A generalizing characteristic of the dynamics of the phenomenon under study is determined using the following average indicators: average row level, average growth theme, average growth rate.

The average level of the series characterizes the generalized value of the absolute levels of the series.

For interval time series, the average level is determined:

a) at equal intervals according to the simple arithmetic mean formula (7.18):

where y 1 …y n - absolute levels of the series;

n - number of levels.

For example, the average level for the interval dynamics series given in paragraph 7.1 is 935 million rubles.

b) for unequal intervals using the weighted arithmetic mean formula (7.19):

where t is the duration of time intervals between the levels of the series.

The average level of moment series of dynamics is determined by:

a) for a series with equally spaced dates using the average chronological simple formula (7.20):

Example, the average level for the moment series of dynamics given in paragraph 7.1 is 195 people.

b) for a series with unequally spaced dates using the average chronological weighted formula (7.21):

The average absolute increase is calculated in two ways:

a) chain (based on chain absolute increases) (7.22):

where m is the number of absolute increments (m = n - 1, n is the number of members of the series);

b) basic (based on the total basic absolute increase) (7.23):

For our moment series of dynamics, the average absolute increase, calculated by the chain method, is 2 people:

Calculation using the basic method gives the same result. In this way, the average increase in headcount per quarter is 2 people.

Average growth rate for series with equal intervals, or with equally spaced dates, calculated:

a) in a chain way (according to the geometric mean formula) (7.24):

where m is the number of growth coefficients (m = n - 1);

b) in the basic way (7.25):

Average growth rate for series with equal intervals and equally spaced dates, is calculated using formula (7.26):

The average growth coefficient for the series under consideration is, i.e. average growth in numbers for the quarter was 101.03%.

Average growth rates (coefficients) are calculated based on average growth rates or coefficients by subtracting 100% or 1 from the latter (7.27 and 7.28):

The average growth rate for our example is 1.03% (101.03%-100%).

When simultaneously analyzing the dynamics of two phenomena, it is of interest to compare the intensity of their changes over time. Such a comparison is made in the presence of time series of the same content, but relating to different territories or objects, or when comparing series of different contents characterizing the same object. Comparison of the intensity of changes in series levels over time is possible using coefficients advance, representing the ratio of the basic growth rates or increment of two dynamics series for the same periods of time (7.29) and (7.30):

For example, the growth rate of production volumes at the enterprise in the reporting year was 126%, and the growth rate of personnel was 120%. Thus, the growth rate of production volumes in the reporting year outpaced the growth of personnel at the enterprise by 1.05 times (126/120).

The lead coefficient can also be calculated based on a comparison of average growth rates or growth rates:

Methods for analyzing the main trend of a time series

The main tendency of a series of dynamics (or trend) was a stable change in the level of a phenomenon over time, caused by the influence of constantly acting factors and free from random fluctuations.

In cases where the levels of a time series are continuously increasing or continuously decreasing, the main trend of the series is obvious. However, quite often the levels of time series undergo various changes (i.e., they either increase or decrease), and the general trend is unclear. The task of statistics is to identify trends in such series. For this purpose, time series are processed using the methods of interval enlargement, moving average, and analytical alignment.

Enlarging intervals is the simplest method. It is based on increasing the time periods to which the levels of a series of dynamics relate. At the same time, the number of intervals decreases. Let's consider the application of this method using the example of monthly data on the output of an enterprise.

Different directions of changes in the levels of the series for individual months make it difficult to draw conclusions about the main trend in production. However, if the monthly levels are combined into quarterly levels, and then the average monthly output is calculated by quarter, then the trend becomes obvious.

5,23 < 5,57 < 5,87 < 6,03.

Thus, the time series shows an upward trend.

The moving average method is as follows. The average level is determined from a certain volume of an odd number of the first levels of the series, and then from the same number of levels, but starting from the second. Then from the third and so on. Thus, the average slides along the dynamics series, moving one level. Let us consider the note of this method using the example of labor productivity at an enterprise.

Year	Annual output per worker, t	Moving average
three-term	five-membered
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999	15,4 14,0 17,6 15,4 10,9 17,5 15,0 18,5 14,2 14,9	- (15,4 + 14,0 + 17,6) : 3 = 15,7 (14,0 + 17,6 + 15,4) : 3 = 15,4 14,6 14,6 14,5 17,0 15,9 15,9 -	- - 14,7 15,1 15,2 17,1 16,8 17,6 - -

The series, smoothed by five-term averages, already allows us to talk about a tendency towards an increase in labor productivity at the enterprise. The disadvantage of the method is the loss of information associated with shortening the series

The considered methods make it possible to determine the general trend of changes in the levels of a number of dynamics. However, they do not allow us to obtain a generalized statistical trend model. For this purpose they use analytical alignment method rows of dynamics. The main content of the method is that the general development trend is presented as a function of time:

Where is the level of the time series, calculated using the corresponding equation at a point in time t.

The determination of the theoretical levels of a series of dynamics is carried out on the basis of the so-called adequate mathematical model, which best reflects the main trend.

The simplest models for displaying socio-economic processes are the following:

Linear

Indicative

Power

Parabola

The function parameters are usually calculated using the least squares method.

The parameters of the equation that satisfy this condition can be found by solving a system of normal equations. Based on the obtained trend equation, theoretical levels are calculated. Thus, leveling a series of dynamics consists of replacing the actual levels y smoothly changing theoretical levels.

To make the final choice of the type of adequate mathematical function, special criteria of mathematical statistics are used (criterion x 2, Kolmogorova - Smirnova and others).

Methods for studying seasonal variations

When comparing quarterly and monthly data for many socio-economic phenomena, we often find periodic oscillations arising under the influence of changing seasons. They are the result of the influence of natural and climatic conditions, general economic factors, as well as other numerous and varied factors that are often regulated.

In statistics, periodic fluctuations that have a definite and constant period equal to an annual interval are called seasonal fluctuations or seasonal waves, and the dynamic series in this case is called the seasonal dynamics series. Seasonal fluctuations are observed in various sectors of the economy, including in the chemical and forestry complex. In some cases, they can negatively affect the results of production activities. Therefore, the question arises about regulating seasonal changes. This regulation should be based on a study of seasonal fluctuations.

In statistics, there are a number of methods for studying and measuring seasonal fluctuations. The simplest of them is to calculate special indicators called seasonality indices I s . The combination of these indicators reflects the seasonal wave.

In order to identify a stable seasonal wave, which would not be affected by the random conditions of one year, seasonal fluctuation indices are calculated using data for several lats (at least three).

If the dynamics series does not contain a pronounced trend in development, then seasonality indices are calculated directly from empirical data without their preliminary alignment.

For each month, the average level is calculated, for example, for three years (), then the average monthly level is calculated for the entire series (). After this, seasonality indices are determined, which are percentages of the averages for each month to the overall average monthly level of the series (7.35):

Example.There are monthly data on the company's sales volume of wall materials, million pieces. conditional brick. It is required to calculate seasonality indices.

Month	Sales volume, million units	Is, %
2000	2001	2002	Average monthly level
1 2 3 4 5 6 7 8 9 10 11 12	10,2 15,2 17,3 19,4 21,2 26,1 28,3 21,4 22,1 14,6 9,5 12,4	9,7 16,1 14,8 22,7 25,4 28,2 25,8 23,3 20,7 15,2 8,6 12,9	11,8 14,4 15,6 16,5 29,1 25,2 23,5 23,6 28,2 26,3 13,3 14,6	10,6 15,2 15,9 19,5 25,2 26,5 25,6 22,8 20,3 15,4 10,5 13,3	57,6 82,5 86,3 105,9 136,8 143,9 140,6 123,8 110,2 83,6 57,0 72,2
TOTAL	217,7	223,4	221,1	221,1	1200,4
Average	18,14	18,61	18,51	18,42	100,0

For clarity, the seasonal wave is depicted as a graph.

