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

Dynamic series (or time series) - these are the numerical values ​​of a certain statistical indicator at successive moments or periods of time (i.e., arranged in chronological order).

The numerical values ​​of one or another statistical indicator that makes up the dynamics series are called series levels and is usually denoted by the letter y. First term of the series y 1 called initial or basic level, and the last one y n - final. The moments or periods of time to which the levels relate are designated by t.

Dynamic series are usually presented in the form or , and the time scale is constructed along the abscissa axis t, and along the ordinate axis - the scale of series levels y.

Example of a dynamics series

Table. Number of residents of Russia in 2004-2009. in million people, as of January 1
Graph of the dynamics of the number of inhabitants of Russia in 2004-2009. in million people, as of January 1

The data clearly illustrates the annual decline in the number of residents of Russia in 2004-2009.

Types of dynamics series

Dynamics series classified according to the following main characteristics:

1. By time — moment and interval series (periodic), which show the level of a phenomenon at a specific point in time or for a certain period. The sum of the levels of an interval series gives a very real statistical value for several periods of time, for example, the total output, the total number of shares sold, etc. Although the levels of a moment series can be summed up, this sum, as a rule, has no real content. So, if you add up the inventory values ​​at the beginning of each month of the quarter, the resulting amount does not mean the quarterly inventory value.
2. According to presentation form — series of absolute, relative and average values.
3. By time intervals — rows uniform and uneven (complete and incomplete), the first of which have equal intervals, while the second does not have equal intervals.
4. According to the number of semantic statistical quantities — isolated and complex series (one-dimensional and multidimensional). The former represent a series of dynamics of one statistical value (for example, the inflation index), and the latter - several (for example, consumption of basic food products).

In our series of dynamics: 1) moment (levels as of January 1 are given); 2) absolute values ​​(in millions of people); 3) uniform (equal intervals of 1 year); 4) isolated.

Indicators of changes in the levels of a series of dynamics

Analysis of time series begins with determining exactly how the levels of the series change (increase, decrease, or remain unchanged) in absolute and relative terms. To track the direction and size of changes in levels over time, dynamics are calculated for series indicators of changes in the levels of a series of dynamics:

• absolute change (absolute increase);
• relative change (growth rate or dynamics index);
• rate of change (growth rate).

All these indicators can be determined basic in a way when the level of a given period is compared with the first (base) period, or chain way - when two levels of neighboring periods are compared.

Base absolute change represents the difference between the specific and the first levels of the series, determined by the formula

i-that) period is greater or less than the first (basic) level, and, therefore, may have a “+” sign (when levels increase) or “-” (when levels decrease).

Chain absolute change represents the difference between the specific and previous levels of the series, determined by the formula

It shows how much (in units of series indicators) the level of one ( i-that) period is greater or less than the previous level, and may have a “+” or “-” sign.

In column 3, basic absolute changes are calculated, and in column 4, chain absolute changes are calculated.

Year	y					, %	,%
2004	144,2
2005	143,5	-0,7	-0,7	0,995	0,995	-0,49	-0,49
2006	142,8	-1,4	-0,7	0,990	0,995	-0,97	-0,49
2007	142,2	-2,0	-0,6	0,986	0,996	-1,39	-0,42
2008	142,0	-2,2	-0,2	0,985	0,999	-1,53	-0,14
2009	141,9	-2,3	-0,1	0,984	0,999	-1,60	-0,07
Total			-2,3		0,984		-1,60

Between basic and chain absolute changes there is relationship: the sum of chain absolute changes is equal to the last basic change, that is

.

Ours confirms the correctness of the calculation of absolute changes: = - 2.3 is calculated in the final line of the 4th column, and = - 2.3 - in the penultimate line of the 3rd column.

Baseline relative change (baseline growth rate or base momentum index) represents the ratio of the specific and first levels of the series, determined by the formula

Chain relative change (chain growth rate or chain dynamics index) represents the ratio of the specific and previous levels of the series, determined by the formula

.

The relative change shows how many times the level of a given period is greater than the level of any previous period (with i>1) or what part of it is (with i<1). Относительное изменение может выражаться в виде coefficients, that is, a simple multiple ratio (if the comparison base is taken as one), and in percent(if the comparison base is taken to be 100 units) by multiplying the relative change by 100%.

