Statistical study of the dynamics of socio-economic phenomena

The development of social phenomena in time is called dynamics. A number of statistical indicators characterizing the development of social phenomena in time are called series of dynamics. The importance of the series of dynamics is that they make it possible to identify the patterns of development of phenomena, facilitate their analysis. Each row consists of 2 graphs: one indicates the periods or dates of time, the second – the numerical characteristic of the phenomenon under study in these periods, called the level of the series. The levels of a series can be expressed in absolute, mean and relative values. Time series consisting of absolute values can be of two types: interval and momentary. In the interval series, data characterizing the state of the phenomenon for a given period of time are given.

Table 17 Resource requirements

Production of passenger cars

Years

1991

1992

1993

1994

1998

Production of passenger cars, kt.

529

730,1

916,7

1119,0

1201,0

A feature of interval series of dynamics is that the data of these series can be summed up and new numerical values related to longer periods of time can be obtained.

The moment series of dynamics consists of indicators characterizing the state of the phenomenon at certain points in time.

Table 18 Resource requirements by component

Fuel capacity in warehouses

Date

as of 1.01

as of 1.02

as of 1.03

as of 1.04

Fuel capacity, t

13,8

12,8

15,0

17,0

The levels of the moment series cannot be added, since the constituent phenomena of the unit are sequentially repeated at different levels of the series, so their sum does not make sense.

Summation of interval series indicators is often resorted to to build series of dynamics with increasing results.

Table 19 Resource requirements

Coal production in January-July

By month

I

II

III

IV

V

VI

VII

By month, t

40

36

42

42

45

45

48

Cumulative total since the beginning of the year

40

76

118

160

205

250

298

Cumulative totals are often given in the reports of enterprises.

If the series of dynamics consists of relative or average values, then they cannot be summed up, but their difference makes real sense.

Table 20 Resource requirements by component

Level of mechanization of loading and unloading operations

(in % of the total amount of work)

Years

1975

1980

1985

1990

1995

Level, %

31

67

79

87

89

The level of mechanization for the period 1975-1995 increased by 58 points (89-31). A point in the statistics is called a difference of 1 percent. The intensity level of this growth is defined as: 89% : 31% = 2.87 (287 – 100 = 187%) or this value can be obtained by assigning the difference in points to the level with which the comparison is made: 58 : 31 = 1.87%.

For the correct construction of rows of dynamics, it is necessary to comply with a number of requirements:

1) All indicators of a number of dynamics should be reliable, accurate, scientifically based;

2) All indicators of the series should be comparable. The main condition for the comparability of statistical indicators is the same methodology for their determination;

(3) Indicators of a number of dynamics should be comparable in the territory to which they relate;

(4) The indicators of a number of dynamics should be comparable in time, i.e. they should be calculated for the same periods of time or on the same date;

(5) All series values shall be given in the same units of measurement.

(6) Comparability of prices should be respected.

When studying the series of dynamics, statistics solves a number of problems:

1) measures the absolute and relative rate of growth or decrease in the level for individual periods of time;

2) gives generalizing characteristics of the level and the rate of its change over a particular period;

3) identifies and numerically characterizes the main trends in the development of the phenomenon at individual stages;

4) gives a comparative numerical description of the development of this phenomenon in different regions or at different stages;

5) identify the factors that determine the change in the phenomenon under study in time;

6) makes forecasts of the development of the phenomenon in the future.

To compare the individual levels of a number of dynamics, the following indicators are calculated: absolute growth rates, growth rates (growth coefficients), growth rates and the absolute value of one percent of growth. The calculation of these indicators is based on a comparison of the levels of a number of dynamics. In this case, the level with which the comparison is made can be basic or chain.

Absolute growth shows how much in absolute terms the level of the reporting period is greater or less than the level of the base period. Absolute growth is calculated with both a constant and variable comparison base. Absolute increase per unit of time measures the absolute rate of growth or decline of a level.

A=yn-yn-1 (variable comparison base).

A=yn-y1(k) (constant comparison base).

       – The level of the series taken as the basis of comparison.

The growth rate, the growth rate and the absolute value of 1% of the increase characterize the intensity of the growth process.

The growth coefficient shows how many times the level of the reporting period is greater or less than the level of the baseline and is calculated with both a variable and a constant comparison base.

With a variable base of comparison , with a constant comparison base. The growth rate can be greater than one, less than one, equal to one. Growth coefficients expressed as a percentage are called growth rates:

,

.

If growth coefficients, calculated with a variable comparison base, characterize the change in the phenomenon from period to period, then growth coefficients with a constant base are a continuous line of development.