Having an idea of ​​the seasonal changes of a particular phenomenon, an enterprise can correctly distribute material, financial and labor resources throughout the year,

In the case when the levels of the time series show a tendency to increase or decrease, the actual data are compared with the aligned ones, i.e., obtained using analytical alignment. Seasonality indices are calculated using formula (7.36).

One of the most important tasks of statistics is the study of changes in the analyzed indicators over time, i.e. their dynamics. This problem is solved using the analysis of dynamics series (time series).

Dynamics series(time series) is a series of numerical values ​​of a statistical indicator, located in chronological sequence, characterizing the change in the phenomenon being studied over time.

A series of dynamics can be depicted graphically, which allows one to visualize the development of a phenomenon over time. Line charts are more often used: time is marked on the x-axis, and series levels are marked on the y-axis. Bar, pie and other charts are also widely used.

Each row of dynamics has two main elements:

1) time indicator t;

2) level of series y.

Time indicators can be periods(year, quarter, month, day) and moments(a specific date at the beginning or end of the period).

Row level- this is the size (volume, value) of a particular phenomenon (indicator) achieved over a certain period of time or at a certain moment. Levels in a time series can be represented absolute, relative or average quantities.

According to time, the series are divided into momentary and interval.

Momentary is called a series of dynamics, the levels of which characterize the state of the phenomenon at certain dates (points in time). For example, the number of outstanding cases in court remaining at the end of the reporting period - as of July 1, 2010, the number of suspended cases as of this date, the number of persons wanted as of the reporting date).

Interval(periodic) dynamics series is a series whose levels characterize the size of a phenomenon for a specific period of time (year, quarter, month). For example, the number of civil cases considered by magistrates in 2009 or the number of persons in respect of whom acquittals were made at first instance in the 1st half of 2010.

To quantify the dynamics of legal phenomena, statistical indicators such as absolute increases, growth rate, growth rate, which are divided into basic, chain and medium. The calculation of these dynamics indicators is based on a comparison of the levels of a series of dynamics. If the comparison is carried out with the same level taken as the basis of comparison, then these indicators are called basic. The base level is either the initial level in the dynamics series, or the level from which some new stage in the development of the phenomenon begins (for example, the number of people convicted under articles of the Criminal Code of the Russian Federation since 1997, the year the new Criminal Code came into force). If the comparison is carried out with a variable base and each subsequent level of the series is compared with the previous one, then the dynamics indicators calculated in this way are called chain

For time series with significant fluctuations in levels, average levels are used as a basis for comparison.

Absolute increase(Δу) is equal to the difference between the two compared levels.

Baseline absolute growth

Δy i b = y i - y b.

Chain absolute increase

Δy i = y i - y i - 1.

Average absolute increase

where y i is the level of the period being compared;

y i -1 - level of the previous period;

y b - level of the base period;

n is the number of levels of the series.

Growth rate is the ratio of the level of a series of one period to the level of a series of another period, expressed as a percentage.

Baseline growth rate T i b =

Chain growth rate T i =

Average growth rate

Comment. If the growth rate and average growth rate are calculated in fractions (not multiplied by 100%), then they are called respectively growth rate And average growth rate.

Rate of increase is calculated as the ratio of the absolute increase (Δу) to the level taken as the basis of comparison. The growth rate shows by what percentage the level being compared has changed relative to the level taken as the base of comparison. It can be positive, negative or zero.

Base growth rate T pr i b =

Chain growth rate T pr i =

Average growth rate.

Comment. If the corresponding growth rate is calculated, then the growth rate is:

T pr. = T r. - 100%.

Using the above formulas, we get:

Baseline absolute growth

Δy b 2002 = y 2002 - y 2004 = 2035 - 2930 = - 895, Δy b 2003 = y 2003 - y 2004 = 2232 - 2930 = - 698,

Δy b 2005 = y 2005 - y 2004 = 3609 - 2930 = 679, Δy b 2006 = y 2006 - y 2004 = 4229 - 2930 = 1299.

Chain absolute increase

Δy 2003 = y 2003 - y 2002 = 2232 - 2035 = 197, Δy 2004 = y 2004 - y 2003 = 2930 - 2232 = 698,

Δy 2005 = y 2005 - y 2004 = 3609 - 2930 = 679, Δy 2006 = y 2006 - y 2005 = 4229 - 3609 = 620.

Average absolute increase

Baseline growth rate

T b 2002 = T b 2003 = T b 2005 = T b 2006 =

Chain growth rate

T 2003 = T 2004 =

T 2005 = T 2006 =

Average growth rate

Base growth rate

T pr b 2002 = T pr b 2003 =

T pr b 2005 = T pr b 2006 =

Chain growth rate

T pr2003 = T pr 2004 =

T pr 2005 = T pr2006 =

Average growth rate

Along with the indicated indicators in the dynamics series, it can be calculated middle row level . It is applicable for any series of dynamics: interval and moment.

In interval series dynamics, the average level () is determined by dividing the sum of the levels of the series by their number, i.e., using the arithmetic mean method:

y i - absolute levels of the series; n - number of levels.

IN moment series with equal intervals time average level - the average chronological moment series - is determined by the formula:

In a moment series with unequal intervals time, the average level of the series is determined by the formula of the arithmetic weighted average

where y i are the levels of the dynamics series that remained unchanged during the period of time t i .

Using the above formula for the interval dynamics series, we obtain:

In practice, it is generally accepted that the values ​​of the levels of series of dynamics of statistical indicators are formed under the influence of the following components: trend, seasonal, cyclical and random components. In most cases, the actual level of a dynamics series can be represented as the sum or product of the above components. A model in which a series of dynamics is presented as the sum of the listed components is called additive model series of dynamics. A model in which a series of dynamics is presented as a product of the listed components is called multiplicative model series of dynamics. The main task of studying a separate dynamics series is to identify and give quantitative expression to each of the components listed above in order to use the information obtained to predict future values ​​of the series.

Under trend understand the smooth change that determines the general direction of development, the main trend of a series of dynamics. This is a systematic component that characterizes the long-term impact of factors on the dynamics of the indicator being studied.

Along with long-term trends in the time series of social processes, there are often more or less regular fluctuations - periodic components of the dynamics series.

If the period of oscillation does not exceed one year, then they are called seasonal . Most often, the cause of their occurrence is considered to be natural and climatic conditions that determine socio-economic phenomena (during the holiday season the number of burglaries increases, the number of lawsuits filed in courts by individuals decreases, etc.).

With a longer period of oscillation, it is believed that in the dynamics series there is cyclic component. Examples include demographic, investment and other cycles.

If the trend and periodic components are removed from a time series, then a random component remains, which is the result of the action of a large number of side factors. The influence of each of these factors is insignificant, but their total impact is felt. In judicial statistics, one of these random factors that can have a significant impact on the dynamics is changes in legislation.

An important task solved using time series is to determine the general development trend, i.e. trend. Identifying a trend in statistics is also called leveling a series of dynamics, and methods for identifying the main trend are called leveling methods.

Alignment can be carried out in different ways: the method of enlarging intervals, smoothing using the moving average method, or analytical alignment.