In ours, column 5 contains basic relative changes, and column 6 contains chain relative changes.

There is a relationship between basic and chain relative changes: the product of chain relative changes is equal to the last basic change, that is

In our example about the number of inhabitants of Russia, the correctness of the calculation of relative changes is confirmed: = 0.995 * 0.995 * 0.996 * 0.999 * 0.999 = 0.984 - calculated according to the data of the 6th column, and = 0.984 - in the penultimate row of the 5th column.

Rate of change(growth rate) of levels - a relative indicator showing how many percent a given level is greater (or less) than another, taken as the basis of comparison. It is calculated by subtracting 100% from the relative change, that is, using the formula:

,

Or as a percentage of the absolute change to the level in comparison with which the absolute change is calculated (baseline level), that is, according to the formula:

.

In our column 7 the basic rates of change are found, and in column 8 the chain rates are found. All calculations indicate an annual decrease in the number of residents in Russia for the period 2004-2009.

Average indicators of the dynamics series

Each series of dynamics can be considered as a certain set n time-varying indicators that can be summarized as averages. Such generalized (average) indicators are especially necessary when comparing changes in a particular indicator over different periods, in different countries, etc.

A generalized characteristic of the dynamics series can serve, first of all, middle row level. The method for calculating the average level depends on whether it is a moment series or an interval series (periodic).

In case interval of a series, its average level is determined by the formula from the levels of the series, i.e.

=
If available moment row containing n levels ( y1,y2, …, yn) With equal intervals between dates (time points), then such a series can be easily converted into a series of average values. In this case, the indicator (level) at the beginning of each period is simultaneously the indicator at the end of the previous period. Then the average value of the indicator for each period (the interval between dates) can be calculated as half the sum of the values at at the beginning and end of the period, i.e. How . The number of such averages will be . As stated earlier, for series of average values, the average level is calculated using the arithmetic mean. Therefore, we can write
.
After transforming the numerator we get
,

Where Y1 And Yn— first and last levels of the row; Yi— intermediate levels.

This average is known in statistics as average chronological for moment series. It received its name from the word “cronos” (time, Latin), since it is calculated from indicators that change over time.

In case unequal intervals between dates, the chronological average for a moment series can be calculated as the arithmetic mean of the average values ​​of levels for each pair of moments, weighted by the distances (time intervals) between dates, i.e.
.
In this case, it is assumed that in the intervals between dates the levels took different values, and we are one of two known ( yi And yi+1) we determine the averages, from which we then calculate the overall average for the entire analyzed period.
If it is assumed that each value yi remains unchanged until the next (i+ 1)- th moment, i.e. If the exact date of change in levels is known, then the calculation can be carried out using the weighted arithmetic average formula:
,

Where is the time during which the level remained unchanged.

In addition to the average level in the dynamics series, other average indicators are calculated - average change in series levels(basic and chain methods), average rate of change.

Baseline mean absolute change is the quotient of the last underlying absolute change divided by the number of changes. That is

Chain mean absolute change levels of the series is the quotient of dividing the sum of all chain absolute changes by the number of changes, that is

The sign of average absolute changes is also used to judge the nature of the change in a phenomenon on average: growth, decline or stability.

By subtracting 1 from the base or chain average relative change, the corresponding average rate of change, by the sign of which one can also judge the nature of the change in the phenomenon under study, reflected by this series of dynamics.

Dynamic series are the values ​​of statistical indicators that are presented in a certain chronological sequence.

Each time series contains two components:

Series levels are expressed in both absolute and average or relative values. Depending on the nature of the indicators, time series of absolute, relative and average values ​​are constructed. Dynamic series from relative and average values ​​are constructed on the basis of derived series of absolute values. There are interval and moment series of dynamics.

Dynamic interval series contains indicator values ​​for certain periods of time. In an interval series, levels can be summed up to obtain the volume of the phenomenon over a longer period, or the so-called accumulated totals.

Dynamic moment series reflects the values ​​of indicators at a certain point in time (date of time). In moment series, the researcher may only be interested in the difference in phenomena that reflects the change in the level of the series between certain dates, since the sum of the levels here has no real content. Cumulative totals are not calculated here.