The growth rate shows by what percentage the level of the reporting period is greater or less than the level of the baseline.

With a variable comparison base:

       or                or .        

With a constant comparison base:

       or .        

Absolute value of 1% increase (A1%):

       or .        

There is a certain relationship between the dynamics indicators calculated with a constant and variable base. Thus, the sum of absolute increases with a variable base gives the total increase for the period under study.

,

where        n is the number of levels of series dynamics.

When comparing the dynamics of the development of two phenomena, it is possible to use an indicator that provides a ratio of growth rates for the same periods of time for two dynamic series. This indicator is called the lead factor:

,

where        , are the corresponding levels of the time series being compared. With this coefficient, time series of the same content but belonging to different territories or to different organizations or series of different content characterizing the same object can be compared.

To obtain a generalizing characteristic of the intensity of the development of the phenomenon over a long period, the average dynamics indicators are calculated.

The average level of the dynamics series is calculated differently depending on the type of series. For the interval series, it is calculated by the formula of the arithmetic mean prime:

,

where         is the number of levels in the series.

For a moment series with equal intervals according to the formula of the average chronological prime:

.

For a moment series with unequal intervals according to the formula of the average chronological weighted:

,

where        t is the time intervals.

All other averages for interval and torque series of dynamics are calculated equally.

The average absolute increment () is determined by the formula arithmetic mean of the absolute increments calculated with a variable base:

       or ,        

where        , are the final and initial levels of the dynamic series.

The average growth rate is determined by the formula geometric mean of the growth coefficient for individual periods:

or        ,

where        n is the number of levels of the series;

The average growth rate is the average growth rate expressed as a percentage:

.

The average growth rate is determined on the basis of the growth rate:

       or .        

In the practice of statistics, it is often necessary to compare the series of dynamics of two or more related or interrelated phenomena (coal and oil, wool and silk). To do this, it is necessary to convert the absolute indicators of the compared series of dynamics into relative ones, taking the indicators of any one year for 1 or 100. Such a transformation of series of dynamics, consisting of absolute values, into comparable series of relative quantities, is called bringing them to a common base.

Table 21

Coal and oil production in the USSR (million tons)

Years

Coal

Petroleum

1998

608

328

1999

624

353

2000

641

377

2001

655

400

As a comparison base, we take the level of 1999.

;                 etc.

Table 22 Resource requirements by component

Coal and oil production in the USSR (in % by 1999)

Years

Coal

Petroleum

1998

97,4

92,9

1999

100

100

2000

102,7

106,8

2001

105,0

113,3

Sometimes the levels of the phenomenon for one year are not comparable with the levels for other years due to territorial, departmental or other changes. In this case, a technique called closing the rows of dynamics is used in statistics.

Although in 2000 there was a change in the boundaries of the region, in connection with which, the data for 1998-1999 turned out to be incomparable with the data of 2001-2002.

To close these series, we will take in each of them a comparison base for the level of 2000, for which there is data both in the old and in the new boundaries of the district. These two rows with the same comparison base can then be replaced by one closed row of speakers. However, it is also possible to calculate the absolute levels of a series within the new boundaries. Next, the resulting coefficient is multiplied by the level of the series in the old boundaries (2100 · 1.03 and 2100 · 1.01). It should be borne in mind that the results obtained by closing the series of dynamics contain some error.

Table 23

Retail turnover, rub. mln.

Years

Before you change the scope boundaries

(in the old borders)

After changing the boundaries of the scope

(within new boundaries)

2000 = 100%

Closed speaker range (2000 = 100%)

Absolute levels within the new boundaries

in the old borders

within new borders

1998

2266

103

103

2163

1999

2222

101

101

2121

2000

2200

2100

100

100

100

2100

2001

2058

98

98

2053

2002

2205

105

105

2205

Graphically, the dynamics of phenomena are most often depicted in the form of bar and line diagrams. Line charts most closely match the nature of moment observations. Interval indicators are better to depict in columns only if the number of levels is small. Pie charts are also used.

One of the most important tasks of dynamics analysis is the identification and quantitative characterization of the main trend in the development of the phenomenon. A trend is understood as a general direction towards the growth, decrease or stabilization of the level of the phenomenon over time. However, the rise and fall of the level can occur either evenly, or accelerated, or slowly.

A uniform increase or decrease here refers to growth (decrease) at a constant absolute rate, when the chain absolute increases are the same. With accelerated growth, systematic chain increases in absolute value, and with slow growth or decline, they decrease. However, in practice, the levels of a number of dynamics very rarely increase or decrease strictly evenly. Therefore, when analyzing the dynamics, we are talking not just about the development trend, but about the main trend that is quite stable and stable during this stage of development.

