentage, then it is called the growth rate.
Growth rate
basic -
or growth rate.
The values of the chain growth rates, each calculated to its own base, differ not only in the number of percentages, but also in the magnitude of the absolute change that makes up each percent. Therefore, it is impossible to add or subtract chain growth rates. The absolute value of 1% increase is equal to one hundredth of the previous level, or baseline.
In general, the growth rate of one of the alternative shares depends on the growth rate of the other share and the value of this share as follows:
The absolute change in shares in points depends on the size of the share and the growth rate in this way:
If there are not two, but more groups in the aggregate, the absolute change in each of the shares in points depends on the share of this group in the base period and on the ratio of the growth rate of the absolute value of the volumetric trait of this group with the average growth rate of the volumetric trait in the entire population. The share of the f-th group in the compared (current) period is determined as
Average indicators of dynamics - the average level of the series, average absolute changes and accelerations, average growth rates - characterize the trend.
Average level the interval series of dynamics is defined as a simple arithmetic average of the levels for equal periods of time:
or as a weighted arithmetic average of the levels for unequal intervals of time, the duration of which is the weights.
A special form of the arithmetic mean called chronological average:
If the exact dates of changes in the levels of the moment series are known, then the average level is determined as
where ti- the time during which the level was maintained.
Average absolute gain (absolute change) is defined as a simple arithmetic average of absolute changes over equal periods of time (chain absolute changes) or as a quotient from dividing the basic absolute change by the number of averaged time intervals from the base to the compared period:
Average rate of change is determined most accurately when the time series is analytically aligned exponentially. If oscillation can be neglected, then the average rate is determined as geometric mean of chain growth rates for NS years or from the general (basic) growth rate for NS years:
Average growth rate() is calculated using the geometric mean formula of the indicators of growth rates for individual periods:
where Кр1, Кр2, ..., Кр n-1 - growth rates in comparison with the previous period; n is the number of levels in the row.
The average growth rate can be defined differently.
All processes and phenomena occurring in the social life of a person are the subject of study of statistical science, they are in constant motion and change.
Dynamic series in statistical science is called statistical data characterizing changes in phenomena over time, they are built to identify and study the emerging patterns in the development of phenomena in various spheres (for example, economic, political and cultural) of the life of society.
There are two main elements in the rows of dynamics:
1) time indicator (g);
2) levels of development of the studied phenomenon (y). In the series of dynamics, certain dates of time or separate periods can act as indicators of time.
The levels that form the series of dynamics determine a quantitative assessment of the development in time of the phenomenon or process under study, they can be expressed in relative, absolute or average values. The levels of the series of dynamics, depending on the nature of the phenomenon under study, can refer to specific dates in time or to separate periods.
The time series consists of comparable statistical indicators. For the correct construction of time series, it is necessary that the composition of the studied statistical population belongs to the same territory, to the same range of objects and was calculated using the same methodology.
Time series data should be expressed in the same units of measurement, and the time intervals between series values should be the same as possible.
2. Types of series of dynamics
Series of dynamics are subdivided into moment, interval and series of average values.
Momentary series of dynamics display the state of the investigated processes for certain dates of time.
Interval series of dynamics reflect the results of the development or functioning of the studied processes for certain periods of time.
Calculation of the average dynamic range. To characterize the process for a certain period, the average level is calculated from all members of the time series.
The methods for calculating it depend on the type of time series. For interval series, the average is calculated using the arithmetic mean formula, and for equal intervals, the simple arithmetic average is used, and for unequal intervals, the weighted arithmetic average.
To find the average values of the moment series, use the average chronological:
The chronological average of the moment series is equal to the sum of all levels of the series, divided by the number of members of the series without one, and the first and last members of the series are taken in half size.
If the intervals between the periods are not equal, then the arithmetic weighted average is used, and the time intervals between the dates are taken as weights, to which the paired averages of the adjacent level values refer.
3. The main indicators of the analysis of time series
For the analysis of time series in statistics, such indicators are used as the level of the series, the average level, the absolute increase, the growth rate, the growth rate, the growth rate, the advance rate, the absolute value of one percent of the increase.
The level of the series is the absolute value of each member of the time series. All levels of the series characterize its dynamics. Distinguish between the initial, final and middle levels of the series. The initial level is the value of the first member of the series. The final level is the value of the last member of the series, the average level is the average of all values of the time series.
Absolute gain- this is one of the most important statistical indicators, it characterizes the size of the increase or decrease in the studied phenomenon for a certain period of time is defined as the difference between this level and the previous or initial one. The level that is compared is called the current level, and the level with which the comparison is made is called the baseline, since it is the base for comparison. If each level of the series is compared with the previous one, then chain indicators are obtained, and if all levels of the series are compared with the same initial level, then the obtained indicators are called basic.
For the time series at 0, at 1, at 2, ..., y n-1, y n, consisting of n+ 1 levels, the absolute gain is determined by the formulas:
1) chain: ?I = y i- at i -1 ;
2) basic ? = y i- at 0,
where y i- the current level of the row;
y i at i;
y 0 - the initial level of the row.
Average absolute growth formula:
where ? y- average absolute growth;
y n- the final level of the row;
y 0 - the initial level of the row.
The indicators of the growth rate and the growth rate are calculated. Growth rate is the most common statistical indicator that characterizes the ratio of a given level of a statistical process to the previous or initial level, expressed as a percentage. Growth rates calculated as the ratio of a given level to the previous one are called chain and to the initial one - basic.
Growth rates are calculated using the formulas:
1) chain:
2) basic:
where y i- the current level of the row;
y i-1 - level preceding at i;
at 0 - the initial level of the row.