The most important condition for the correct construction of time series is comparability of series levels belonging to different periods. The levels must be presented in homogeneous quantities, and there must be equal completeness of coverage of different parts of the phenomenon.

In order to avoid distortion of the real dynamics, in statistical research preliminary calculations are carried out (closing the dynamics series), which precede the statistical analysis of the time series. Under closing the series of dynamics refers to the combination into one series of two or more series, the levels of which are calculated using different methodology or do not correspond to territorial boundaries, etc. Closing the dynamics series may also imply bringing the absolute levels of the dynamics series to a common basis, which neutralizes the incomparability of the levels of the dynamics series.

Indicators of changes in the levels of time series

To characterize the intensity of development over time, statistical indicators are used, obtained by comparing the levels with each other, as a result of which we obtain a system of absolute and relative dynamics indicators: absolute growth, growth coefficient, growth rate, growth rate, absolute value of 1% growth. To characterize the intensity of development over a long period, average indicators are calculated: the average level of the series, the average absolute increase, the average growth rate, the average growth rate, the average growth rate, the average absolute value of 1% increase.

If during the study it is necessary to compare several successive levels, then you can get either a comparison with a constant base (basic indicators) or a comparison with a variable base (chain indicators).

Basic indicators characterize the final result of all changes in the levels of the series from the period of the base level to the given (i-th) period.

Chain indicators characterize the intensity of level changes from one period to another within the time period being studied.

Absolute increase expresses the absolute rate of change in a series of dynamics and is defined as the difference between a given level and the level taken as the basis of comparison.

Absolute increase (basic)

(9.1)

where y i is the level of the period being compared; y 0 - level of the base period.

Absolute growth with a variable base (chain), which is called the growth rate,

(9.2)

where y i is the level of the period being compared; y i-1 - level of the previous period.

Growth rate K i is defined as the ratio of a given level to the previous or basic level; it shows the relative rate of change of the series. If the growth rate is expressed as a percentage, it is called the growth rate.

Base growth rate

Chain growth factor

Growth rate

(9.5)

The growth rate of TP is defined as the ratio of the absolute increase of a given level to the previous or base level.

Base growth rate

(9.6)

Chain growth rate

(9.7)

1) T p = T p - 100%; 2) T p = K i - 1. (9.8)

Absolute value of one percent increase A i. This indicator serves as an indirect measure of the baseline level. It represents one hundredth of the base level, but at the same time it also represents the ratio of absolute growth to the corresponding growth rate.

This indicator is calculated using the formula

(9.9)

To characterize the dynamics of the phenomenon being studied over a long period, a group of average dynamics indicators is calculated. Two categories of indicators in this group can be distinguished: a) average levels of the series; b) average indicators of changes in the levels of the series.

Average row levels are calculated depending on the type of time series.

For an interval series of dynamics of absolute indicators, the average level of the series is calculated using the simple arithmetic average formula:

where n is the number of levels of the series.

For a moment dynamic series, the average level is determined as follows.

The average level of the moment series at equal intervals is calculated using the average chronological formula:

(9.11)

where n is the number of dates.

The average level of a moment series with unequal intervals is calculated using the weighted arithmetic average formula, where the duration of time intervals between time points of changes in the levels of the dynamic series is taken as weights:

where t is the duration of the period (days, months) during which the level did not change.

Average absolute increase(average growth rate) is defined as the arithmetic average of the growth rate indicators for individual periods of time:

(9.13)

where y n is the final level of the series; y 1 - initial level of the row.

Average growth rate() is calculated using the geometric mean formula of the growth coefficients for individual periods:

(9.14)

where K p1, K p2, ..., K p n-1 are growth coefficients compared to the previous period; n is the number of levels of the series.

The average growth rate can be defined differently:

Average growth rate,%. This is the average growth rate, which is expressed as a percentage:

Average growth rate,%. To calculate this indicator, the average growth rate is initially determined, which is then reduced by 100%. It can also be determined by decreasing the average growth rate by one:

Average absolute value of 1% increase can be calculated using the formula

Methods for processing time series

When processing a time series, the most important task is to identify the main tendency in the development of the phenomenon (trend) and smooth out random fluctuations. To solve this problem in statistics, there are special methods called alignment methods.

