Dynamics is the change in socio-economic phenomena over time. To study the dynamics of phenomena, time series are constructed and analyzed. A dynamics series is a series of values of a statistical indicator arranged in chronological order. The components of a dynamics series are indicator values, called series levels, and time indicators - periods or points in time to which the levels relate. If the dynamics series consists of levels, then its form is where - the level of the dynamics series at the moment or over a period of time. The classification of dynamics series is presented in Figure 17.
Conditions for correct construction of a dynamics series:
- 1) periodization of development, i.e. dividing it in time into homogeneous stages, within which the indicator obeys one law of development;
- 2) the levels of the series must be comparable in terms of territory, range of objects covered, units of measurement, registration time, prices, calculation methodology;
- 3) the levels of the series must correspond to the intensity of the processes being studied;
- 4) the levels of the series must be ordered in time.
When studying time series, statistics are faced with the following tasks: to characterize the intensity of development of a phenomenon from period to period (from date to date), as well as the average intensity of development for the period under study, to identify the main trend in the development of the phenomenon, to forecast development for the future, and also to study seasonal fluctuations.
Rice. 17.
Dynamics series indicators
To characterize the intensity of development of a phenomenon over time, the following indicators of a series of dynamics are calculated: absolute growth, growth rates, growth rates, growth rates, growth rates, absolute values of 1% growth. Their calculation is based on comparing the levels of the series with each other. In this case, the level being compared is called the current (reporting) level, and the level with which the comparison is made is called the base level. The listed indicators can be calculated on a variable or constant basis. If each level is compared with previous level, then we obtain dynamics indicators with a variable base (chain indicators of dynamics).
If each level is compared with the initial level or some other level taken as the basis of comparison, then indicators of dynamics with a constant base are obtained ( basic indicators speakers).
The basis of comparison should be chosen reasonably, depending on economic features phenomena and research objectives. Formulas for calculating dynamics indicators are presented in Table 17.
Table 17
Dynamics series indicators
|
Index |
Basic |
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1. Absolute growth shows how many units the level of the series has increased or decreased over a given period of time |
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2. The growth coefficient shows how many times the current level of the series is greater than the base level (if the 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) |
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3. Growth rate, % |
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4. Growth rate |
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5. Rate of increase, % shows by what fraction (or percentage) the level of the current period is greater (or less) than the base level |
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6. The absolute value of 1% increase shows how many absolute units are accounted for by 1% increase (decrease) |
Note. - the level of any period (except the first), called the level of the current (reporting) period. - level of the period preceding the current one. - level taken as a constant base of comparison (often the first level).
There is a relationship between chain and basic indicators of absolute growth and growth rates:
To characterize the intensity of development over a long period, average dynamics are calculated. The formulas for their calculation are presented in Table 18.
Table 18
Average indicators of the dynamics series
|
Index |
Calculation formula |
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1. Middle row level: for an interval series with equal intervals |
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for an interval series with unequal intervals |
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for a moment series with equal intervals |
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 for moment series with unequal intervals |
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2. Average absolute growth shows how many units the level has increased or decreased compared to the previous one on average per unit of time (on average annually, monthly, etc.) |
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3. Average growth rate |
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4. Average growth rate |
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5. The average growth rate shows by what percentage the level increased or decreased compared to the previous one on average per unit of time (on average annually, monthly, etc.) |
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6. Average absolute value of 1% increase |
Note.
Number of row levels. - duration of the time interval between levels. - last level row.
Indicators of a number of dynamics are divided into groups (Fig. 18).
Rice. 18. Grouping indicators of a series of dynamics
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”.
Exist different kinds series of dynamics, which are classified according to the following signs :
- 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 are usually not stationary in time, 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 increases in level changes will be chain-like. 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. Find the growth rate as the ratio of 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 in 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.
| Index | 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 row; 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 enterprise assets 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.
All processes and phenomena occurring in human social life are the subject of study of statistical science; they are in constant motion and change.
In statistical science, time series are statistical data that characterize changes in phenomena over time; they are constructed to identify and study emerging patterns in the development of phenomena in various fields(for example, economic, political and cultural) life of society.
There are two main elements in the dynamics series:
1) time indicator (g);
2) levels of development of the phenomenon being studied (y). In dynamics series, specific time dates or individual periods can serve as time indicators.
The levels that form the dynamics series determine the quantitative assessment of the development over time of the phenomenon or process under study; they can be expressed as relative, absolute or average values. The levels of time series, depending on the nature of the phenomenon under study, can relate to certain time dates or to individual periods.
The time series consists of comparable statistical indicators. For the correct construction of time series, it is necessary that the composition of the statistical population under study belongs to the same territory, to the same range of objects and was calculated using the same methodology.
Time series data should be expressed in the same units of measurement, and the time intervals between series values should be the same as possible.
2. Types of dynamics series
Dynamics series are divided into moment, interval and average series.
Moment dynamics series display the state of the processes under study at certain dates.
