Moment and interval dynamics series example. Dynamic series, their meaning. Types of dynamics series: instant and interval. Dynamic series of absolute and relative values, average values. Average values of time series indicators
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, average level for moment series dynamics given in clause 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, the time series are processed by the methods of enlarging intervals, moving average and analytical leveling.
Enlargement of 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 production 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 Is . 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):
187. Indicate which of the indexes is the general cost index:
4) I = ∑ Z1 Q1 / ∑ Z0 Q1;
188 Test. Which of the following statements does not characterize incomplete observation?
2) Solid;
189. The Law “On State Statistics” does not include the following section...
4) Annual statistics.
190. What is the normal moment of the fourth order, if the normal distribution is taken as the basis of comparison?
191. The general yield index has the form:
1) I = ∑ Y1*P1 / ∑у0*P1;
192. Which of the listed rules for constructing statistical tables does not meet the requirements?
3) with different units of measurement, there is no point in assigning a separate column, and also not indicating units of measurement in columns or lines;
193. What is the name of the area that is occupied by crops at the end of spring sowing, and from which products are expected to be obtained in a given year?
2) spring productive area;
194. What term can be used to determine the amount of production per hectare of crops?
2) productivity;
195. How is the livestock safety indicator determined?
3) the ratio of the number of livestock in circulation to the number of dead animals;
196. If the total energy capacity is divided by the size of the area of agricultural land and multiplied by 100, we get:
2) Energy supply indicator;
197. Which of the following indicators is calculated by dividing the total volume of work performed by tractors in standard hectares by the average annual number of conditional standard tractors?
3) Average annual production;
198. Which answer goes beyond the question about the types of labor productivity index?
3) direct, indirect;
199. How to determine the production of total farm products per 100 hectares of agricultural land?
1) production of products (cost of production) of crop production and livestock breedingstvadivide by the area of agricultural land and multiply the result by 100;
200. What cost is called actual?
1) cost, reflecting actual costs and determined based on data accounting at the end of the year;
201. What is the object of statistical observation?
1) The set of social phenomena and processes that are subject to statistical observation;
202. A survey of budgets, incomes, and expenses of the population covering population units is an observation:
3) examination of the main massif;
203. What type of grouping solves the problem of determining cause-and-effect relationships between the characteristics under study?
3) Analytical;
Test - 204. The division of a homogeneous population according to the value of a varying characteristic is carried out in statistics using groupings:
2) structural;
205. Relative values of structure:
A) characterize the composition of the phenomenon and show what specific weight each part of it makes up in the total;
B) characterize the relationship between the individual components of the phenomenon.
Relative coordination values:
C) characterize the composition of the phenomenon and show what proportion each part of it makes up in total;
D) characterize the relationship between the individual parts of the phenomenon.
Answers: 4) b, d.
206. A series of dynamics may consist of:
A) from absolute total values;
B) from relative and average values.
Answers: 3) a, b;
207. For 2003 - 2005. capital commercial bank increased by 20%, the absolute value of 1% increase is 12 thousand UAH. Determine the bank's capital in 2005 (thousand UAH).
Answers: 3) 2400;
208 Test. What is the ability of a sample population to renew a population called?
2) Representativeness;
209. What formula should you choose to calculate the harmonic simple mean?
1) XWed =N / ∑1/ X
210. What is meant by a statistical hypothesis?
3) Scientific assumption about the properties of random variables, which is verified based on the results of statistical observation;
211. What types of diagrams are there?
2) Linear, column, tape, rectangular, circular, sector, radial, curly;
212. The coefficient of variation is calculated as:
1) percentage ratio of the standard deviation to the arithmetic mean;
Statistics test - 213. The essence of analytical alignment is:
1) application of certain analytical alignment equations;
214. What is the value of the correlation coefficient if the connection is weak, not close?
1) 0 ≤ R ≤ 0,2;
215. 3 plots of land covered with natural grassy vegetation and used for haymaking are called:
3) hayfields;
216. The average number of animals is calculated as:
2) by dividing the sum of feed days for a certain period by the number of days of this period;