There are three main ways to process time series:

a) enlargement of intervals of a time series and calculation of averages for each enlarged interval;

b) moving average method;

c) analytical alignment (alignment using analytical formulas).

Enlargement of intervals- the simplest way. It consists in transforming the initial dynamics series into longer time periods, which makes it possible to more clearly identify the effect of the main trend (main factors) of changes in levels.

For interval series, the totals are calculated by simply summing the levels of the initial series. For other cases, the average values ​​of the enlarged series are calculated ( variable average). The average variable is calculated using the simple arithmetic average formulas.

Moving average- this is a dynamic average that is sequentially calculated when moving one interval for a given period duration. If, suppose, the duration of the period is 3, then moving averages are calculated as follows:

(9.19)

With even periods of the moving average, you can center the data, i.e. determine the average of the averages found. For example, if the moving average is calculated with a period duration of 2, then the centered averages can be defined as follows:

(9.20)

The first calculated centered one is assigned to the second period, the second to the third, the third to the fourth, etc. Compared to the actual one, the smoothed series becomes shorter by (m - 1)/2, where m is the number of interval levels.

The most important way to quantitatively express the general trend of changes in the levels of a time series is analytical alignment of the dynamics series, which allows us 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 specific 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. Theoretical analysis is based on calculated dynamics indicators. Practical analysis - on the study of a line diagram.

The task of analytical alignment is to determine not only the general trend of 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 levels located next to the interpolated ones;

2) by average absolute growth;

3) by growth rate.

The method of determining quantitative values ​​outside the series is called extrapolation. Extrapolation is used to predict those factors that not only determine the development of a phenomenon in the past and present, but may also influence its development in the future.

You can extrapolate using the arithmetic mean, the average absolute growth, or the average growth rate.

Seasonal unevenness ( seasonal fluctuations), which is understood as stable intra-annual fluctuations caused by numerous factors, including natural and climatic ones. Seasonal variations are measured using seasonality indices, which are calculated in two ways depending on the nature of dynamic development.

With a relatively constant annual level of the phenomenon seasonality index can be calculated as a percentage of the average value from the actual levels of the same months to the overall average level for the period under study:

(9.23)

In conditions of variability of the annual level, the seasonality index is defined as the percentage ratio of the average value from the actual levels of the same months to the average value from the leveled levels of the same months.

Changes in socio-economic phenomena over time are studied by statistics using the method of constructing and analyzing time series. Dynamic series are the values ​​of statistical indicators that are presented in a certain chronological sequence.

Each time series contains two components:

1) t- indicators of time periods (years, quarters, months, days or dates);

2) y- indicators characterizing the object under study for time periods or on corresponding dates, which are called series levels.

Series levels are expressed in both absolute and average or relative values. Depending on the nature of the indicators, time series are built absolute, relative and average values. There are interval and moment series of dynamics.

Interval series contains indicator values ​​for certain periods of time. In an interval series, levels can be summed up to obtain the volume of the phenomenon over a longer period, or the so-called accumulated totals.

Moment series reflects the values ​​of indicators at a certain point in time (date of time). In moment series, the researcher may only be interested in the difference in phenomena that reflects the change in the level of the series between certain dates.

The most important condition for the correct construction of time series is the comparability of the levels of the series belonging to different periods. The levels must be presented in homogeneous quantities, and there must be equal completeness of coverage of different parts of the phenomenon.

In order to avoid distortion of the real dynamics, in a statistical study, the closure of the dynamics series is carried out - combining into one series two or more series, the levels of which are calculated using different methodology or do not correspond to territorial boundaries, as well as bringing the absolute levels of the dynamics series to a common basis.

Average series levels are calculated depending on the type of time series.

For an interval series of dynamics of absolute indicators, the average level of the series is calculated using the simple arithmetic average formula, where n is the number of levels of the series.

For a moment dynamic series, the average level is determined as follows.

at regular intervals calculated using the average chronological formula, where n is the number of dates.

Average level of moment series at irregular intervals is calculated using the formula of the arithmetic weighted average, where the duration of time intervals between time points of changes in the levels of the dynamic series is taken as weights, where t is the duration of the period (days, months) during which the level did not change.

Average absolute increase(average growth rate) is defined as the arithmetic mean of the growth rate indicators for individual periods of time, where y n is the final level of the series; y 1 - initial level of the row.