Interval enlargement method consists in transforming the initial series of dynamics into a series of longer periods (monthly into quarterly, quarterly into annual, etc.).

Moving average method lies in the fact that the average level is calculated from a certain number of the first levels, then from the same number of levels, but starting from the second, then starting from the third, etc. Thus, the average “slides” along the series speakers, moving one level. For example,

The first two methods make it possible to determine only the general trend in the development of a phenomenon, but it is impossible to obtain a generalized statistical model of the trend using these methods. In order to give a quantitative model expressing the main trend of changes in the levels of the dynamic series over time, it is used analytical alignment of the dynamics series.

The main content of the method is that the mathematical model of the trend is presented in the form of a certain function of time, which best reflects (approximates) the main trend in the development of a series of dynamics. The choice of model type should be based on a theoretical analysis that reveals the nature of the development of the phenomenon, as well as on a graphical representation of a series of dynamics (linear diagram). The selection of an adequate function is carried out using the least squares method - the minimum sum of squared deviations between the calculated and actual y i levels of the dynamics series:

The main models of the general trend of time series are the following:

1. Even development displayed by the linear function equation,

where a o and a 1 are the parameters of the equation, t is time.

Parameter a 1 determines the direction of development. If a 1 > O, then the levels of the dynamics series increase uniformly, if a 1< О - происходит их равномерное снижение.

The model of uniform development of the general trend is applied to time series with constant absolute increases.

2. Equally accelerated (equally slow) development represented by the second order parabola equation

Parameter a 2 characterizes a constant change in the intensity of development (per unit time). The levels of time series for which such a general development trend model is used change with constant growth rate.

3. Exponential development displayed by exponential function

where a 1 is the rate of growth (decrease) of the phenomenon being studied per unit of time, i.e., the intensity of development. For this model of the general development trend, the levels of a number of dynamics are inherent constant growth rate.

Other mathematical functions are also used.

The patterns identified during the analysis of time series can serve as a basis for predicting the development of the phenomenon being studied in the future. The basis of forecasting is the assumption that the pattern operating within the analyzed time series, which serves as the basis for forecasting, is preserved in the future.

A rough forecast can be obtained based on the average of the series.

When forecasting based on a series of dynamics with constant absolute increase the formula is applied:

where is the predicted level of the series,

The actual value of the last level of the dynamics series,

Average absolute increase,

k - forecast period (lead period).

When forecasting based on a series of dynamics with constant growth rates the following formula applies:

where is the average growth coefficient (chain) of the dynamics series, serving as a forecasting base.

For a more accurate forecast, for example, statistical forecasting methods such as the growth curve method and adaptive methods are used.

Example. Taking into account that the chain growth rates in the number of people convicted of bribery are approximately the same, let us build a rough forecast for 2007.

Using the appropriate formula, we get:

Thus, the number of people convicted of bribery (Articles 290, 291 of the Criminal Code of the Russian Federation) in 2007 should have been approximately 5,075 people. (According to the statistical collection “Crime and Delinquency (2004-2008)”, the number of people convicted under sentences that entered into force in 2007 according to basic qualifications was 4869.)

1. Federal Law “On official statistical accounting and the system of state statistics” dated November 29, 2007 No. 282-FZ.

2. Decree of the President of the Russian Federation dated March 30, 1998 No. 328 “On the development of a unified state system for registering and recording crimes.”

3. Decree of the Government of the Russian Federation dated June 2, 2008 No. 420 “On the Federal State Statistics Service”

4. Instructions for judicial records management in a district court, approved by order of the Judicial Department at the Supreme Court of the Russian Federation dated April 29, 2003 No. 36

5. Instructions for judicial records management in the supreme courts of republics, regional and regional courts, courts of federal cities, courts of the autonomous region and autonomous districts”, approved by order of the Judicial Department under the Supreme Court of the Russian Federation dated December 12, 2004 No. 161

6. Order of the Judicial Department at the Supreme Court of the Russian Federation dated October 16, 2009 No. 187 “On approval of the statistical card for the defendant”

7. Instructions for maintaining judicial statistics, approved by order of the Judicial Department at the Supreme Court of the Russian Federation dated December 29, 2007 No. 169

8. Resolution of the Federal State Statistics Service of January 15, 2008 No. 4 “On approval of statistical tools for organizing statistical monitoring of the registration of criminal cases and recording of crimes”

9. Order of the Judicial Department at the Supreme Court of the Russian Federation dated May 20, 2009 No. 97 “On approval of the Table of statistical reporting forms on the activities of federal courts of general jurisdiction and justices of the peace, samples of statistical reporting forms”, as amended by order of the Judicial Department No. 130 dated June 23, 2010 (Orders and samples of statistical reporting forms are posted on the website of the Judicial Department www.cdep.ru section “Judicial Statistics”).

Main

1. Lovtsov D.A., Bogdanova M.V. Legal statistics: Texts of lectures. - M.: RAP, 2007.

2. Luneev V.V. Legal statistics - M.: Yurist, 2007.

3. Savyuk L.K. Legal statistics. -M.: Yurist, 2007

Various economic and other indicators, which are given for a certain period of time or as of a certain point, are widely used in practical statistics. Information based on these indicators is called rows of dynamics . The absolute values ​​of the phenomenon under study in the dynamics series as of the corresponding period of time or moment are called the levels of the dynamics series. On their basis, the most important indicators of the dynamics series are calculated and in mathematical terms you only need to add, subtract, divide, multiply and extract the root, and also remember that you cannot change the chronological sequence of the levels of the dynamics series. And based on the already recorded levels of a series of dynamics, it is possible to predict the values ​​of the levels for future periods, and this is where “adult” mathematics begins.

Chain and basic indicators of dynamics series and their calculation

The main indicators characterizing absolute and relative changes in the dynamics series are: absolute increase (decrease), growth rate, growth rate, growth rate, absolute value of one percent increase (decrease) .

Indicators of dynamics series by the nature of their calculations are divided into chain and basic .

Chain indicators of dynamics series characterize the intensity of changes from one period to another period. Chain indicators are obtained by comparing (subtracting or dividing) two adjacent levels of a series of dynamics - the next level and the previous level. Chain indicators do not depend on the length of the dynamics series and on what level is taken as its beginning.

Basic indicators of time series - these are indicators with a constant base (beginning). They characterize the final results of all changes in a series of dynamics in comparison with the period (moment), which is taken as the base period (moment).

Basic indicators are calculated by comparing each level of the dynamics series with the same level taken as the basis. Usually this is the first (initial) level of the series, although, if this is dictated by the task of analysis, any other level can be taken as the basic level. If the initial level of the dynamics series for the phenomenon or process being studied is atypically high or atypically low, then the indicators of the dynamics series calculated in comparison with it may turn out to be of little use for the analysis task.

Let us introduce the following notation:

• Y- designation of the level of a series of dynamics in general form;
• Y1 - first (initial) level of the dynamics series;
• Yn- the last level of the dynamics series;
• Ym- any level of dynamics.

We will calculate the indicators for the series of dynamics given in the following table:

Table. Export volumes of enterprise "X", in millions of rubles.

Year	2013	2014	2015	2016	2017
Volume	1256,4	1408,8	1650,6	2150,0	2888,2

Absolute increase (decrease) expresses absolute changes in the levels of dynamics series - increase or decrease - compared to any achieved level. There is a difference between chain and basic absolute growth (decrease).

Chain absolute increase (decrease) is calculated by subtracting the dynamics of the previous level of the same series from any level of a series.