Interval time series display the results of the development or functioning of the processes under study for individual periods of time.
Calculation of the average time series. To characterize the process for a certain period, the average level is calculated from all members of the time series.
The methods for calculating it depend on the type of time series. For interval series, the average is calculated using the arithmetic mean formula, and for equal intervals, the simple arithmetic mean is used, and for unequal intervals, the weighted arithmetic mean is used.
To find the average values of a moment series, the chronological average is used:
The average chronological moment series is equal to the sum of all levels of the series, divided by the number of members of the series minus one, and the first and last members of the series are taken in half.
If the intervals between periods are not equal, then the weighted arithmetic mean is used, and the time intervals between dates, which include paired averages of adjacent level values, are taken as weights.
3. Main indicators of time series analysis
To analyze time series in statistics, indicators such as series level, average level, absolute growth, growth rate, growth coefficient, growth rate, lead coefficient, absolute value of one percent of growth are used.
The level of the series is the absolute value of each member of the dynamic series. All levels of a series characterize its dynamics. There are initial, final and middle levels of the series. The initial level is the value of the first term of the series. The final level is the value of the last member of the series, the average level is the average of all values of the dynamic series.
Absolute increase- this is one of the most important statistical indicators; it characterizes the size of the increase or decrease in the phenomenon being studied over a certain period of time, defined as the difference between a given level and the previous or initial one. The level that is compared is called the current one, and the level with which the comparison is made is called the base level, since it is the basis for comparison. If each level of a series is compared with the previous one, then chain indicators are obtained, and if all levels of a series are compared with the same initial level, then the resulting indicators are called basic.
For the dynamic series y 0, y 1, y 2,…, y n-1 , y n , consisting of n+ 1 levels, absolute growth is determined by the formulas:
1) chain: ?I = y i– y i -1 ;
2) basic ? = y i– 0 ,
Where y i– current row level;
y i y i ;
y 0 – initial level of the series.
Formula for average absolute growth:
Where ?y– average absolute increase;
y n– final level of the row;
y 0 – initial level of the series.
Calculate indicators of growth rate and growth rate. The growth rate is the most common statistical indicator, which characterizes the ratio of a given level of the statistical process to the previous or initial one, expressed as a percentage. Rates of growth, calculated as the ratio of a given level to the previous one, are called chain and to the initial level - basic.
Growth rates are calculated using the formulas:
1) chain:
2) basic:
Where y i– current row level;
y i-1 – level previous y i ;
at 0 – initial level of the series.
If the comparison base for growth rates is taken as 1, then the resulting statistical indicators are called growth coefficients.
The growth rate is the ratio of absolute growth to the previous or initial level, expressed as a percentage. The growth rate can be calculated from the growth rate data. To do this, you need to subtract 100 from the growth rate or 1 from the growth coefficient, in the latter case we get the growth coefficient Kpr.
Growth rates are calculated using the following formulas:
1) chain: Tpr. = (y – y i -1); y i-1 = Tr.ts. – 100 or (Kr.ts. – 1) x 100;
2) basic: Tpr. = (y i– y 0); y 0 = Tr.b. – 100 or (Cr.b. – 1) x 100.
To characterize the growth rate and increase on average for the entire period, calculate average tempo growth and gain. The average growth rate (coefficient) is determined by the geometric mean formula, when the average growth rate is calculated from the absolute data of the first and last members of the dynamic series, the following geometric mean formula is applied:
Where at 1 - First level;
y n– final level;
n– number of members of the series.
If there are chain growth coefficients, then the average growth coefficient is determined by the formula:
Where TO 1 , TO 2 , K 3 …K n– growth rates for any period.
Advance coefficient is the ratio of the basic growth rates of two dynamic series for the same periods of time. Denoting the advance coefficient K op, the basic growth rates of the first dynamic series - through K 1, the second - K 11, Then:
TO op = K 1 / By 11.
This coefficient shows how many times faster the level of one series of dynamics will grow compared to another. The ratio of absolute growth to the growth rate is the absolute value of one percent according to the formula:
A% = ? (absolute increase) / Tpr.
Interpolation and extrapolation
To solve unknown intermediate values of a time series, the interpolation method is used.
Interpolation– a method for determining unknown intermediate values of a time series.
Interpolation essentially consists of an approximate reflection of the existing pattern within a certain period of time - in contrast to extrapolation, which requires going beyond this period of time.
Extrapolation– a method of determining quantitative characteristics for populations and phenomena that have not been observed, by extending to them the results obtained from observing similar populations over the past time, into the future, etc.
The average level of a series of dynamics characterizes the typical value of absolute levels.
The average level y in interval time series is calculated by dividing the sum of levels y; by their number n.
In a moment series of dynamics with equal time dates, the level will be determined as follows:
In a moment series of dynamics with unequal dates, the average level is determined:
The characteristic of generalizing individual absolute increases in a series of dynamics is called average absolute increase.