217. What is animal productivity?
3) this is the average product yield per animal;
218. Average dynamics indicator wages calculated by the aggregate index formula:
2) I = ∑ X1 T1: ∑ X0 T;
219. What area is called spring productive?
2) the area that remained at the end of spring sowing;
220. What products are called commercial products?
1) Part of the gross output that is sold;
221. What is the unit of statistical observation?
1) The primary element of the research object, which is the bearer of essential features and especiallyCmTo her, which are subject to registration;
222. In terms of completeness of coverage of observation units - observation happens...
3) continuous, not continuous;
223. What relative value characterizes the change in processes and phenomena over time?
4) relative magnitude of dynamics.
224 Statistics test. Relative dynamics are obtained by comparing the indicators of each subsequent period:
A) with the previous one;
B) with the original.
Answers: 3) a, 6;
225. The dynamics series characterizes:
A) the structure of the population according to some characteristic;
B) changes in population characteristics over time.
The level of the dynamics series is:
C) a certain value of a varying characteristic in the aggregate;
D) the value of the indicator on a certain date or for a certain period.
Answers: 4) B, G;
226. An individual index is the result of a comparison of two values of the same name related to:
A) different periods of time;
B) different territories.
Answers: 1) a;
227. Define the correlation coefficient...
3) a meter for the tightness of connection with a simple linear relationship;
228. What type of average is the option that falls in the middle of the variation series?
2) Median;
229. What method of selection requires the previous gradation of the general population into qualitatively homogeneous groups?
2) Serial;
230. What formula is used to calculate the pair correlation coefficient?
1) R = Yx – Y* X / Gy* Gx;
231. The simple arithmetic mean is calculated using the formula:
2) XAv = ∑Xi / N
232. What is the growth rate?
1) the ratio of each subsequent level to the previous or to the initial level;
233. What is the formula for the general labor index?
2) I = ∑ T0 Q1: ∑ T1 Q1;
234 Test. What are deposits?
1) these are lands that were previously used for agricultural crops. crops, but have not been sown for several years;
235. What is the name of the indicator that is determined by the ratio of the number of calves obtained per year only from cows to the number of cows at the beginning of the year?
3) offspring yield per 100 cows;
236. The average egg production of chickens is calculated...
2) by dividing the gross collection of eggs (excluding pullet eggs) by the average number of laying hens for the corresponding period;
237. By what means do enterprises reimburse the cost of depreciation of fixed assets?
2) depreciation charges;
1) divide the total volume of work performed by tractors in reference hectares by the number of tractor days worked;
239. What area is called seeded?
1) the area where the seeds were sown;
240. What production is called gross?
2) products obtained on the farm;
241. What is the subject of statistics as a social science?
3) the quantitative side of mass social phenomena in specific conditions of place and time;
242. The germination of grain can be determined by observation...
2) selective;
243. What relative value characterizes the ratio of the planned indicator to another value taken as the basis of comparison?
3) the relative amount of fulfillment of the planned target;
244. Distribution series are:
A) attribute;
B) variational.
Answers: 3) a, b;
245 Statistics test. The number of cows on the farms during the quarter changed as follows (heads) by:
1.01-614 1.02-588 1.03-610 1.04-620
Determine the average number of cows per quarter.
Answers: 3) 605;
246. Over the past year, industrial production volumes increased by 2,5%, And wholesale prices for industrial products decreased by an average of 1.2%. The growth rate of industrial production was, %:
A) 102.5; b)97.5;
Wholesale prices:
B) 101.2; d) 98.8.
Answers: 2) a, d;
Statistics test - 247. Which scientist discovered the law of normal distribution?
3) Gauss;
248. What rule is used in practice when studying a population for its compliance with the normal law?
2) The 3 sigma rule;
249. Which of the following mathematical functions is used to align the dynamics series if the growth (chain) coefficient is stable?
3) Yt= ao*a1T;
250 Test. The formula for the mean square deviation will look like this...
2) G2 = ∑(Xi – XWed)2* Fi / ∑ Fi
When analyzing a time series, the following indicators are calculated:
- average level of dynamic series;
- absolute growth: chain and basic, average absolute growth;
- growth rates: chain and base, average growth rate;
- growth rates: chain and basic, average growth rate;
- the absolute value of one percent increase.