Example 1. Let's calculate the chain absolute increase:

Δ a(c) = Y m − Y m−1

Δ c (2014) = 1408.8 − 1256.4 = 152.4.

Δ c (2015) = 1650.6 − 1408.8 = 241.8.

Δ c (2016) = 2150.0 − 1650.6 = 499.4.

Δ c (2017) = 2888.2 − 2150.0 = 738.2.

The total export volume of enterprise "X" from 2013 to 2017 is Δ ts (2014) + Δts (2015) + Δts (2016) + Δts (2017) = 1631.8 million rubles.

The basic absolute growth is calculated by subtracting from any level of the series the dynamics of the initial level of the series, which is taken as the basis.

Example 2. Let's calculate the basic absolute increase:

Δ a(b) = Y m − Y 1

Δ b (2014) = 1408.8 − 1256.4 = 152.4.

Δ b (2015) = 1650.6 − 1256.4 = 394.2.

Δ b (2016) = 2150.0 − 1256.4 = 893.6.

Δ b (2017) = 2888.2 − 1256.4 = 1631.8.

There is a mathematical relationship between chain and basic absolute growth : the sum of chain absolute increases (decreases) is equal to the basic absolute increase (decrease) corresponding to the last level of the dynamics series:

The indicator of the intensity of change in a series of dynamics, depending on whether it is expressed as a coefficient or as a percentage, is called the growth coefficient or growth rate.

Growth rate shows how many times the corresponding level of the dynamics series is greater than the base level (if the coefficient is greater than one) or what part of the base level represents the level of the reporting period (if it is less than one).

Growth rate characterizes the rate of development of the phenomenon under study.

Growth coefficient and growth rate are two forms of expressing the intensity of change and the difference between them is only in units of measurement.

Growth rate × 100 = growth rate, %.

If the absolute levels of the phenomenon under study decrease, then growth rate less than one (less than 100%), but it never cannot be a negative number . There are chain and base growth rates. The chain growth rate is calculated by dividing the level of the dynamics series by the previous level of the series:

Overall growth rate for the entire period calculated by multiplying all growth rates:

Example 3. Let's calculate the chain growth rates:

Tc (2014) = 1408.8: 1256.4 = 1.121 = 112.1%.

Tc (2015) = 1650.6: 1408.8 = 1.172 = 117.2%.

Tc (2016) = 2150.0: 1650.6 = 1.303 = 130.3%.

Tc (2017) = 2888.2: 2150.0 = 1.343 = 134.3%.

Overall growth rate for the entire period:

Tts (2014-2017) = 1.121 × 1.172 × 1.303 × 1.343 = 2.299 = 229.9%.

Baseline growth rate calculated by dividing any level of the dynamics series by the initial level, which is considered basic:

Example 4. Let's calculate the basic growth rates:

Tb (2014) = 1408.8: 1256.4 = 1.121 = 112.1%.

Tb (2015) = 1650.6: 1256.4 = 1.319 = 131.9%.

Tb (2016) = 2150.0: 1256.4 = 1.711 = 171.1%.

Tb (2017) = 2888.2: 1256.4 = 2.299 = 229.9%.

There is a mathematical relationship between chain and base growth rates : the product of chain growth rates is equal to the base growth rate for the last level of the dynamics series:

Growth rate shows by what part of the whole the corresponding level of a series of dynamics has increased or decreased in comparison with any achieved level, and rate of increase - by what percentage? The growth rate is calculated by subtracting one from the growth rate (if a growth rate is used) or 100 percent (if the growth rate is expressed as a percentage).

Thus, the formulas for calculating the growth rate are:

Kpr (ts) = T p(t) − 1

Kpr(b) = T p(b) − 1.

For example,

1,299 = 2,299 − 1,0 .

Formulas for calculating the growth rate:

Tpr (ts) = T r (ts) − 100%

Tpr(b) = T p(b) − 100%.

For example,

129,9 = 229,9 % − 100,0 % .

In contrast to the growth rate, growth rates can also be negative numbers . In this case, they show by what part of the whole or by what percentage the level of the phenomenon under study has decreased.

There is no mathematical relationship between the chain and base growth rates.

Absolute value of 1 percent increase (decrease) expresses the real content of the growth (decrease) rate. In practice, there may be significant growth rates, but a very insignificant absolute increase in the phenomenon, and vice versa - small growth rates, but a significant increase. The absolute value of 1 percent increase (decrease) is calculated by dividing the sum of chain absolute increases or basic absolute growth by the growth rate:

.

For example,

Average values ​​of time series indicators

Average values ​​of time series indicators express the levels and typical values ​​of their changes in a certain period of time. Before considering the average values ​​of indicators of time series, we will distinguish between the concepts of interval and moment time series.

Interval time series characterize the values ​​of the phenomenon under study for a certain period of time, for example, for a month, for a year, for five years. Moment dynamics series characterize the values ​​of the phenomenon being studied at a certain point in time, for example, at the beginning or end of the month, the beginning or end of the year, and so on. In the previous paragraph we looked at the interval dynamics series and its indicators.

Average level of interval dynamics series is calculated by dividing the sum of the levels of the series by the number of levels:

.

Example 5. Calculate the average annual export volume of enterprise "X".

Solution. Let's calculate the average level using the formula for the interval dynamics series:

Average level of the moment series of dynamics , if there are equal time intervals between moments, it is calculated using the average chronological formula:

.

Example 6. Calculate the average number of employees of enterprise "X" at the beginning of the year. The table below shows the number of employees at the beginning of each year from 2013 to 2017.

Solution. We calculate using the chronological average formula:

If there are unequal time intervals between the moments of the dynamics series, the average level of the moment series is calculated using the average chronological weighted formula:

In this formula y1 - yn- levels of dynamics series, t1 - tn- time periods, for example, 1 month, 2 months, 1 year, 2 years, 3 years... All time periods must be expressed in the same unit of measurement (days, months, years, etc.).

Average absolute increase (decrease) expresses the absolute value by which, on average, in each unit of time during the corresponding period, the indicators of the phenomenon under study increased or decreased. It is calculated by dividing the sum of chain absolute increases by the number of absolute increases:

,

where is the number of absolute increases.

If there is no data on chain absolute increases, but the initial and final levels of the dynamics series are known, then the average absolute increase can be calculated through the basic absolute increase using the formula

Example 7. Using data on exports of enterprise "X", calculate the average annual increase in exports.

Solution. Let's calculate the indicator we are interested in through the sum of chain absolute increases:

.

Let's calculate it through the base absolute growth:

As you can see, we got the same result.

Average growth rate is an indicator of changes in the intensity of changes in the levels of a series of dynamics. It characterizes the average intensity of development of the phenomenon under study, showing how many times on average the levels of a series of dynamics have changed per unit time. The average growth rate can be expressed as ratios or percentages.

The chain average growth rate is calculated using the geometric mean formula:

,

Where n- number of chain growth rates,

T- individual chain growth rates, expressed in coefficients.

If there is no information about each chain growth rate, the average growth rate can be calculated using the formula using the last and first levels of the dynamics series

Example 8. Calculate the average growth rate of exports of enterprise "X".

Solution. We calculate using the geometric mean formula:

We calculate using the formula using the last and first levels of the dynamics series:

.

We got the same result.

Average growth rate shows by what percentage on average the level of the phenomenon under study increased (if it has a plus sign) or decreased (if it has a minus sign) during the entire period under consideration. The average growth rate is calculated by subtracting 100% (if expressed as a percentage) or one (if expressed as a coefficient) from the average growth rate.