Average absolute increase at is defined as follows: the sum of chain absolute increases (y n) is divided by their number (n):
The average absolute increase can also be determined using absolute dynamics series; for this purpose, the difference between the final at P and basic at 0 levels of the period under study, which is divided into m– 1 subperiods.
The average absolute growth rate is determined by the formula:
Average growth rate (T R ) – these are individual growth rates of a series of dynamics that have a general characteristic, its formula:
The average growth rate, which is determined by absolute levels of dynamics, is as follows:
Based on the relationship between the basic and chain growth rates, the average growth rate is determined by the formula:
Average growth rate T P is based on the relationship between growth rates and increments. If there is information about average growth rates T, then the dependence is used to obtain the average growth rate of Tp.
To find average value of a moment series with equal levels use the average chronological: .
Average chronological for different levels of moment series:
Purpose of the service. Using this online calculator can be calculated average value of moment series according to the average chronological formulas.
Instructions. Select the amount of data and specify whether it is days, months or years
Example No. 1. The population of the city was:
- as of January 1 – 80,500 people,
- as of February 1 – 80,540 people,
- as of March 1 – 80,550 people,
- as of April 1 – 80,560 people,
- as of July 1 – 80,620 people,
- as of October 1 – 80,680 people,
- as of January 1 of the next year - 80,690 people.
Solution.
The data presented is a moment series. We find the average using the chronological average formula.
Average chronological for different levels of the moment series:
y av = (80500+80540)*1 + (80540+80550)*1 + (80550+80560)*1 + (80560+80620)*3 + (80620+80680)*3 + (80680+80690)*3 /(2*12) = 1934790/(2*12) = 80616.25 ≈ 80616 people
Average for the first quarter: 
Human
Average for the second quarter: 
Human
Average for the third quarter: 
Human
Average for the first half of the year:
Human
Example No. 2. According to Tables 7(Appendix 2) select the dynamic series corresponding to your option, for which:
1. Calculate:
a) the average annual level of the dynamics series;
b) chain and basic indicators of dynamics: absolute growth, growth rate, growth rate;
c) average absolute growth, average growth rate, average growth rate.
Guidelines
To characterize the dynamics, a system of dynamics indicators is calculated.
| Dynamics indicator | Calculation formulas | |
| on a chain basis | on a basic basis | |
| Absolute increase (+), decrease (-) | Δ c =y i -y i-1 | Δ b =y i -y 1 |
| Growth rate | ||
| Growth rate |  |  |
| Rate of increase | T pr c = T r c - 100% | T pr b = T r b - 100% |
| Absolute value of one percent increase | A1%=0.01·y i-1 | - |
- average row levels;
- average indicators of changes in series levels.
To find the average level of a moment series, use the chronological average:
.
Average absolute increase calculated depending on the initial data in the following ways:
or
Average growth rate(decrease):
or, . Average growth rate(decrease): T pr = T r - 100%.
In the following example, we will find the average fund size wages(for interval series).
The average level of the interval series is calculated using the formula:
The average size of the salary from 1994 to 2004 was 548.45 thousand rubles.
Average growth rate
On average, for the entire period from 1994 to 2004, the growth in wages was 1.1 (increased by 10% annually).
Average growth rate
T pr = T r - 1 = 1.1-1 = 0.1
Average absolute increase
On average, over the entire period, the wage fund increased by 50 thousand rubles. from year to year.
In the following example, we will find the average number of production personnel (for the moment series).
Chain indicators of a series of dynamics.
| Period | number of PPP | Absolute increase | Growth rate, % | Rates of growth, % | Absolute content of 1% increase | Rate of increase, % |
| 1994 | 470 | 0 | 0 | 100 | 4.7 | 0 |
| 1995 | 500 | 30 | 6.38 | 106.38 | 4.7 | 6.38 |
| 1996 | 505 | 5 | 1 | 101 | 5 | 1.06 |
| 1997 | 533 | 28 | 5.54 | 105.54 | 5.05 | 5.96 |
| 1998 | 540 | 7 | 1.31 | 101.31 | 5.33 | 1.49 |
| 1999 | 589 | 49 | 9.07 | 109.07 | 5.4 | 10.43 |
| 2000 | 577 | -12 | -2.04 | 97.96 | 5.89 | -2.55 |
| 2001 | 594 | 17 | 2.95 | 102.95 | 5.77 | 3.62 |
| 2002 | 640 | 46 | 7.74 | 107.74 | 5.94 | 9.79 |
| 2003 | 628 | -12 | -1.88 | 98.13 | 6.4 | -2.55 |
| 2004 | 646 | 18 | 2.87 | 102.87 | 6.28 | 3.83 |
To find the average level of a moment series, use the chronological average:
The average number of industrial personnel of the enterprise for the analyzed period was 566.4 people. 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 built. 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 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: average level of the series, average absolute growth, average growth rate, average growth rate, average growth rate, average absolute value of 1% growth.
If during the study it is necessary to compare several successive levels, then it is possible to obtain 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 one.
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 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, the cause of which are 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 average from the aligned levels of the months of the same name.