The process of development, the movement of socio-economic phenomena over time in statistics is usually called dynamics. To display the dynamics, they build dynamics series (chronological, temporal), which represent a series of time-varying values ​​of a statistical indicator, arranged in chronological order.

The components of a dynamics series are indicators of series levels and time indicators (years, quarters, months, days) or moments (dates) of time. The levels of the series are usually designated by “y”, the moments or periods of time to which they refer are indicated by “t”.

There are different types of time series, which are classified according to the following criteria: :

• Depending on the way the levels are expressed, the dynamics series are divided into series of absolute, relative and average values .
• Depending on whether the levels of the series express the state of the phenomenon at certain points in time (at the beginning of the month, quarter, year, etc.) or its value over certain time intervals (for example, per day, month, year, etc.), differentiate accordingly moment and interval time series .
• Depending on the distance between levels, the dynamics rows are divided into series with equally spaced levels and unequally spaced levels in time . Dynamic series of periods following one another or dates following at certain intervals are called equidistant. If the series contains interrupted periods or uneven intervals between dates, then the series are called unequally spaced.
• Depending on the presence of the main tendency of the process being studied, the dynamics series are divided into stationary and non-stationary . If the mathematical expectation of the value of the attribute and the dispersion (the main characteristics of a random process) are constant and do not depend on time, then the process is considered stationary, and the dynamics series are also called stationary. Economic processes over time are usually not stationary, because contain the main development trend, but they can be converted into stationary ones by eliminating trends.

Indicators of changes in the levels of a series of dynamics

Analysis of the speed and intensity of development of a phenomenon over time is carried out using statistical indicators that arise as a result of comparing levels with each other. These indicators include: absolute growth, growth and growth rate, 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 takes place is basic .

Absolute increase (Δу) characterizes the size of the increase (or decrease) in the level of the series over 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 level y i-1 is the previous one for this level, and absolute increments in level changes will be chain ones. If k are constant for a given series, then the absolute increases will be basic.

The indicator of the intensity of change in the level of a series, depending on whether it is expressed as a coefficient or as a percentage, is usually called the growth coefficient (growth rate). Growth rate (t) shows how many times a given level of a 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 = y i / y i-1 or t = y i / 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 fraction (or percentage) the level of a given period or point in time is greater (or less) than the base level. The growth rate is found 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, or zero.

In statistical practice, instead of calculating and analyzing growth rates and increments, they often consider absolute value of one percent increase (A) . It represents one hundredth of the base level and at the same time the ratio of absolute growth to the corresponding growth rate: A = Δy /(Δt*100) = y i-1 /100

Average level of dynamics series calculated according to the chronological average. Middle chronological is called the average, calculated from values ​​that change over time. Such averages summarize chronological variation. The chronological average reflects the totality of the conditions under which the phenomenon under study developed in a given period of time. Formulas for calculating the average indicators of the dynamics series are presented in the table.

Table - Formulas for calculating average indicators of a series of dynamics

Indicator	Designation and formula
Average level of interval dynamics series
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 “Dynamic series in statistics”

Problem 1 . Data on areas under potatoes before and after changing the boundaries of the region, thousand hectares:

Close the series, expressing the area under potatoes in the context of changes in the boundaries of the region.

Solution

Let us take the third period as a basis for comparison - a period for which there is data both within the former and within the old boundaries of the region. Then we close these two rows with the same base into one.

Problem 2 . There is information on exports of products from the region for a number of years:

Determine: 1) chain and basic: a) absolute increases; b) growth rate; c) growth rate; 2) the absolute content of one percent of growth; 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 us remind you that:
- if each current level is compared with the previous one, we will get chain indicators;
- if each current level is compared with the initial one, we will obtain basic indicators.

To solve this, let’s expand the proposed table.

The average level of the series is determined by the simple arithmetic mean: 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 . Using the following information, determine the average size of the enterprise’s property for the quarter:

Solution

The average size of an enterprise's property for a quarter is determined by the formula:

= (30/2 +40 +50 +30/2) / (4-1) = 40 million rubles.

A chronological series (dynamic series, dynamic series) is a series of statistical indicators, the sequential change of which reflects the development of social phenomena over time. The dynamics series contains two elements: an indicator of time, to which statistical indicators relate; level of series y.