In our example:

Forecasting the levels of time series

Models based on averages can be used when the level value of a time series fluctuates around the average value and there is no clear tendency in the series.

Moving average method

In forecasting, the value of the moving average (let’s denote it Mt) is calculated by the formula

,

Where N- length of the smoothing interval.

In this case, the average value that is used for the forecast is the adaptive average. When pronosing, it is assumed that this adaptive average value is the best (most probable) value for the next period. Let us denote the predicted value by Ft. Then

Ft+1 = M t.

Example 9. Let's consider an example with data on sales volumes of refrigerators of enterprise "X" by month.

When examining the sales volume graph, it is clear that changes in volume are not subject to any long-term tendency or trend; sales volumes fluctuate around the average value.

Therefore, when calculating the forecast, you can use the average value. Let's calculate the values ​​of the moving average using the above formula:

for the third month - ,

for the fourth month -

The results are given in the third column of the table (it is impossible to calculate moving averages for the first two months using this formula).

Months t	Refrigerator sales volumes yt	Moving average Mt
1	113	-
2	117	-
3	112	114
4	113	114
5	108	111
6	112	111
7	116	112
8	120	116
9	121	119
10	113	118
11	111	115
12	118	114

Forecast Ft	Forecast error ε t
-	-
-	-
-	-
114	-1
114	-6
111	1
111	5
112	8
116	5
119	-6
118	-7
115	3

The most probable forecasts for each month using the corresponding formula are given in the fourth column of the table. Forecast for the first month of next year F13 = 114 can be done based on data from the last three months.

When using the average value model, the forecasts depend on the length of the smoothing interval. Therefore, the natural question is: how to choose an interval and what value is the “best” for the interval? To answer this question, you need to estimate the forecast error of the average for different smoothing intervals and select the one with the smallest random forecast error.

The forecast error for each point in time is calculated using the formula

ε t = y t − F t.

The average forecast error based on a moving average is usually calculated as the mean absolute deviation, which is denoted MAD (Mean Absolute Deviation):

Where n- number of calculated errors.

When assessing the forecast, you can also use the mean square error and the mean absolute percentage error.

The mean squared error MSE (Mean Squared Error) is calculated using the formula

.

Mean Absolute Percenting Error MAPE (Mean Absolute Percenting Error) is calculated using the formula

.

Example 10. In our case, when N=3, MAD=4.67. For values N From 2 to 6 the MAD values ​​are as follows:

N	MAD
2	4,50
3	4,67
4	4,78
5	4,11
6	4,42

Based on these error values, we can conclude that using a smoothing interval of five periods gives the best forecast in terms of minimum mean absolute deviation. Using this smoothing interval, we obtain a forecast: it is most likely that 116 refrigerators will be sold in the first month of next year:

F13 = (118 + 111 + 113 + 121 + 120)/5 = 116,6 .

When using the moving average value formula, each level of the dynamics series within the smoothing period is assigned the same weight. So, if N=3, then the weight corresponds to 1/3, so the formula in this case can be written as follows:

Mt = (1/3) y t+(1/3) y t−1 + (1/3) y t−2.

But you can also use moving averages with different weights - the so-called weighted moving averages. In this case, the following condition must be observed: the sum of the weights is equal to one. For example, when N=3 you can use scales 3/5, 1/5, 1/5. In this case

Mt = (3/5) y t+(1/5) y t−1 + (1/5) y t−2.

Forecasting models based on the moving average and moving weighted average have a significant drawback: only the latter are used to calculate the predicted value N levels of a series of dynamics and only the previous ones are used to calculate the error n−N levels. Therefore, other methods are used to predict the average values ​​of time series.

Exponential average method (exponential smoothing)

The basic formula for the value of exponential average:

Ft+1 = αy t + (1 − α )F t,

Where α - exponential smoothing parameter, which can take values ​​from 0 to 1.

Thus, the forecast for each subsequent period is based on the weighted average of the previous level of the dynamics series and the value of the previous forecast. For example, to predict the value of the fourth level of the dynamics series, the formula will be as follows:

F4 = αy 3 + (1 − α )F 3 ,

for third level forecast

F3 = αy 2 + (1 − α )F 2 ,

for second level forecast

F2 = αy 1 + (1 − α )F 1 . ,

That is, the forecast uses a weighted average of y3 And F3 with scales α and 1 − α .

In general, the forecast for each subsequent period is a weighted average of all previous levels of the dynamics series.

Let's return to the equations for predicting the values ​​of the third and fourth levels of the dynamics series. Substituting each next equation into the previous one, we get

or in general

.

Thus, in the general case, the predicted value is calculated using all levels of the dynamics series by multiplying them by the corresponding coefficients (weights): or .

Since the exponential smoothing parameter α takes values ​​from 0 to 1, these coefficients form a decreasing geometric progression with the first term a1 = α and quotient q = 1 − α . That is, they are subject to the exponential distribution law. For example, if α = 0.5, then α (1 − α ) = 0,25 , α (1 − α )² = 0.125 and so on. If α = 0.2, then α (1 − α ) = 0,16 , α (1 − α )² = 0.128 and so on. On the graph you can see that the scales decrease exponentially, but in the first case more rapidly, and in the second - more slowly.

Depending on the value of the exponential smoothing parameter α Different levels of a dynamics series can be assigned different weights. For example, if it is known about the predicted indicator that its future values ​​are more influenced by the closest previous levels of the series, then the parameter α should be greater than in the case where earlier values ​​of the dynamics series have a greater influence. And if earlier values ​​have a greater influence, then the parameter α should be less.

In practical calculations it is assumed that F1 = y 1 , since those necessary for the calculation F1 values y0 And F0 unknown.

Example 11. Let's make a forecast using the exponential smoothing method for a series of dynamics containing data on sales volumes of refrigerators for enterprise "X" from the previous examples.

F1 = y 1 = 113,0

F2 = 0,2⋅113 + (1 − 0,2)⋅113,0 = 113,0

F3 = 0,2⋅117 + (1 − 0,2)⋅113,0 = 113,8

F4 = 0,2⋅112 + (1 − 0,2)⋅113,8 = 113,44

Forecast for the first month of next year:

F13 = 0,2⋅118 + (1 − 0,2)⋅114,33 = 115,06 .

The values ​​of the exponential average, if we assume that α = 0.2 are given in the third column of the table.

Forecast ( α = 0,2 ) Ft	Forecast error ε t
113,0	0
113,0	4
113,8	-1,8
113,44	-0,44
114,35	-6,35
112,28	-0,28
112,23	3,77
112,98	7,02
114,38	6,62
115,71	-2,71
115,17	-4,17
114,33	3,67

The forecast can be clarified if you choose a more optimal value α : one in which the average forecast error is the smallest. Let us choose MSE as a value characterizing the error. This error for different values α next:

α	MSE
0,01	4,01
0,02	4,00
0,05	3,97
0,10	3,97
0,15	3,98
0,20	4,02
0,25	4,05
0,30	4,08
0,35	4,13
0,40	4,16
0,45	4,20
0,50	4,23

We see that the smallest forecast error for a given time series when using the exponential equalization method corresponds to the values α from 0.05 to 0.15. Let us take the optimal value to be in the middle between these two, that is, 0.1. Then we get the following sales volume forecast: 114.3.