If the comparison base for the growth rates is taken as 1, then the obtained statistical indicators are called growth factors.
The growth rate is the ratio of the absolute growth to the previous or initial level, expressed as a percentage. The growth rate can be calculated from the growth rate data. To do this, it is necessary to subtract 100 from the growth rate or from the growth coefficient - 1, in the latter case, we obtain the growth coefficient Kpr.
Growth rates are calculated using the following formulas:
1) chain: Tp. = (y - y i -1); y i-1 = Tr.c. - 100 or (Cr.c. - 1) x 100;
2) basic: Тпр. = (y i- at 0); y 0 = Tr.b. - 100 or (Cr.b. - 1) x 100.
To characterize the rates of growth and gain on average for the entire period, the average rate of growth and gain is calculated. The average growth rate (coefficient) is determined by the geometric mean formula, when the average growth rate is calculated according to the absolute data of the first and last members of the time series, the following geometric mean formula is applied:
where at 1 - First level;
y n- final level;
n- the number of members of the series.
If there are chain growth rates, then the average growth rate is determined by the formula:
where TO 1 , TO 2 , K 3 ... K n- growth rates for any period.
Lead coefficient Is the ratio of the basic growth rates of two time series for the same time intervals. Having denoted the advance coefficient K op, the basic growth rates of the first row of dynamics - through K 1, the second - K 11, Then:
TO op = K 1 / K 11.
This coefficient shows how many times the level of one series of dynamics will grow faster in comparison with another. The ratio of the absolute growth to the growth rate is the absolute value of one percent according to the formula:
A% =? (absolute growth) / Тпр.
Interpolation and extrapolation
To solve unknown intermediate values of the dynamic series, the interpolation method is used.
Interpolation- a method for determining unknown intermediate values of the time series.
Interpolation is essentially an approximate reflection of the existing pattern within a certain time interval - in contrast to extrapolation, which requires going beyond this time interval.
Extrapolation- a method for determining quantitative characteristics for populations and phenomena that have not been observed, by extending to them the results obtained from observation of similar populations in the past, in the future, etc.
The average level of a number of dynamics characterizes the typical value of the absolute levels.
The average level y in the interval series of dynamics is calculated by dividing the sum of the levels y; by their number n.
In the momentary series of dynamics with equal time dates, the level will be determined as follows:
In the momentary series of dynamics with unequal dates, the average level is determined by:
The characteristic of generalizing individual absolute increments of a number of dynamics is called the average absolute increment.
Average absolute growth at is defined as follows: the sum of the absolute chain increments (at n) is divided by their number (n):
The average absolute growth can also be determined by the absolute series of dynamics, for this, the difference between the final at NS and basic at 0 levels of the studied period, which is divided into m- 1 sub-periods.
The indicator of the average absolute growth is determined by the formula:
Average growth rate (T R ) - these are the individual growth rates of a number of dynamics, which have a generalizing characteristic, its formula:
The average growth rate, which is determined by the absolute levels of dynamics, is as follows:
Based on the relationship between the basic and chain growth rates, the average growth rate is determined by the formula:
Average growth rate T NS is based on the relationship between growth and growth rates. If there is information about average growth rates T, then the dependence is used to obtain the average growth rate of Тп.
The process of development, the movement of socio-economic phenomena in time in statistics is usually called dynamics. To display the dynamics, the series of dynamics are built, which are the series of the values of the statistical indicator that change over time, arranged in chronological order. Here, the process of economic development is depicted as a set of continuous interruptions, allowing a detailed analysis of the features of development using characteristics that reflect the change in the parameters of the economic system over time.
The constituent elements of a series of dynamics are indicators of the levels of the series and periods of time or points in time.
There are different types of dynamics series. They can be classified according to the following criteria:
1. Depending on the way of expressing the levels, the series of dynamics are subdivided into the series of absolute, relative and average values.
2. Depending on how the levels of a series are expressed, the state of the phenomenon at certain moments of time or its magnitude for certain intervals of time, the moment and interval series of dynamics are distinguished, respectively.
3. Depending on the distance between the levels, the series of dynamics are subdivided into series of dynamics with equally spaced levels and unequal levels in time.
4. Depending on the presence of the main tendency of the process under study, the series of dynamics are subdivided into stationary and non-stationary.
The most important condition for constructing a series of dynamics is the comparability of all levels included in it. This condition is solved either in the process of collecting and processing data, or by recalculating them.
The problem of data comparability is especially acute in time series, because they cover significant periods of time during which changes could occur, leading to incomparability of statistical data.
The comparability of the levels of a number of dynamics is directly influenced by the methodology of accounting or calculation of indicators.
The condition for the comparability of the levels of a number of dynamics is the periodization of the dynamics. In the process of development in time, first of all, quantitative changes in phenomena occur, and then, at certain stages, qualitative leaps occur, leading to a change in the regularity of the phenomenon. Therefore, the national approach to the study of the series of dynamics is to break the series covering large periods of time into those that would combine only periods of the same quality in the development of a population characterized by one pattern of development.
The process of identifying homogeneous stages in the development of the series of dynamics is called the periodization of dynamics.
The need to form series of dynamics for strictly homogeneous periods or stages does not mean denying the possibility of constructing and studying series of dynamics covering long historical periods of time, including various stages of the development of the phenomenon.
It is also important that in the series of dynamics the intervals or moments for which the levels are determined have the same economic meaning.
The condition for the comparability of the levels of the interval series is the presence of equal intervals for which the levels are given.
The levels of a number of dynamics may turn out to be incomparable in the circle of covered objects due to the transition of objects from one subordination to another.