Based on the time reflected in the dynamics series, moment and interval chronological series are distinguished.

In a moment series of dynamics, statistical indicators characterize the state of a phenomenon at a certain point in time. For a moment series of dynamics, it is characteristic that each subsequent one, therefore the sum of indicators of such a series does not make economic sense.

An interval series of dynamics consists of indicators characterizing the size of a phenomenon over a certain period of time. The indicators of such a series can be summed up, resulting in a new series of dynamics, each indicator of which characterizes the size of the phenomenon over a longer period of time.

According to the way in which dynamics series are expressed, they can be series of absolute, relative and average values.

To characterize the intensity of changes in social phenomena over time, the following indicators are calculated: absolute growth, growth rate, growth rate, absolute value of 1% growth, advance coefficient.

Depending on the comparison base, they can be basic (one, constant level is taken as the comparison base) and chain (the previous level is taken as the comparison base).

The absolute increase in y is the difference between the levels of the series, which is expressed in units of measurement of indicators of the dynamics series:

y basic = yi - yo;

y chain = yi - yi-1,

where уi are the levels of the dynamics series;

уо - basic level;

ush-1 - previous level.

Growth rates Tr - the ratio of one level to another, taken as the basis of comparison, are expressed as coefficients or percentages:

Tr basic = ;

Tr chain = .

Growth rate Tpr - the ratio of absolute growth to the level taken as the basis of comparison, expressed in coefficients or percentages:

T pr basic = ;

T pr chain =

The absolute value of 1% growth A shows what absolute value is contained in 1%, and is defined as the ratio of the chain absolute growth to the chain growth rate, expressed as a percentage:

Those. the absolute value of a 1% increase can also be defined as 0.01 of the previous level.

To generalize the dynamics of social phenomena, the average level of a series of dynamics, the average absolute growth, the average growth rate and the average growth rate are determined.

The average level of a series of dynamics is called average chronological, which gives a general characteristic of the development of phenomena over time.

In the interval dynamics series, the average level y is determined by the formula:

where n is the number of levels of the series;

y - levels.

In the moment series of dynamics:

1) with equal intervals between points in time, the average level is determined by the formula:

where n is the number of levels;

2) with unequal intervals between points in time, the average level is determined by the formula:

where ti is the value of the intervals between time points.

The average absolute increase is determined by individual values ​​of chain absolute increases:

The average growth rate is determined by the geometric mean formula:

where Ti is the growth rate;

m is the number of growth rates.

If the levels of the dynamics series are known, then the average growth rate can be determined as

where уо, уn are the level of the first and last period (moment) of time in the dynamics series.

The average growth rate is determined based on the average growth rate:

Tpr = Tr - 1 (100%).

One of the tasks solved when analyzing dynamics is to establish a pattern (trend) in the development of a phenomenon over time.

For this purpose, the methods of enlarging intervals, moving averages and analytical leveling are used.

The method of enlarging intervals is that the original dynamics series is transformed and replaced by another, in which the indicators relate to longer periods of time. This method is used only for interval time series.

The moving average method consists in forming enlarged intervals consisting of the same number of levels. In this case, we obtain each subsequent interval by gradually shifting from the initial interval of the dynamics series by one interval; using enlarged intervals, the average of the levels included in each interval is determined. When using the analytical leveling method to identify the trend in the development of a phenomenon over time, the actual levels are replaced by theoretical ones, calculated on the basis of the equation of a curve or straight line reflecting the general trend.

If the series is aligned with the equation of a straight line, then the general trend will be expressed by the equation:

where a and b are the parameters of the equation;

yt - theoretical levels of the dynamics series;

t - periods or moments in time.

To calculate yt for known t, it is necessary to first determine the parameters of the equation. To do this, the least squares method is used, which gives a system of linear equations:

where y are the actual levels of the dynamics series;

n is the number of these levels.

This system of equations can be simplified if we number the time periods t in such a way that their sum is equal to 0 (t = 0). To do this, in a dynamics series with an even number of levels, numbering must begin from the middle of the series with the numbers -1, +1; in a dynamics series with an odd number of levels, numbering must begin from the middle of the series from 0, then