The main trend (trend) is a fairly smooth and stable change in the level of the phenomenon in time, more or less free from random fluctuations. The main trend can be represented either analytically – in the form of trend management (model), or graphically. In cases where the level of a number of dynamics then rises, then decreases or changes in one direction, but very unevenly, it is advisable to compare and use in the calculation of dynamics indicators not annual, but more typical average annual levels, the change of which better reflects the main development trend. Other ways and techniques for identifying and characterizing the main trend are also possible.

Method of enlarging intervals. This method consists in the transition from intervals shorter to longer (the transition from days to weeks, from months to quarters or years, from annual intervals to multi-year intervals). If the levels of a series of dynamics fluctuate with a more or less certain periodicity, in waves, then it is advisable to take the enlarged interval equal to the period of oscillation (the wavelength of the cycle). If there is no such periodicity, then the enlargement is carried out gradually from small intervals to larger and larger ones, until the general direction of the trend becomes apparent.

Moving average method. It consists in the fact that the average level is calculated from a certain number of the first in order levels of the series, then the average level from the same number of levels starting from the second, etc. The resulting average refers to the middle of the enlarged interval.

;        ;        .

The moving average has disadvantages:

1) the impossibility of obtaining levels for the ends of the smoothed series;

2) arbitrariness of the choice of the smoothing interval.

Aligning a series in a straight line provides a solution to the following equation:

,

where        t is the time (the sequence number of the interval or the moment of time).

The calculation of parameters is greatly simplified if .

               

,        therefore , the middle level of the row;

,        therefore , the average absolute increase.

With an odd number of row levels, the level in the middle of the row is assumed to be zero. Dates above this time will be indicated by natural numbers with the sign ” – : – 1; – 2; – 3; below with the sign ” + “: +1; +2; +3. The sum of t must be zero.

If the number of levels in a series is even, the values of t will be as follows:

1998

1999

2000

2001

2002

2003

-3

-2

-1

1

2

3

After finding the parameters of the equation, the equation of the straight series of dynamics is drawn up. According to it, the levels of the aligned series of dynamics are calculated. The correctness of the calculation of the levels of the aligned series of dynamics can be checked as follows: the sum of the values of the empirical series must coincide with the sum of the calculated values of the levels of the aligned series .

Forecasting is the determination of the approximate size of phenomena in the future. Extending a past trend in the future is called extrapolation. The possibility of extrapolation is provided by two circumstances:

1) the general conditions that determine the development trend in the past do not undergo significant changes in the future;

2) the tendency of the development of the phenomenon is characterized by one or another analytical equation.

Interpolation is an approximate calculation of the levels lying inside a series of dynamics, but for some reason unknown. When the levels of a series change in the arithmetic progression:

,

where        , are adjacent levels.

If the levels change exponentially, then .

Seasonal fluctuations are relatively stable intraannual fluctuations of a phenomenon. They are caused by a number of objective reasons (natural and climatic) and lead to a deterioration in the performance of enterprises. Seasonal variance analysis is necessary to improve operational planning and the development of interventions to reduce their negative impacts.

When studying seasonal unevenness, statistics have two tasks:

identify seasonal unevenness; determine its size (calculate the seasonal wave).

The presence of seasonal unevenness is detected using the graphic method. To do this, line charts are used, on which data on the average daily production of goods are applied by months, but not less than for 3 years. Several broken lines will be plotted on the chart. The average daily production may increase from year to year, but if the maximum and minimum volumes for all three years fall on the same months, then we can talk about the presence of seasonal unevenness.

The following methods have been proposed for measuring seasonal variations:

the method of absolute differences; the method of relative differences; construction of seasonality indices.

When using the method of absolute differences, the average level of the phenomenon for each month is first determined according to 3-year data, then the average for the entire period under consideration is determined.

Next, the absolute deviation of the average for each month from the total average is determined.

The method of relative differences is a development of the method of absolute differences. To find relative differences, absolute deviations are divided by the total average and expressed as a percentage.

The depth of seasonal variations is measured by seasonality indices (Ixez):

,

where         is the average of the actual average levels of the months of the same name;

– the total average for the study period.

The seasonality index shows how much the average monthly consumption of each month differs from the total average for the entire period.

Security questions

What is called dynamics? Types of time series. How to determine the growth rate? How to determine the average level of a moment series with unequal intervals? How to determine the average growth rate? Methods of identification and characteristics of the main trend. How do I find the seasonality index?