The exponential averaging formula can be transformed so that the forecast takes into account the forecast error for the previous period:

Ft+1 = αy t + (1 − α )F t

Ft+1 = αy t+ F t − α F t

Ft+1 = F t+ α (y t − F t)

Ft+1 = F t+ α ε t).

As the last expression shows, a forecast using the exponential average method is formed from a forecast with an exponential average of the previous period with the addition of an error margin multiplied by a smoothing parameter α . If the error is greater than zero, it means that the previous forecast was less than the actual value and the next forecast will be increased accordingly. If the error is less than zero, then the forecast was less than the actual value and the forecast for the next period will be reduced accordingly.

Confidence interval for forecasts based on mean values

The confidence interval of the forecast is determined by calculating the standard error sε .

In fact, we can assume that in 68% of cases the forecasts are in the range Ft ± sε , and in 95% of cases - in the interval Ft± 2 sε .

To calculate sε , you can use the value of the mean absolute error MAD or the value of the mean square error MSE:

.

Example 12. In our example with refrigerator sales volumes, the standard error for forecasting using the moving average method is if N=5, equal to . This means that for 8 months (0.68⋅12) the forecast should be rounded in the range from 112 to 122 (116.6±5.1), and for 11 months (0.95⋅12) - in the range from 106 to 127 (116.6±2⋅5.1).

Standard forecast error using the exponential smoothing method, if α =0.1, is . This means that in 68% of cases the forecast with rounding should be within the range from 110 to 118 (114.3±4.1), and in 95% - within the range from 106 to 123 (114.3±2⋅4.1 ).

Dynamics series- these are a series of statistical indicators characterizing the development of natural and social phenomena over time. Statistical collections published by the State Statistics Committee of Russia contain a large number of dynamics series in tabular form. Dynamic series make it possible to identify patterns of development of the phenomena being studied.

Dynamics series 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 dynamics series can be expressed in absolute values ​​(product production in tons or rubles), relative values ​​(share of the urban population in %) and average values ​​(average wages of industry workers by year, etc.). A dynamics row contains two columns or two rows.

Correct construction of time series requires the fulfillment of a number of requirements:

1. all indicators of a series of dynamics must be scientifically based and reliable;
2. indicators of a series of dynamics must be comparable over time, i.e. must be calculated for the same periods of time or on the same dates;
3. indicators of a number of dynamics must be comparable across the territory;
4. indicators of a series of dynamics must be comparable in content, i.e. calculated according to a single methodology, in the same way;
5. indicators of a number of dynamics should be comparable across the range of farms taken into account. All indicators of a series of dynamics must be given in the same units of measurement.

Statistical indicators can characterize either the results of the process being studied over a period of time, or the state of the phenomenon being studied at a certain point in time, i.e. indicators can be interval (periodic) and momentary. Accordingly, initially the dynamics series can be either interval or moment. Moment dynamics series, in turn, can be with equal or unequal time intervals.

The original dynamics series can be transformed into a series of average values ​​and a series of relative values ​​(chain and basic). Such time series are called derived time series.

The methodology for calculating the average level in the dynamics series is different, depending on the type of the dynamics series. Using examples, we will consider the types of dynamics series and formulas for calculating the average level.

Interval time series

The levels of the interval series characterize the result of the process being studied over a period of time: production or sales of products (for a year, quarter, month, etc.), the number of people hired, the number of births, etc. The levels of an interval series can be summed up. At the same time, we get the same indicator over longer time intervals.

Average level in interval dynamics series() is calculated using the simple formula:

• y— series levels ( y 1 , y 2 ,...,y n),
• n— number of periods (number of levels of the series).

Let's consider the methodology for calculating the average level of an interval dynamics series using data on the sale of sugar in Russia as an example.

	Sugar sold, thousand tons

This is the average annual volume of sugar sales to the Russian population for 1994-1996. In just three years, 8137 thousand tons of sugar were sold.

Moment dynamics series

The levels of moment series of dynamics characterize the state of the phenomenon being studied at certain points in time. Each subsequent level includes, in whole or in part, the previous indicator. For example, the number of employees on April 1, 1999 fully or partially includes the number of employees on March 1.

If we add up these indicators, we get a repeat count of those workers who worked throughout the month. The resulting amount has no economic content; it is a calculated figure.

In moment series of dynamics with equal time intervals, the average level of the series calculated by the formula:

• y-moment series levels;
• n-number of moments (series levels);
• n - 1— number of time periods (years, quarters, months).

Let's consider the methodology for such calculation using 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 - an enterprise:

The calculation was made using the average chronological formula. The average number of employees of the enterprise for the 1st quarter was 155 people. The denominator is 3 months in a quarter, and the numerator (465) is a calculated number that has no economic content. In the vast 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 using the weighted arithmetic mean formula. The length of time (t-days, months) is taken as the average weight. Let's perform the calculation using this formula.

The list of employees of the enterprise for October is as follows: on October 1 - 200 people, on October 7, 15 people were hired, on October 12, 1 person was fired, on October 21, 10 people were hired, and until the end of the month there were no hiring or dismissal of workers. This information can be presented as follows:

When determining the average level of a series, it is necessary to take into account the duration of the periods between dates, i.e. apply:

In this formula, the numerator () has economic content. In the example given, the numerator (6665 person-days) is the company’s employees in October. The denominator (31 days) is the calendar number of days in the month.

In cases where we have a moment series of dynamics with unequal time intervals, and the specific dates of change in the indicator are unknown to the researcher, then first we need to calculate the average value () for each time interval using the simple arithmetic average formula, and then calculate the average level for the entire series of dynamics, by weighing the calculated average values ​​over the duration of the corresponding time interval. The formulas are as follows:

The dynamics series discussed above consist of absolute indicators obtained as a result of statistical observations. The initially constructed series of dynamics of absolute indicators can be transformed into derivative series: series of average values ​​and series of relative values. Series of relative values ​​can be chain (in % of the previous period) and basic (in % of the initial period taken as the basis of comparison - 100%). The calculation of the average level in the derivative time series is performed using other formulas.

A series of averages

First, we transform the above moment series of dynamics with equal time intervals into a series of average values. To do this, we calculate the average 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 present this in tabular form.

Average level in derivative series average values ​​are calculated by the formula:

Note that the average payroll number of employees of the enterprise for the 1st quarter, calculated using the chronological average formula based on the database on the 1st day of each month and the arithmetic average - according to the derived series - are equal to each other, i.e. 155 people. A comparison of the calculations allows us to understand why in the average chronological formula the initial and final levels of the series are taken in half size, and all intermediate levels are taken in full size.

Series of average values ​​derived from moment or interval series of dynamics should not be confused with series of dynamics in which levels are expressed by an average value. For example, the average wheat yield by year, the average salary, etc.

Series of relative quantities

In economic practice, series are 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 series with relative values ​​of dynamics.

The average level of the series in relative dynamics series is called the average annual growth rate. Methods for its calculation and analysis are discussed below.

Analysis of time series

For a reasonable assessment of the development of phenomena over time, it is necessary to calculate analytical indicators: absolute growth, growth coefficient, growth rate, growth rate, absolute value of one percent of growth.

The table shows a numerical example, and below are calculation formulas and economic interpretation of the indicators.

Analysis of the dynamics of production of product "A" by the enterprise for 1994-1998.

	Produced thousand tons	Absolute gains,		Growth rates		Pace growth, %		Growth rate, %		Value of 1% increase, thousand tons.
			basic		basic		basic		basic
		3	4	5	6	7	8	9	10	11

Absolute increases (Δy) show how many units the subsequent level of the series has changed compared to the previous one (gr. 3. - chain absolute increases) or compared to the initial level (gr. 4. - basic absolute increases). The calculation formulas can be written as follows:

When the absolute values ​​of the series decrease, there will be a “decrease” or “decrease”, respectively.