Chain and basic indicators are calculated to characterize changes in the levels of a dynamic series and differ from each other in their comparison bases: chain indicators are calculated in relation to the previous level (variable comparison base), basic indicators are calculated in relation to the level taken as the comparison base (constant comparison base).
Average indicators represent generalized characteristics of a series of dynamics. With their help, the intensity of development of a phenomenon is compared in relation to various objects, for example, countries, industries, enterprises, etc., or time periods.
9.2.1. Average level of dynamics series
A specific numerical value of a statistical indicator relating to a moment or period of time is called dynamics series level and is denoted by y i (where i- indicator of time).
The method for calculating the average level depends on the type of time series, namely: whether it is momentary or interval, with equal or unequal time intervals between adjacent dates.
If an interval series of dynamics of absolute or average values with equal periods of time is given, then to calculate the average level, the simple arithmetic average formula is used:
where y 1, y 2, y i, ..., y n - levels of the dynamic series;
n is the number of levels of the series.
Example 9.2. According to the table, we determine the average monthly amount of insurance compensation paid by the insurance company, per one damaged object for the six months:
If the time intervals of the interval time series are unequal, then the value of the average level is found using the weighted arithmetic average formula, in which the length of time periods corresponding to the levels of the time series (t i) is used as weights
Example 9.3. Based on the data presented in the table, we will determine the average monthly amount of insurance compensation paid by the insurance company per damaged object:
In moment series of dynamics with equal time intervals between dates, the average level of the series is calculated using the formula for the average chronological simple
where y n are the values of the indicator at the end of the period under review.
Example 9.4. According to the size data below Money on the depositor's account at the beginning of each month we will determine the average size deposits in the first quarter of 2006:
The average level of the moment series of dynamics is equal to:
Although the first quarter includes three months (January, February, March), four levels of the series must be used in the calculation (including data as of April 1). This is easy to prove. Indeed, if we calculate the average levels by month, we get:
in January
in February
The calculated averages form an interval series of dynamics with equal time intervals, in which the average level is calculated, as we saw above, using the simple arithmetic average formula:
Similarly, if you want to calculate the average level of a moment series of dynamics with equal intervals between dates for the first half of the year, then as the last level in the formula for the average chronological downtime you should take data for July 1, and if for a year, data for January 1 of the next year.
In moment series of dynamics with unequal intervals between dates, the chronological weighted average formula is used to determine the average level:
where t i is the length of the time period between two adjacent dates.
Example 9.5. Based on data on inventories of goods at the beginning of the month, we determine the average size inventory in 2006
| date | 01.01.06 | 01.02.06 | 01.03.06 | 01.07.06 | 01.09.06 | 01.12.06 | 01.01.07 |
|---|---|---|---|---|---|---|---|
| Inventories of goods, thousand rubles. | 1 320 | 1 472 | 1 518 | 1 300 | 1 100 | 1 005 | 920 |
The average level of the series is:
Distance between dates
If there is complete information about the values of a momentary statistical indicator for each date, then the average value of this indicator for the entire period is calculated using the weighted arithmetic average formula:
where y i - indicator values
t i is the length of the period during which this value of the statistical indicator remained unchanged.
If we supplement example 9.4 with information about the dates of changes in funds in the depositor’s account in the first quarter of 2006, we obtain:
- cash balance as of January 1 - 132,000 rubles;
- January issued - 19,711 rubles;
- January 28 deposited - 35,000 rubles;
- February 20 deposited - 2000 rubles;
- February 24 deposited - 2581 rubles;
- Issued on March 3 - 3370 rubles. (no other changes occurred in March).