The incomparability of the levels of a series may arise due to changes in the territorial boundaries of regions, districts, etc.
That. Before we analyze the time series, it is necessary, based on the purpose of the study, to make sure that the levels of the series are comparable and, in the absence of the latter, achieve it using additional calculations.
In order to bring the levels of a series of dynamics to a comparable form, sometimes one has to resort to a technique called "closing the series of dynamics." Closing is understood as the unification in one row of two or more rows of dynamics, the levels of which are calculated according to different methodologies or different territorial boundaries. For the implementation of the closure, it is necessary that for one of the periods there are data calculated according to different methodologies.
Another way of closing the series of dynamics is that the levels of the year in which the changes took place, both before the changes and after the changes, are taken as 100%, and the rest are recalculated as a percentage in relation to these levels, respectively.
The same problem of reducing to a comparable form arises in the analysis of the development over time of the economic indicators of individual countries, administrative and territorial regions. This is, firstly, the question of the comparability of prices of the compared countries, and secondly, the comparability of the methodology for calculating the compared indicators. In such cases, the series of dynamics lead to one base, that is, to the same period or point in time, the level of which is taken as the comparison base, and all other levels are expressed as coefficients or as a percentage in relation to it.
The analysis of the rate and intensity of the development of the phenomenon in time is carried out using statistical indicators, which are obtained as a result of comparing the levels with each other. These indicators include: absolute growth, rate of growth and sprouting, the absolute value of one percent of growth. In this case, it is customary to call the compared level the reporting level, and the level with which the comparison is made - the basic one.
The absolute increase characterizes the size of the increase (decrease) in the level of the series for a certain period of time. It is equal to the difference between the two compared levels and expresses the absolute growth rate.
The indicator of the intensity of changes in the level of a series, depending on whether it is expressed as a coefficient or as a percentage, is usually called the growth rate or growth rate. The growth rate and the growth rate are two forms of expressing the intensity of level changes. The growth rate shows how many times the given level of the series is greater than the baseline level or what part of the baseline level is the level of the current period for a certain period of time. As a basic level, depending on the purpose of the study, a certain level constant for all or for each subsequent previous level can be taken. In the first case, they talk about basic growth rates, in the second - about chain growth rates.
Along with the growth rate, it is possible to calculate the growth rate indicator, which characterizes the relative rate of change in the level of the series per unit time. The growth rate shows by what proportion (or percentage) the level of a given period or point in time is more (or less) than the base level.
In statistical practice, instead of calculating and analyzing growth and growth rates, the absolute value of one percent of growth is often considered. It represents one hundredth of the baseline level and at the same time - the ratio of the absolute growth to the corresponding growth rate. The absolute value of one percent of the increase serves as an indirect measure of the base level and, together with the growth rate, allows you to calculate the absolute increase in the level for the period under consideration.
The average level of a number of dynamics is calculated according to the average chronological. The chronological average is called the average calculated from values that change over time. These averages summarize the chroological variation. The chronological average reflects the totality of those conditions in which the studied phenomenon developed in a given period of time.
The methods for calculating the average level of the interval and moment series of dynamics are different. For interval series with equally spaced levels, the average level is found according to the simple arithmetic mean formula, and for unequally spaced levels - according to the weighted arithmetic mean.
A generalizing indicator of the rate of change of a phenomenon in time is the average absolute increase, which makes it possible to establish how much, on average, per unit time the level of the series should increase in order to, starting from the initial level for a given number of periods, reach the final level.
A free generalizing characteristic of the intensity of changes in the levels of a series of dynamics is the average growth rate, which shows how many times the level of a dynamic series has changed on average per unit time. The need to calculate the average growth rate arises due to the fact that the growth rate fluctuates from year to year. In addition, the average growth rate often needs to be determined when level data are available at the beginning of a period and at the end of it, but intermediate data are not available.
A number of dynamics can be influenced by factors of an evolutionary and oscillatory nature, as well as be influenced by factors of different influences.
Evolutionary influences are changes that determine a certain general direction of development, which makes its way through other systematic and random fluctuations. Such changes in the time series are called a development trend, or trend.
Oscillatory influences are cyclical (market) and seasonal fluctuations. Cyclic ones are that the knowledge of the trait under study increases over time, reaches a certain maximum, then decreases, reaches a certain minimum, increases again to the previous value, etc. Seasonal fluctuations are fluctuations that periodically repeat at a certain time every year, day of the month, or hour of the day.
Let us consider irregular fluctuations, which for socio-economic phenomena can be divided into 2 groups: a) sporadically occurring changes caused, for example, by an ecological catastrophe; b) random fluctuations resulting from the action of a large number of relatively weak secondary factors.
That. The initial values of a number of dynamics are subject to a wide variety of influences. Let's distinguish its 4 main components: the main trend (T), cyclical (K), seasonal (S), random fluctuations (E). Depending on their relationship with each other, an additive or multiplicative model of a number of dynamics can be built.
The additive model of a series of dynamics y = T + K + S + E is characterized mainly by the fact that the nature of cyclical and seasonal fluctuations remains constant.
A multiplicative model of a series of dynamics y = T * K * S * E. In this model, the nature of cyclical and seasonal fluctuations remains constant only in relation to the trend.
A trend is a long-term component of a series of dynamics. In the socio-economic series of dynamics, trends of 3 types can be observed: average level, variance, autocorrelation.
The average level trend is analytically expressed using a mathematical function around which the actual levels of the phenomenon under study vary.
The variance trend is the trend in the variance between the empirical levels and the deterministic component of the series.
The tendency of autocorrelation is the tendency to change the relationship between the individual levels of a series of dynamics. This change is not visible graphically.