Indicators of absolute growth indicate that, for example, in 1998, the production of product “A” increased by 4 thousand tons compared to 1997, and by 34 thousand tons compared to 1994; for other years, see table. 11.5 gr. 3 and 4.

Growth rate shows how many times the level of the series has changed compared to the previous one (gr. 5 - chain coefficients of growth or decline) or compared to the initial level (gr. 6 - basic coefficients of growth or decline). The calculation formulas can be written as follows:

Rates of growth show what percentage the next level of the series is compared to the previous one (gr. 7 - chain growth rates) or compared to the initial level (gr. 8 - basic growth rates). The calculation formulas can be written as follows:

So, for example, in 1997, the production volume of product “A” compared to 1996 was 105.5% (

Growth rate show by what percentage the level of the reporting period increased compared to the previous one (column 9 - chain growth rates) or compared to the initial level (column 10 - basic growth rates). The calculation formulas can be written as follows:

T pr = T r - 100% or T pr = absolute growth / 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% more, and compared to 1994 - by 9% (109% - 100%).

If the absolute levels in the series decrease, then the rate will be less than 100% and, accordingly, there will be a rate of decline (the rate of increase with a minus sign).

Absolute value of 1% increase(column 11) shows how many units must be produced in a given period so that the level of the previous period increases 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.

The absolute value of 1% growth can be determined in two ways:

• divide the level of the previous period by 100;
• chain absolute increases are divided by the corresponding chain growth rates.

Absolute value of 1% increase =

In dynamics, especially over a long period, a joint analysis of the growth rate with the content of each percentage increase or decrease is important.

Note that the considered methodology for analyzing time series is applicable both for time series, the levels of which are expressed in absolute values ​​(t, thousand rubles, number of employees, etc.), and for time series, the levels of which are expressed in relative indicators (% of defects , % ash content of coal, etc.) or average values ​​(average yield in c/ha, average wage, etc.).

Along with the considered analytical indicators, calculated for each year in comparison with the previous or initial level, when analyzing dynamics series, 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 dynamics series we are considering, the average level of the series is calculated using a simple formula:

Average annual production volume of the product for 1994-1998. amounted to 218.4 thousand tons.

The average annual absolute growth is also calculated using the simple arithmetic average formula:

Annual absolute increases 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 series level indicators given in the table.

Average annual growth rate and average annual growth rate

First of all, we note that the growth rates shown in the table (columns 7 and 8) are series of 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 annual growth rates? This value is called the average annual growth rate.

The average annual growth rate is calculated in the following sequence:

The average annual growth rate ( is determined by subtracting 100% from the growth rate.

The average annual growth (decrease) coefficient using geometric mean formulas can be calculated in two ways:

1) based on the absolute indicators of the dynamics series according to the formula:

• n— number of levels;
• n - 1- number of years in the period;

2) based on annual growth rates according to the formula

• m— number of coefficients.

The calculation results using 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 11.5, column 6, 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 production volume of product "A" increased by 4.0% on average per year. Annual growth rates ranged from 1.7% in 1998 to 5.5% in 1997 (for each year’s growth rates, see Table 11.5, group 9).

The average annual growth rate (growth) allows you to compare the dynamics of development of interrelated phenomena over a long period of time (for example, the average annual growth rate of the number of workers in sectors of the economy, the volume of production, etc.), to compare the dynamics of a phenomenon in different countries, to study the dynamics of some or phenomena according to periods of historical development of the country.

Seasonal analysis

The study of seasonal fluctuations is carried out in order to identify regularly recurring differences in the level of time series depending on the time of year. For example, the sale of sugar to the population in the summer increases significantly due to the canning of fruits and berries. The need for labor in agricultural production varies depending on the time of year. The task of statistics is to measure seasonal differences in the level of indicators, and in order for the identified seasonal differences to be natural (and not random), it is necessary to build an analysis on the basis of data for several years, at least for at least three years. In table 11.6 shows the initial data and methodology for analyzing seasonal fluctuations using the simple arithmetic average method.

The average value for each month is calculated using the simple arithmetic average formula. For example, for January 2202 = (2106 +2252 +2249):3.

Seasonality index(Table 11.5, column 7.) is calculated by dividing the average values ​​for each month by the total average monthly value, taken as 100%. The average monthly for the entire period can be calculated by dividing the total fuel consumption for three years by 36 months (1188082 tons: 36 = 3280 tons) or by dividing the average monthly sum by 12, i.e. total total for gr. 6 (2022 + 2157 + 2464, etc. + 2870) : 12.

Table 11.6 Seasonal fluctuations in fuel consumption in agricultural enterprises in the region for 3 years

	Fuel consumption, tons			Amount for 3 years, t (2+3+4)	Average monthly for 3 years, t	Seasonality index,
September

Rice. 11.1. Seasonal fluctuations in fuel consumption in agricultural enterprises over 3 years.

For clarity, a seasonal wave graph is constructed based on seasonality indices (Fig. 11.1). Months are located on the abscissa axis, and seasonality indices in percentage are located on the ordinate axis (Table 11.6, group 7). The overall average monthly for all years is located at the 100% level, and the average monthly seasonality indices in the form of points are plotted on the graph field in accordance with the accepted scale along the ordinate axis.

The points are connected by a smooth broken line.

In the example given, the annual fuel consumption differs slightly. If, in the dynamics series, along with seasonal fluctuations, there is a pronounced tendency of growth (decrease), i.e. levels in each subsequent year systematically significantly increase (decrease) compared to the levels of the previous year, then we obtain more reliable data on the extent of seasonality as follows:

1. for each year we calculate the average monthly value;
2. Let's calculate the seasonality indices for each year by dividing the data for each month by the average monthly value for that year and multiplying by 100%;
3. for the entire period, we calculate the average seasonality indices using the simple arithmetic average formula from the monthly seasonality indices calculated for each year. So, for example, for January we will obtain the average seasonality index if we add up the January values ​​of seasonality indices for all years (let’s say for three years) and divide by the number of years, i.e. on three. Similarly, we calculate the average seasonality indices for each month.

The transition for each year from absolute monthly values ​​of indicators to seasonality indices makes it possible to eliminate the tendency of growth (decrease) in the dynamics series 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 population’s demand for certain types of goods. The results of the study of seasonal fluctuations are important for the effective management of economic processes.

Reducing dynamics series to the same base

In economic practice, there is often a need to compare several series of dynamics (for example, indicators of the dynamics of electricity production, grain production, passenger car sales, etc.). To do this, you need to transform the absolute indicators of the compared time series into derived series of relative basic values, taking the indicators of any one year as one or 100%. Such a transformation of several time series is called bringing them to the same base. Theoretically, the absolute level of any year can be taken as the basis of comparison, but in economic research, for the basis 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 basis for comparison.

Methods for aligning time series

To study the pattern (trend) of development of the phenomenon under study, data over a long period of time is required. The development trend of a particular phenomenon is determined by the main factor. But along with the action 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, drought years, etc.). Almost all series of dynamics of economic indicators on the graph 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 from actual data from a series of dynamics and from a graph. But statistics must not only determine the general trend in the development of a phenomenon (growth or decline), but also provide quantitative (digital) characteristics of development.