So, from January 1 to January 4 (four days) the value of the indicator remained equal to 132,000 rubles, from January 5 to January 27 (23 days) its value was 112,289 rubles, from January 28 to February 19 (23 days) - 147,289 rubles, from February 20 to 23 (four days) - 149,289 rubles, from February 24 to March 2 (seven days) - 151,870 rubles, from March 3 to 31 (29 days) - 148,500 rubles. For convenience of calculations, we present these data in the table:
| Period length, days | 4 | 23 | 23 | 4 | 7 | 29 |
|---|---|---|---|---|---|---|
| Cash balance, rub. | 132 00 | 112 289 | 147 289 | 149 289 | 151 879 | 148 500 |
Using the weighted arithmetic mean formula, we find the value of the average level of the series
As you can see, the average value is different from that obtained in example 9.4, it is more accurate, since more accurate information was used in the calculations. In example 9.4, only the data at the beginning of each month was known, but it was not specified when exactly the changes in the indicator occurred; the chronological average formula was applied.
In conclusion, we note that calculating the average level of a series loses its analytical meaning in cases of large variability of the indicator within the series, as well as in cases of a sharp change in the direction of development of the phenomenon.
9.2.2. Indicators of absolute changes in time series levels
Absolute increases are calculated as the difference between two values of adjacent levels of a dynamic series (chain increases) or as the difference between the values of the current level and the level taken as the basis of comparison (basic increases). Absolute growth indicators have the same units of measurement as the levels of the time series. They show how many units the indicator has changed during the transition from one moment or period of time to another.
Basic absolute increases are calculated using the formula
where y i - i-th current row level,
y 1 - the first level of the dynamics series, taken as the basis of comparison.
The formula for determining chain absolute increases has the form
where i - 1 is the level preceding the i-th level of the dynamic series.
Average absolute growth shows how many units on average monthly, or quarterly, or annually, etc. the value of the indicator changed during the period of time under consideration. Depending on what data we have, it can be calculated in the following ways:
Example 9.6. Using the table data, we will determine the indicators of absolute increases in the amount of insurance compensation paid by the insurance company.
* The sum of all calculated chain absolute increases gives the basic absolute increase of the last period.
The average monthly absolute increase for the half year is equal to
Thus, on average, the monthly amount of insurance compensation payments increased by 1.2 thousand rubles.
9.2.3. Indicators of relative changes in time series levels
Characteristics of the relative change in the levels of a series of dynamics are the coefficients and growth rates of the indicator values and the rate of their growth.
The growth coefficient is the ratio of two levels of a time series, expressed as a simple multiple ratio. It shows how many times the value of the indicator has changed in one period (point) of time compared to another. The growth rate is the growth rate expressed as a percentage. It shows what percentage the value of the indicator is in a given period, if the level with which the comparison is being made is taken as 100%.
Just like absolute increases, growth coefficients and rates can be chain and basic.
The chain coefficient and growth rate measure the relative change in the current level of the indicator compared to its previous level:
growth factor:
growth rate:
The basic coefficient and growth rate characterize the relative change in the current level of the indicator compared to the basic (most often the first) level:
growth rate
growth rate
Chain and basic growth coefficients have the following relationship with each other:
The average growth rate and growth coefficient in time series with equally spaced levels are calculated using the simple geometric mean formula
Chain growth factors;
- chain growth rates.
These formulas can be reduced to the following form:
In order to determine what percentage current level indicator is greater or less than the value of the previous or baseline level, the growth rate is calculated. They are calculated by subtracting 100% from the corresponding growth rates:
The average growth rate is calculated in a similar way: 100% is subtracted from the average growth rate:
Example 9.7. The table shows the calculated growth coefficients, growth rates and increments of the indicator characterizing the average monthly amount of insurance compensation paid by the company for the period from January to June.
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.
For general characteristics 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 increment, 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 living wage amounted to 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 |
||||||
|
Growth rate T pr,% |
Basic |
||||||
|
Ab. Meaning 1% increase A |
Let's calculate seasonality indices
Table 5 Final index calculations
Let's depict the wave of seasonality
Fig.1
From the beginning of the year, sales begin to gradually decline, but after the middle they rise again. Trade turnover peaks in January and reaches its minimum in August.
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 - the scale of the 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:
- 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.
- According to presentation form — series of absolute, relative and average values.
- 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.
- 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 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 the basic absolute changes are calculated, and in column 4 the 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 the series is momentary or interval (periodic).
When 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 (times), 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.
When 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.