About a dozen methods are used to check for a trend. Consider 2 of them: a method based on checking the difference between the means of two different parts of the same series and the Foster-Stewart method.
In the first case, the series of dynamics is divided into 2 equal or post-equal parts and the hypothesis of the existence of a difference in means is tested.
The Foster-Stewart method, in addition to determining the presence of a trend of the phenomenon, allows us to detect the trend of dispersion of the levels of a number of dynamics, which is important to know when analyzing and forecasting economic phenomena.
After the presence of a trend in a series of dynamics is established, it is described using smoothing methods. Anti-aliasing methods are divided into 2 main groups:
1.smoothing or mechanical alignment of individual members of a series of dynamics using the actual values of adjacent levels
2. alignment using a curve drawn between specific levels in such a way that it reflects the trend inherent in the series, at the same time relieving it of minor fluctuations.
Let's consider each of them.
Averaging method for the left and right half. A number of dynamics are divided into 2 parts, an average value is found for each of them, and a trend line on the chart is drawn through the obtained points.
Interval coarsening method. If we consider the levels of economic indicators for short periods of time, then due to the influence of various factors acting in different directions, in the series of dynamics, there is a decrease and increase in these levels.
Simple moving average method. Smoothing a number of dynamics using a moving average consists in calculating the average level from a certain number of the first in the order of levels, then - the average level of the same number of levels, starting from the second, then starting from the third, and so on. when calculating the average level, it is as if they "slide" along a series of dynamics from its beginning to the end, each time dropping one level at the beginning and adding one next. Hence the name - moving average.
Each link of the moving average is the average level for the corresponding period, which refers to the middle of the selected period. For each specific series of dynamics, the algorithm for calculating the moving average is as follows:
1. Determine the smoothing interval, that is, the number of levels included in it m (m 2. Calculate the average value of the levels that form the smoothing interval, which is at the same time the smoothing value of the level located in the center of the smoothing interval, provided that m is an odd number. 3. Shift the smoothing interval one point to the right, then calculate the smoothed value for t + 1 term using the formula, perform the shift again, etc. Weighted Moving Average Method. A weighted moving average differs from a simple moving average in that the levels included in the averaging interval are summed up with different weights. The most important way to quantify the general tendency of changes in the levels of the dynamic series is the analytical alignment of the series of dynamics, which allows you to obtain a description of the smooth line of development of the series. In this case, empirical levels are replaced by levels that are calculated on the basis of a certain curve, where the equation is considered as a function of time. The form of the equation depends on the specific nature of the dynamics of development. It can be defined both theoretically and practically. The theoretical analysis is based on the calculated dynamics. Practical analysis - on the study of a line chart. The analysis of the series of dynamics also presupposes the study of seasonal irregularities (seasonal fluctuations), which are understood as stable intra-annual fluctuations, which are caused by numerous factors, including natural and climatic ones. The task of analytical alignment is to determine not only the general trend in the development of the phenomenon, but also some missing values both within the period and beyond. The method of determining unknown values within a time series is called interpolation. These unknown values can be determined: 1) using the half-sum of the levels located next to the interpolated ones; 2) by the average absolute growth; 3) by the growth rate. The way to quantify values outside the range is called extrapolation. Extrapolation is used to predict those factors that not only in the past and present determine the development of a phenomenon, but can also influence its development in the future. You can extrapolate by the arithmetic mean, by the average absolute growth, by the average growth rate. Multivariate statistical analysis is a section of mathematical statistics that develops mathematical methods for identifying the nature and structure of relationships between phenomena characterized by a large number of different properties. Usually, for the analysis, the results of measuring the components of a multidimensional attribute for each object from the studied population are used. Wikimedia Foundation. 2010. Rows of dynamics- These are the series of sequentially arranged in chronological order of statistical indicators that characterize the development of the phenomenon in time. The series of dynamics can consist of absolute, relative and average values. Depending on the… … Official terminology SERIES OF DYNAMICS- in statistics, time series, successive series of values characterizing the change in any phenomenon over time (in dynamics). Baseline R. d. (Usually the result of statistical summaries, for example, the number of livestock at the beginning of the year) ... ... Agricultural encyclopedic dictionary Statistical series characterizing the change (development) of socio-economic phenomena over time. For example, the data on the production of electricity in the USSR for the period 1928 73 represent the production of electricity in the USSR, ... ... Great Soviet Encyclopedia Probabilistic models of economic dynamics- (probabilistic models of economic dynamics) - models of economic development, proceeding from its understanding as a probabilistic system. The possibility of carrying out the corresponding calculations is based on the properties of the hierarchical structure of the construction ... ... Economics and Mathematics Dictionary probabilistic models of economic dynamics- Models of economic development, proceeding from its understanding as a probabilistic system. The possibility of making appropriate calculations is based on the properties of the hierarchical structure of building the economy (more and more, it becomes ... ... Technical translator's guide The process of development, the movement of socio-economic phenomena in time in statistics is usually called dynamics. To display the dynamics build rows of dynamics
(chronological, temporal), which are series of time-varying values of a statistical indicator, arranged in chronological order. The constituent elements of a series of dynamics are indicators of the levels of the series and indicators of time (years, quarters, months, days) or moments (dates) in time. The levels of the series are usually denoted by "y", moments or periods of time to which they refer - by "t". There are various types of series of dynamics, which are classified according to the following criteria