Trends in the development of phenomena are studied by methods of aligning dynamics series:

• Interval enlargement method
• Moving average method

In table Table 11.7 (column 2) shows actual data on grain production in Russia for 1981-1992. (in all categories of farms, in weight after modification) and calculations for leveling this series using three methods.

Method of enlarging time intervals (column 3).

Considering that the dynamics series is small, three-year intervals were taken and the averages were calculated for each interval. The average annual volume of grain production for three-year periods is calculated using the simple arithmetic average 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 calculation results in gr. 3.

Given in gr. 3 indicators of the average annual volume of grain production in Russia indicate a natural increase in grain production in Russia for the period 1981 - 1992.

Moving average method

Moving average method(see groups 4 and 5) is also based on the calculation of average values ​​for aggregated periods of time. The goal is the same - to abstract from the influence of random factors, to cancel out their influence in individual years. But the calculation method is different.

In the example given, five-tier (over five-year periods) moving averages are calculated and assigned to the middle year in the corresponding five-year period. Thus, for the first five years (1981-1985), using the simple arithmetic average formula, the average annual volume of grain production was calculated and recorded in table. 11.7 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 eliminating the initial year and adding the year following the five-year period and dividing the resulting amount 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 researcher decides the question. In principle, the longer the period, the more smoothing occurs. But we must take into account the length of the dynamics series; do not forget that the moving average method leaves 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.

Table 11.7 Alignment of data on grain production in Russia for 1981 - 1992

	Produced, million tons	Average for 3 years, million tons	5-year rolling total, million tons		Estimated indicators

Analytical alignment method

Analytical alignment method(gr. 6 - 9) is based on calculating the values ​​of the aligned series using the corresponding mathematical formulas. In table 11.7 shows calculations using the equation of a straight line:

To determine the parameters, it is necessary to solve the system of equations:

The necessary quantities for solving the system of equations have been calculated and given in the table (see groups 6 - 8), let’s substitute them into the equation:

As a result of the calculations we get: α= 87.96; b = 1.555.

Let's substitute the values ​​of the parameters and get the equation of the straight line:

For each year we substitute the value t and get the levels of the aligned series (see column 9):

Rice. 11.2. Grain production in Russia for 1981-1982.

In the leveled series, there is a uniform increase in series levels on average per year by 1.555 million tons (the value of the “b” parameter). The method is based on abstracting the influence of all other factors except the main one.

Phenomena can develop in dynamics evenly (increase or decrease). In these cases, the straight line equation is most often suitable. If the development is uneven, for example, at first very slow growth, and from a certain moment a sharp increase, or, conversely, first a sharp decrease, and then a slowdown in the rate of decline, then the leveling must be performed using other formulas (equation of a parabola, hyperbola, etc.). If necessary, one should turn to textbooks on statistics or special monographs, where the issues of choosing a formula to adequately reflect the actual trend of the dynamics series being studied are described in more detail.

For clarity, we will plot the indicators of the levels of the actual dynamics series and the aligned series on a graph (Fig. 11.2). The actual data is represented by a broken black line, indicating increases and decreases in the volume of grain production. The remaining lines on the graph show that the use of the moving average method (line with cut ends) allows you to significantly align the levels of the dynamic series and, accordingly, make the broken curved line on the graph smoother and smoother. However, straight lines are still crooked lines. Constructed on the basis of theoretical values ​​of the series obtained using mathematical formulas, the line strictly corresponds to a straight line.

Each of the three methods discussed has its own advantages, but in most cases the analytical alignment method is preferable. However, its application is associated with large computational work: solving a system of equations; checking the validity of the selected function (form of communication); calculating the levels of the aligned series; plotting. To successfully complete such work, it is advisable to use a computer and appropriate programs.

The dynamics series is a series of numbers that characterize changes in a social phenomenon over time. The values ​​of the indicators that form the dynamics series are called the level of the series.

To generally characterize the level of a phenomenon for a given period, the average level of the series is calculated. The method for calculating the average level of a series depends on the nature of the series. There are moment and interval dynamics series.

A moment series is a series that is formed by indicators characterizing the state of a phenomenon at a particular point in time.

An interval series of dynamics is a series that is formed by indicators characterizing a phenomenon for a particular period of time.

The average level of the interval series is determined by the formula:

where n is the number of terms of the dynamics series.

The average level of the moment series is determined by the average chronological formula:

The absolute increase shows by how many units the analyzed level of the series has increased (or decreased) relative to the basic level (according to the basic scheme) or the level of the previous year (according to the chain scheme). Accordingly, it is determined by the formulas:

(according to the basic scheme),

(according to a chain diagram).

The growth rate shows how many times the analyzed level of the series has increased (or decreased) compared to the level taken as the basis of comparison (according to the basic scheme) or the previous level (according to the chain scheme). The growth rate is expressed as a percentage or abstract numbers (growth coefficient). It is determined by the formula:

(according to the basic scheme),

(according to a chain diagram).

The growth rate shows by what percentage the analyzed level of the series has increased (or decreased) compared to the base (according to the basic scheme) or the previous level of the series (according to the chain scheme). It is defined as the ratio of absolute growth to the level taken as the basis of comparison using the formulas:

(according to the basic scheme),

(according to a chain diagram).

The rates of growth and gain are interconnected, as can be seen from the formulas for their calculation:

This makes it possible to determine the growth rate through the growth rate:

The average growth rate and the average growth rate characterize, respectively, the growth and growth rates for the period as a whole. The average growth rate is calculated from data from the dynamics series using the geometric mean formula:

where is the number of chain growth coefficients.

Based on the ratio of growth rates and growth, the average growth rate is determined:

The absolute value of one percent of growth A is the ratio of the chain absolute growth to the chain growth rate expressed as a percentage. It is determined by the formula:

As can be seen from the calculation, the absolute value of one percent of growth is equal to 0.01 of the previous level.

Using a series of dynamics, phenomena that are seasonal in nature are studied. Seasonal fluctuations are stable intra-annual fluctuations in the dynamics series, caused by specific conditions of production, consumption or sale of products or services. For example, fuel or electricity consumption for domestic needs, transportation of passengers, sale of goods, etc.

The level of seasonality is assessed using seasonality indices. The seasonality index shows how many times the actual level of a series at a moment or time interval is greater than the average level. It is determined by the formula:

where is the level of seasonality;

The current level of the dynamics series;

Average row level.

Graphically, the seasonality index can be represented using a polygon - the main type of graphs used to graphically represent dynamic series.

Task 3

According to Table 2, calculate:

1. Main analytical indicators of dynamics series (according to chain and basic schemes):

Absolute increase;

Rates of growth;

Growth rate;

Absolute value of 1% increase.

2. Average indicators:

Average level of the dynamics series;

Average annual growth rate;

Average annual growth rate.

Table 2 Key indicators

3. Based on the data in Table 3, calculate the seasonality index and graphically depict the seasonal wave.

Table 3 Store turnover, thousand rubles.

Absolute increase

According to the basic scheme

According to the chain diagram

Let's calculate the growth rate

According to the basic scheme

According to the chain diagram

Let's calculate the growth rate:

According to the basic scheme

According to the chain diagram

Let's calculate the average growth rate

In general, during the period the cost of living increased to 128.35%.

Let's calculate the average growth rate

Conclusion: In general, during the period the increase in the cost of living was 28.35%.

Let's calculate the absolute value of one percent increase

Table 4 Main analytical indicators of the dynamics series

Indicators	Calculation scheme
Row level Y i
Absolute increase?Y	Basic
Growth rate T r,%	Basic