: The analysis of the rate and intensity of the development of the phenomenon in time is carried out using statistical indicators arising from the comparison of levels among themselves. These indicators include: absolute growth, rate of growth and growth, the absolute value of one percent of growth. In this case, it is customary to call the compared level reporting
, and the level with which the comparison is made is basic
. Absolute gain (Δy)
characterizes the size of the increase (or decrease) in the level of the series for a certain period of time. It is equal to the difference between the two compared levels and expresses the absolute growth rate: Δy = y i -y i-k (i = 1,2,3, ..., n). If k = 1, then the level y i-1 is the previous one for this level, and the absolute increments of the level change will be chain. If k are constant for a given series, then the absolute increments will be basic. The indicator of the intensity of the change in the level of the series - depending on whether it is expressed as a coefficient or as a percentage, it is customary to call the growth coefficient (growth rate). Growth rate (t)
shows how many times the given level of the series is greater than the base level (if this coefficient is greater than one) or what part of the base level is the level of the current period for a certain period of time (if it is less than one): t = yi / y i-1 or t = yi / y 1 Growth rate (Δt)
, characterizes the relative rate of change in the level of the series per unit time. The growth rate shows by what proportion (or percentage) the level of a given period or point in time is more (or less) than the base level. Find the growth rate as the ratio of the absolute growth to the level of the series taken as the base: Δt = Δy / y i-1 or Δt = Δy / y 1 or Δt = t-1 (Δt = t-100%). If the growth rate is always a positive number, then the growth rate can be positive, negative and zero. In statistical practice, instead of calculating and analyzing growth and growth rates, they often consider absolute value of one percent increase (A)
... It represents one hundredth of the basic level and at the same time - the ratio of the absolute growth to the corresponding growth rate: A = Δy / (Δt * 100) = y i-1/100 Average level of a number of dynamics
calculated by chronological average. Middle chronological
is called the average calculated from the values that change over time. These averages summarize the chronological variation. The chronological average reflects the totality of those conditions in which the studied phenomenon developed in a given period of time. Formulas for calculating the average indicators of a number of dynamics are presented in the table. Problem 1
... Data on the areas under potatoes before and after changing the boundaries of the district, thousand hectares: Close the row by expressing the area under the potatoes under the conditions of changing the borders of the region. Solution
Let's take the third period as a comparison base - the period for which there are data both in the former and in the old boundaries of the district. Then we merge these two rows with the same base into one. Task 2
... There is information on the export of products from the region for a number of years: Determine: 1) chain and basic: a) absolute increments; b) growth rates; c) growth rates; 2) the absolute content of one percent of the increase; 3) average indicators: a) average level of the series; b) average annual absolute growth; c) average annual growth rate; d) average annual growth rate. Solution
Let's remind that: To solve this, we will expand the proposed table. The average level of the series is determined by the arithmetic average simple: Usr = 202467: 4 = 50616.75 thousand US dollars. The average annual absolute growth is determined by the formula: = (64344-42376) / (4-1) = 7322.67 thousand US dollars. The average annual growth rate is determined by the formula: 3 √(64344:42376) = 1,15=115% The average annual growth rate is determined by the formula: 1,15-1=0,15=15%. Problem 3
... Based on the following information, determine the average size of the company's property for the quarter: Solution
The average size of the property of the enterprise for the quarter is determined by the formula: = (30/2 +40 +50 +30/2) / (4-1) = 40 million rubles. Rows of dynamics- a series of statistical indicators characterizing the development of natural and social phenomena in time. Statistical compilations published by the Goskomstat of Russia contain a large number of series of dynamics in tabular form. The series of dynamics make it possible to reveal the patterns of development of the phenomena under study. The series of dynamics contain two types of indicators. Time indicators(years, quarters, months, etc.) or points in time (at the beginning of the year, at the beginning of each month, etc.). Row level indicators... Indicators of the levels of the series of dynamics can be expressed in absolute values (production of a product in tons or rubles), relative values (share of the urban population in%) and average values (average wages of workers in the industry by years, etc.). A dynamic row contains two columns or two rows. Statistical indicators can characterize either the results of the studied process over a period of time, or the state of the studied phenomenon at a certain point in time, i.e. indicators can be interval (periodic) and momentary. Accordingly, the initial series of dynamics can be either interval or momentary. The momentary series of dynamics, in turn, can be with equal and unequal time intervals. The original series of dynamics can be transformed into a series of average values and a series of relative values (chain and basic). Such series of dynamics are called derived series of dynamics. The methodology for calculating the average level in the series of dynamics is different, due to the type of series of dynamics. Using examples, we will consider the types of series of dynamics and formulas for calculating the average level. The levels of the interval series characterize the result of the process under study for a period of time: production or sales of products (for a year, quarter, month, etc. periods), the number of people employed, the number of births, etc. The levels of an interval series can be summed up. In this case, we get the same indicator for longer intervals of time. Average level in interval series of dynamics() is calculated by the simple formula: Let us consider the methodology for calculating the average level of the interval series of dynamics using the example of data on sugar sales in Russia. Sugar sold, thousand tons This is the average annual volume of sugar sales to the population of Russia for 1994-1996. In just three years, 8137 thousand tons of sugar were sold. The levels of moment series of dynamics characterize the state of the studied phenomenon at certain points in time. Each subsequent level includes in whole or in part the previous indicator. For example, the number of employees as of April 1, 1999, in whole or in part, includes the number of employees as of March 1. If we add up these indicators, we get a repeated count of those workers who worked during the entire month. The resulting amount has no economic content, it is a calculated indicator. In momentary series of dynamics with equal time intervals, the average level of the series calculated by the formula: Let us consider the methodology for such a calculation based on the following data on the payroll number of employees of the enterprise for the 1st quarter. It is necessary to calculate the average level of a series of dynamics, in this example - enterprises: The calculation was carried out according to the average chronological formula. The average payroll number of employees of the enterprise for the 1st quarter was 155 people. In the denominator - 3 months in the quarter, and in the numerator (465) - this is a calculated number, it has no economic content. In the overwhelming majority of economic calculations, months, regardless of the number of calendar days, are considered equal. In moment series of dynamics with unequal time intervals, the average level of the series is calculated according to the formula of the arithmetic weighted average. The weights of the average are taken as the duration of time (t- days, months). Let's perform the calculation using this formula. The listed number of employees of the enterprise in October is as follows: as of October 1 - 200 people, October 7 hired 15 people, October 12 dismissed 1 person, October 21 hired 10 people and until the end of the month there were no employees hired or fired. This information can be presented as follows: When determining the average level of the series, it is necessary to take into account the duration of the periods between the dates, that is, apply: In this formula, the numerator () has economic content. In this example, the numerator (6,665 person-days) is the plant's employees for October. The denominator (31 days) is the calendar number of days in a month. In cases where we have a momentary series of dynamics with unequal time intervals, and the specific dates of the change in the indicator are unknown to the researcher, then first it is necessary to calculate the average value () for each time interval using the formula of the arithmetic simple average, and then calculate the average level for the entire series of dynamics, after weighing the calculated average values by the duration of the corresponding time interval. The formulas look like this: The series of dynamics considered above consist of absolute indicators obtained as a result of statistical observations. The originally constructed series of dynamics of absolute indicators can be transformed into series of derivatives: series of mean values and series of relative values. The series of relative values can be chain (in% to the previous period) and basic (in% to the initial period taken as the comparison base - 100%). The calculation of the average level in the derived series of dynamics is performed using other formulas. First, we transform the above momentary series of dynamics with equal time intervals into a series of average values. To do this, we calculate the average payroll number of employees of the enterprise for each month, as the average of the indicators at the beginning and end of the month (): for January (150 + 145): 2 = 147.5; for February (145 + 162): 2 = 153.5; for March (162 + 166): 2 = 164. Let's represent it in tabular form. Average level in derived series average values are calculated by the formula: Note that the average payroll number of employees of the enterprise for the 1st quarter, calculated according to the chronological average formula based on the data on the 1st day of each month and according to the arithmetic average - according to the derived series - are equal to each other, i.e. 155 people. Comparison of the calculations makes it possible to understand why in the chronological average formula the initial and final levels of the series are taken at half size, and all intermediate levels are taken in full size. Series of averages derived from moment or interval series of dynamics should not be confused with series of dynamics in which the levels are expressed by the average. For example, the average wheat yield by years, average wages, etc. In economic practice, series are very widely used. Almost any initial series of dynamics can be converted into a series of relative values. In essence, transformation means replacing the absolute indicators of a number with the relative values of the dynamics. The average level of the series in the relative series of dynamics is called the average annual growth rate. The methods for its calculation and analysis are discussed below. For a reasonable assessment of the development of phenomena in time, it is necessary to calculate the analytical indicators: absolute growth, growth rate, growth rate, growth rate, absolute value of one percent of growth. The table provides a numerical example, and below are the calculation formulas and economic interpretation of the indicators. Produced, Absolute Growth rates The pace Growth rate,% Value of 1% at-growth, thousand tons baseline baseline baseline baseline Absolute gains (Δy) show how many units the next level of the series has changed in comparison with the previous one (column 3. - absolute chain increments) or in comparison with the initial level (column 4. - basic absolute increments). Calculation formulas can be written as follows: With a decrease in the absolute values of the series, there will be, respectively, "decrease", "decrease". The indices of absolute growth indicate that, for example, in 1998 the production of product "A" increased by 4 thousand tons as compared to 1997, and by 34 thousand tons as compared to 1994; for the rest of the years see table. 5 gr. 3 and 4. Growth rate shows how many times the level of the series has changed in comparison with the previous one (column 5 - chain growth or decline coefficients) or compared to the initial level (column 6 - basic growth or decline coefficients). Calculation formulas can be written as follows: Rates of growth show how many percent is the next level of the series in comparison with the previous one (column 7 - chain growth rates) or in comparison with the initial level (column 8 - basic growth rates). Calculation formulas can be written as follows: So, for example, in 1997 the volume of production of product "A" in comparison with 1996 amounted to 105.5% ( Growth rate show how many percent the level of the reporting period has increased in comparison with the previous one (column 9 - chain growth rates) or in comparison with the initial level (column 10 - basic growth rates). Calculation formulas can be written as follows: T pr = T p - 100% or T pr = absolute increase / level of the previous period * 100% So, for example, in 1996, compared to 1995, product "A" was produced by 3.8% (103.8% - 100%) or (8: 210) x100%, and compared to 1994 - by 9% (109% - 100%). If the absolute levels in a row decrease, then the rate will be less than 100% and, accordingly, there will be a rate of decline (growth rate with a minus sign). Absolute value of 1% gain(column 11) shows how many units must be produced in a given period in order for the level of the previous period to increase by 1%. In our example, in 1995 it was necessary to produce 2.0 thousand tons, and in 1998 - 2.3 thousand tons, i.e. much bigger. There are two ways to determine the magnitude of the absolute value of a 1% increase: Absolute value of 1% gain = In dynamics, especially over a long period, a joint analysis of the growth rates with the content of each percentage of increase or decrease is important. Note that the considered method of analyzing the series of dynamics is applicable both for the series of dynamics, the levels of which are expressed in absolute values (t, thousand rubles, the number of employees, etc.), and for the series of dynamics, the levels of which are expressed by relative indicators (% of scrap ,% ash content of coal, etc.) or average values (average yield in centners / ha, average wages, etc.). Along with the analytical indicators considered, calculated for each year in comparison with the previous or initial level, when analyzing the series of dynamics, it is necessary to calculate the average analytical indicators for the period: the average level of the series, the average annual absolute increase (decrease) and the average annual growth rate and growth rate. Methods for calculating the average level of a series of dynamics were discussed above. In the interval series of dynamics we are considering, the average level of the series is calculated by the simple formula: Average annual production of a product for 1994-1998 amounted to 218.4 thousand tons. The average annual absolute growth is also calculated using the simple arithmetic mean formula: Annual absolute increments varied over the years from 4 to 12 thousand tons (see column 3), and the average annual increase in production for the period 1995 - 1998. amounted to 8.5 thousand tons. Methods for calculating the average growth rate and average growth rate require more detailed consideration. Let us consider them using the example of the annual indicators of the series level shown in the table. First of all, we note that the growth rates shown in the table (columns 7 and 8) are series of the dynamics of relative values - derivatives of the interval series of dynamics (column 2). Annual growth rates (column 7) vary from year to year (105%; 103.8%; 105.5%; 101.7%). How to calculate the average from the annual growth rate? This value is called the average annual growth rate. The average annual growth rate is calculated in the following sequence: Average annual growth rate (determined by subtracting 100% from the growth rate. The average annual growth (decline) rate according to the geometric mean formulas can be calculated in two ways: 1) on the basis of absolute indicators of a number of dynamics according to the formula: 2) based on the annual growth rates according to the formula The results of the calculation by the formulas are equal, since in both formulas the exponent is the number of years in the period during which the change occurred. And the radical expression is the growth rate of the indicator for the entire period of time (see Table 5, column 6, for the line for 1998). The average annual growth rate is The average annual growth rate is determined by subtracting 100% from the average annual growth rate. In our example, the average annual growth rate is Consequently, for the period 1995 - 1998. the volume of production of product "A" on average for the year increased by 4.0%. Annual growth rates ranged from 1.7% in 1998 to 5.5% in 1997 (for each year, see the growth rates in Table 5, column 9). The average annual growth rate (growth) allows one to compare the dynamics of the development of interrelated phenomena over a long period of time (for example, the average annual growth rate of the number of employees by industry, the volume of production, etc.), to compare the dynamics of a phenomenon in different countries, to study the dynamics of a certain or phenomena according to the periods of the country's historical development. The study of seasonal fluctuations is carried out in order to identify regularly recurring differences in the level of the series of dynamics depending on the season. For example, the sale of sugar to the population in the summer period significantly increases due to the preservation of fruits and berries. Labor requirements for agricultural production differ depending on the season. The task of statistics is to measure seasonal differences in the level of indicators, and in order for the revealed seasonal differences to be regular (and not random), it is necessary to build an analysis on a database for several years, at least for at least three years. Table 6 shows the initial data and methodology for the analysis of seasonal fluctuations by the method of simple arithmetic mean. The average value for each month is calculated using the simple arithmetic mean formula. For example, for January 2202 = (2106 +2252 +2249): 3. Seasonality index(Table 5, column 7.) is calculated by dividing the average values for each month by the total average monthly value, taken as 100%. The monthly average for the entire period can be calculated by dividing the total fuel consumption for three years by 36 months (1,188,082 tons: 36 = 3280 tons) or by dividing by 12 the sum of the monthly average, i.e. total total for gr. 6 (2022 + 2157 + 2464 etc. + 2870): 12. Fuel consumption, tons Amount for 3 years, t (2 + 3 + 4) Average monthly over 3 years, t Seasonality index, September Rice. 1. Seasonal fluctuations in fuel consumption in agricultural enterprises over 3 years. For clarity, based on the seasonality indices, a seasonal wave graph is constructed (Fig. 1). Months are placed on the abscissa, and the seasonality indices in percent are placed on the ordinate (Table 6, column 7). The total monthly average for all years is at the level of 100%, and the average monthly seasonality indices are plotted in the form of dots on the chart field in accordance with the accepted scale along the ordinate. The points are connected with each other by a smooth broken line. In the given example, the annual volumes of fuel consumption differ slightly. If, in the series of dynamics, along with seasonal fluctuations, there is a pronounced upward (downward) tendency, i.e. levels in each subsequent year systematically significantly increase (decrease) in comparison with the levels of the previous year, then more reliable data on the size of seasonality will be obtained as follows: The transition for each year from absolute monthly values of indicators to seasonality indices makes it possible to eliminate the upward (downward) trend in the series of dynamics and more accurately measure seasonal fluctuations. In market conditions, when concluding contracts for the supply of various products (raw materials, materials, electricity, goods), it is necessary to have information about the seasonal needs for means of production, about the demand of the population for certain types of goods. The results of the study of seasonal fluctuations are important for the effective management of economic processes. In economic practice, it is often necessary to compare several series of dynamics with each other (for example, indicators of the dynamics of electricity production, grain production, sales of cars, etc.). To do this, it is necessary to transform the absolute indicators of the compared series of dynamics into derived series of relative basic values, taking the indicators of any one year as a unit or as 100%. Such a transformation of several series of dynamics is called bringing them to the same base. Theoretically, the absolute level of any year can be taken as the base of comparison, but in economic research, for the base of comparison, it is necessary to choose a period that has a certain economic or historical significance in the development of phenomena. At present, it is advisable to take, for example, the 1990 level as a comparison base. To study the patterns (tendencies) of the development of the phenomenon under study, data are needed over a long period of time. The development trend of a specific phenomenon is determined by the main factor. But along with the effect of the main factor in the economy, the development of the phenomenon is directly or indirectly influenced by many other factors, random, one-time or periodically recurring (years favorable for agriculture, dry years, etc.). Almost all series of dynamics of economic indicators on the chart have the shape of a curve, a broken line with ups and downs. In many cases, it is difficult to determine even the general trend of development based on the actual data of a series of dynamics and on the schedule. But statistics should not only determine the general trend of the development of the phenomenon (increase or decrease), but also provide quantitative (digital) characteristics of development. Table 7 (column 2) shows actual data on grain production in Russia for 1981-1992. (in all categories of farms, in weight after revision) and calculations for leveling this row by three methods. Taking into account that the number of dynamics is small, the intervals were taken for three years and for each interval the averages were calculated. The average annual volume of grain production for three-year periods is calculated using the arithmetic average simple formula and referred to the average year of the corresponding period. So, for example, for the first three years (1981 - 1983) the average was recorded against 1982: (73.8 + 98.0 + 104.3): 3 = 92.0 (million tons). Over the next three-year period (1984 - 1986), the average (85.1 +98.6 + 107.5): 3 = 97.1 million tons was recorded against 1985. For other periods, the results of the calculation in gr. 3. Given in gr. 3 indicators of the average annual grain production in Russia indicate a natural increase in grain production in Russia for the period 1981 - 1992. Moving average method(see columns 4 and 5) is also based on the calculation of averages over aggregated periods of time. The goal is the same - to abstract from the influence of random factors, to mutually extinguish their influence in certain years. But the calculation method is different. In the given example, five-bar (for five-year periods) moving averages are calculated and referred to the middle year in the corresponding five-year period. So, for the first five years (1981-1985), according to the arithmetic average simple formula, the average annual grain production was calculated and recorded in table. 7 versus 1983 (73.8+ 98.0+ 104.3+ 85.1+ 98.6): 5 = 92.0 million tons; for the second five-year period (1982 - 1986) the result was recorded against 1984 (98.0 + 104.3 +85.1 + 98.6 + 107.5): 5 = 493.5: 5 = 98.7 million tons For subsequent five-year periods, the calculation is made in a similar way by excluding the initial year and adding the year following the five-year period and dividing the amount received by five. With this method, the ends of the row are left empty. How long should the time periods be? Three, five, ten years? The question is decided by the researcher. In principle, the longer the period, the more smoothing occurs. But one must take into account the length of the dynamics series; do not forget that the moving average method leaves the cut ends of the aligned series; take into account the stages of development, for example, in our country for many years, socio-economic development was planned and, accordingly, analyzed according to five-year plans. Produced, million tons Average for Rolling sum over 5 years, mln.t Estimated indicators Analytical alignment method(gr. 6 - 9) is based on calculating the values of the aligned series according to the corresponding mathematical formulas. Table 7 shows the calculations using the straight line equation: To determine the parameters, you need to solve the system of equations: The necessary values for solving the system of equations are calculated and given in the table (see gr. 6 - 8), we substitute them into the equation: As a result of calculations, we get: α = 87.96; b = 1.555. Substitute the value of the parameters and get the equation of the straight line: For each year, we substitute the t value and obtain the levels of the aligned series (see column 9): Rice. 2. Grain production in Russia for 1981-1982. In the leveled row, there is a uniform increase in the levels of the row on average per year by 1.555 million tons (value of parameter "b"). The method is based on abstracting the influence of all other factors, except for the main one. Phenomena can develop in dynamics evenly (increase or decrease). In these cases, the straight line equation is most often appropriate. If the development is uneven, for example, first a very slow growth, and from a certain moment a sharp increase, or, conversely, first a sharp decline and then a slowdown in the rate of decline, then the alignment must be performed according to other formulas (the equation of a parabola, hyperbola, etc.). If necessary, it is necessary to refer to textbooks on statistics or special monographs, where the issues of choosing a formula for an adequate reflection of the actually existing trend of the studied series of dynamics are described in more detail. For clarity, the indicators of the levels of the actual series of dynamics and aligned series will be plotted on the graph (Fig. 2). The actual data is a broken black line indicating ups and downs in grain production. The rest of the lines on the chart show that the use of the moving average method (line with cut ends) allows you to substantially align the levels of the time series and, accordingly, make the broken curve line smoother and smoother on the chart. However, aligned lines are still curved lines. Constructed on the basis of the theoretical values of the series obtained by mathematical formulas, the line strictly corresponds to a straight line. Each of the three considered methods has its own merits, but in most cases the analytical alignment method is preferable. However, its application is associated with large computational work: solving a system of equations; verification of the validity of the selected function (form of communication); calculation of the levels of the aligned row; building a schedule. For the successful implementation of such work, it is advisable to use a computer and appropriate programs.See what "Rows of dynamics" are in other dictionaries:
Indicators of changes in the levels of a number of dynamics
Table - Formulas for calculating the average indicators of a series of dynamics
Index Designation and formula
The average level of the interval series of dynamics
The average level of the moment series of dynamics
Average absolute growth for the entire period
Average growth rate
Average growth rate
Examples of solving problems on the topic "Series of dynamics in statistics"
- if each current level is compared with the previous one, then we will get chain indicators;
- if each current level is compared with the initial one, then we get the basic indicators.Time series of dynamics
Momentary series of dynamics
A range of average values
Relative series
Time series analysis
thousand tons
gains,
growth,%
3
4
5
6
7
8
9
10
11
Average annual growth rate and average annual growth rate
Seasonal Analysis
Bringing rows of dynamics to the same base
Time series alignment methods
The method of enlarging the time intervals (column 3).
Moving average method
3 years,
million tons
Analytical alignment method
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